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Bruce J. MacLennan 




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The work reported herein was supported in part by the Foundation 
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the Chief of Naval Research. 

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4. TITLE (and Subtltlm) 

A Commentary on Mill's Logic 

Book I - Of Names and Propositions 


Technical Report 



Bruce J. MacLennan 



Naval Postgraduate School 
Monterey, CA 93943 


61152N: RR000-01-10 


Chief of Naval Research 
Arlington, VA 22217 


October 1983 

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16. DISTRIBUTION STATEMENT (of thla Report) 

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19. KEY WORDS (Continue on revetee aide If nacaaaary and Identify by block number) 

Mill's Logic, J. S. Mill, epistemology, philosophy of science, scientific 
method, logic, propositions, definitions, universals, intension, extension, 
connotation, denotation. 

20. ABSTRACT (Continue on raverae aide It naceeaary mnd Identify by block ^timber) 

Mill's Logic is the cornerstone of scientific method; yet, aside from 
Mill's Methods of Induction, its contents are not well known. This report 
attempts to make Book I of Mill's Logic more accessible to students of 
science and the philosophy of science. Each section of Mill's work is 
summarized. Most sections also include comments that criticize Mill's 
position, or relate the topic to more recent developments in the philosophy 
of science. 


70 2 ^ 

>. Tl ' s 


1 CD 


S/N 0102- LF-0U-6601 




Book I 
Of Names and Propositions 

Bruce J. MacLennan 





1. Of the Necessity of Commencing with an Analysis of Language 

2. Of Names 6 

1. Names are Names of Things 6 

2. General and Singular Names 6 

3. Concrete and Abstract 8 

4. Connotat ve and Non-Connotative 9 

5. Positive and Negative 13 

6. Relative and Absolute 16 

3. Of the Things Denoted by Names 1? 

- 1 

1. Necessity of an enumeration of Nameable Things 17 

2. Feelings, or states of Consciousness 18 

3. Feelings must be distinguished from their physical antecedents 18 

4. Perceptions 19 

5. Volitions and Actions 20 

6. Substance and Attribute 21 

7. Body 21 

8. Mind 22 

9. Qualities 22 

10. Relations 23 

11. Resemblance 23 

12. Quantity 24 

13. Attributes Concluded 25 

14. Recapitulation 25 

4. Of Propositions 28 

1. Nature and office of the copula 28 

2. Affirmative and Negative Propositions 28 

3. Simple and Complex 23 

- li 

4. Universal, Particular, and Singular 30 

5. Of the Import of Propositions 33 

1. Is Proposition a Relation Between Two Ideas? 33 

2. Is a Proposition a Relation Between the Meanings of Two Names? 33 

3. Is a Proposition an Expression of Class Membership? 34 

4. What it Really Is 35 

5. What it Is that Propositions Assert or Deny 36 

6. Propositions with Abstract Terms 39 

6. Of Propositions Merely Verbal 40 

1. Essential and Accidental Propositions 40 

2. Essential Propositions are Identical Propositions 40 

3. Individuals Have No Essences 43 

4. Real Propositions, How Distinguished from Verbal 43 

5. Two Modes of Representing the Import of a Real Proposition 44 

7. On Classification and the Predicables 45 

1. Classification, How Connected with Naming 45 

2. The Predicables 46 

3. Genus and Species 47 

- Ill 

4. Kinds Have a Real Existence in Nature 47 

5. Differentia 49 

6. Property 50 

7. Accident 51 

B. Of Definition 52 

1. A Definition, What 52 

2. What Names can be Defined? 53 

3. Complete versus Incomplete Definitions 54 

4. Complete Definitions versus Descriptions 54 

5. Real Definitions versus Nominal Definitions 55 

6. Mathematical Definitions 57 

7. Definitions Grounded on Knowledge of Corresponding Things 58 


- IV 


Mill's Logic is he corne :tone of scientific method. Yet, except for Mill' Methods of 
Mduction, its contents are largely unknown. Although Mill's less important works on 
sociological and political topics are widely reprinted, it is often difficult to find a copy 
of his Logic. The present work attempts to make Mill's magnum opus more accessible 
to students of science and the philosophy of science. 

Mill's Logic presents serious difficulties to the modern reader. The work is long and 
dwells on many issues whose importance has declined. Conversely, there have been 
many developments in the philosophy of science since Mill's time. These, of course, he 
can't discuss. Finally, Mill's Logic makes heavy use of terms which are no longer 
current. These characteristics all decrease its accessibility to students. 

This work follows the same outline as Mill's Logic. In each section 1 have summar- 
ized Mill's major points. Most sections also include comments that criticize Mill's posi- 
tion, or relate the topic to more recent developments in the philosophy of science. 

The main references we have used are Mill's Logic (Mill, 1843), Nagel's editing of 
Mill's work (Nagel, 1950), and Killick's Student's Handbook to Mill's Logic (Killick, 
1909). The latter was a model for this work. 

The preparation of this report was supported in part by the Office of Naval Research 
under contract number N00014-B2-WR-20162. 


Mill's Logic: Of Names and Propositions 

Summary: A good starting place for a definition of logic is Whately's (1868): Logic is the 
science and art of reasoning. As a science it studies the mental processes that must 
take place whenever we reason; as an art it lays down rules, based on this analysis, that 
must be followed if we are to reason correctly. 

Logic is concerned with inferences, not intuitive truths. By the latter Mill means 
our direct sensations, whether of the external world or our own mental states. These 
intuitive truths are beyond doubt, and no science is required to establish their truth. 
Further, no science can make us more confident of them. However, we must be careful 
not to mistake very rapid inferences for these intuitive truths. For instance, judging 
the distance to something we see is a very rapid inference that must be learned. 

Mill states, "The province of logic must be restricted to that portion of our 
knowledge which consists of inferences from truths previously known, whether those 
antecedent data be general propositions or particular observations and perceptions. 
Logic is not the science of belief, but the science of proof or evidence. Insofar as belief 
professes to be founded on proof, the office of logic is to supply a test for ascertaining 
whether or not the belief is well grounded." 

Mill explains the relation of logic to the other sciences as follows: "Logic, however, 
is not the same thing with knowledge, though the field of logic is co-extensive with the 
field of knowledge. Logic is the common judge and arbiter of all particular investiga- 
tions. It does not undertake to find evidence, but to determine whether it has been 
found." Thus logic is the science of science itself. 

Comments: Mill's use of the term logic is much wider than is usual now. It is the sci- 
ence of inference, which includes both inference from generals to particulars, or 
deduction, and inference from particulars to generals, or induction. In contemporary 
usage, logic is taken to mean deductive logic, and usually symbolic logic at that. A 


better term for what Mill calls 'logic' would be 'scientific method'. 

Mill, i. the traditic n of the sensationalistic empiricists (Locke, Jerkeley, and Hume) 
takes sensations as the starting point for knowledge. This is also the position of the 
logical empiricists and logical positivists (such as Mach and Schlick), who followed Mill. 
This view commits the very error that Mill has cautioned us against. The data that is 
directly given to us is not sensations, but perceptions, i.e., organized sensations. That 
is, the basis for knowledge is not classified objects, such as trees, since classification is 
a process of inference. Nor is the basis unorganized "patches of color." Rather we 
perceive organized, but unclassified entities. It is only by a later process of analysis 
that we abstract out the "patches of color" that the sensationalists say are the basis of 
knowledge. We must start our analysis with perceptions because the integration of sen- 
sations into perceptions is an automatic process performed by our sensory apparatus. 
Processes not under our control are not the proper province of logic (or any art). 

Mill's Logic: Of Names and Propositions 

1. Of the Necessity of Commencing with an Analysis of Language 

Summary: Why should a study of scientific method be concerned with language? 
"Logic is a portion of the art of thinking; language is evidently, and by the admission of 
all philosophers, one of the principal instruments or helps of thought; and any imper- 
fection in the instrument or in the mode of employing it is confessedly liable, still more 
than in almost any other art, to confuse and impede the process and destroy all ground 
of confidence in the result." An even more fundamental reason for studying language is 
that we need this study to examine a central topic in logic, the import of propositions . 

This is a central topic because "Whatever can be an object of belief or even of disbe- 
lief must, when put into words, assume the form of a proposition." That is, subjects can 
be conceived but not believed; only propositions can be believed or disbelieved. A pro- 
position is "discourse in which something is affirmed or denied of something." A pro- 
position has three parts: 

• A predicate, which is a name indicating what is ffirmed or denied; 

. A subject, which is a name denoting the thing which the predicate is affirmed or 
denied of; 

• A copula (link), which is the sign indicating whether there is affirmation or denial. 

Comments: 1 can add little to this except to note that since Mill's time the study of 
language has virtually replaced the study of logic. Hence, in the following sections 1 will 
often emphasize the non-linguistic aspects of the subject. 

At this point 1 will mention that it is a common fallacy that Aristotle's analysis of the 
form of propositions is inadequate, since it does not cover all cases. First, it is said 
that it excludes relations, such as less than. But, if we consider a proposition such as 
'Two is less than three", it is clear that this fits the subject-copula-predicate form: 'Two' 

Necessity of Analysis of Language 
is the subject, 'is' is the copula and 'less than three' is the predicate. Further exam- 
ples will ye discussed later. 


Mill's Logic: Of Names and Propositions 
2. Of Names 

1. Names are Names of Things 

Summary: Mill begins with Hobbes' definition of a name: "A name is a word taken at 
pleasure to serve for a mark which may raise in our mind a thought like to some 
thought we had before, and which, being pronounced to others, may be to them a sign 
of what thought the speaker had or had not before in his mind." Thus names serve 
both to identify our own thoughts and to communicate our thoughts to others. 

This suggests the question, "Are words the names of things or of our ideas of 
things?" It seems most proper to consider names to be the names of things. For 
example, when we say 'The sun set' we intend to convey something about the sun not 
our idea of the sun. In fact, we have specific linguistic mechanisms for talking about 
our ideas, as when we say 'The idea of the sun entered my mind.' In other words, pro- 
positions don't just inform the hearer of certain conjunctions of ideas in our mind; they 
also inform the hearer about what we believe about the things in reality. 

Comments: This interpretation is necessary if science is to be of any value to us. If the 
propositions of science are to be valuable to technology, and life in general, they must 
be propositions about the world, not just our ideas of the world. 

2. General and Singular Names 

Summary: They are many ways that words can be categorized. First we can distin- 

• Predicable (categorematic) terms, which can be used alone, either as the subject or 
the predicate of a proposition. 

• Non-pre die able (syncategorematical) terms, which can only form the parts of other 

Thus, predicable terms name things, while non-predicable terms do not. Prepositions 

Of Names 
and adverbs are examples of non-predicable words. 

Comments; This is mostly a syntactic distinction. For example, adverbs and adjecth ss 
denote concepts just as much as nouns, it is just that the syntax of our language 
prevents us from using them as subject or predicate. Instead of 'Heavy is a burden, ' 
we must say "Heavy things are a burden.' There is no essential difference in the propo- 
sitions expressed. 

Even prepositions and conjunctions have a conceptual meaning; it is syntactic limi- 
tations that always require them to be used with other words. There are some words 
that serve a syntactic function only; these are the only truly non-predicable words. 

Summary: There are two broad classes of names: individual and general. An individual 
name "is a name which is only capable of being truly affirmed, in the same sense, of 
one thing." For example, proper names are individual names. Now, it would not be 
possible to give everything, real or imaginary of which we might have cause to think, an 
individual name. Hence, we give general names to broad classes of things (such as 
'stone') and indicate the individual in which we are interested by phrases such as 'this 
stone' or 'the stone on the table.' 

This is not the most important function of general names, however. "It is by their 
means that we are enabled to assert general propositions, to affirm or deny any predi- 
cate of an indefinite number of things at once." A general name is "a name which is 
capable of being truly affirmed, in the same sense, of each of an indefinite number of 

It is necessary to distinguish general names from collective names. Collec ive 
names are really individual names in which the individual named is a composite entity 
made of other individuals. For example 'The US Navy' is a collective name for a partic- 
ular collection of people; it is predicable of this collection as a whole, and only this col- 
lection; it is not predicable of the individual persons in this collection. On the other 

Mill's Logic: Of Names and Propositions 
hand, 'member of the US Navy' is a general name that is predicable of each individual 
in this collection and not of the collection as a whole. 

Comments: General names are the most important class of names, because it is these 
names that denote concepts or universals. Scientific knowledge would not be very 
applicable if its principles and laws were only applicable to the individuals for which the 
laws and principles had been verified. If this were the case, scientific laws would only 
summarize the result of yesterday's experiments; they would not give us principles 
that can be applied tomorrow. Hence, scientific principles are expressed in terms of 
concepts or universals that subsume an indefinite number of individuals. 

Certainly one of the major values of general names (and the concepts they name) is 
economy. Just as it is impossible to have an individual name for everything, so it is 
impossible to have an unlimited number of propositions to express the properties of an 
unlimited number of things. General names allow us to put in finite form knowledge 
about an infinite number of things. 

Finally, note the important distinction between general and collective names; 
modern symbolic logic has essentially obliterated this distinction by calling them both 
sets. The importance of general names is that they are applicable to an indefinite (i.e., 
infinite) number of individuals. Hence propositions involving general names are true 
universally. This is not the case for collective names. I may make a true statement 
about the present members of the Navy which will be invalidated by the very next 
recruit. Hence propositions concerning collections (whether finite or infinite) do not 
have the universality of propositions concerning concepts. Therefore, in science and 
logic we are mostly concerned with general names and general propositions. 

3. Concrete and Abstract 

Summary: Mill makes a number of other distinctions in names, most of which are 
based on traditional scholastic logic. He defines a concrete name as a name which 

Of Names 
stands for a thing, and an abstract name as a name which stands for an attribute of a 
thing. This distinction is orthogonal to that between general and singular names. We 
can have singular abstract names, such as 'visibleness' and 'squareness', which denote 
a single attribute. We can also have general abstract names, such as 'redness', which 
apply to a number of different shades of red. Note that 'whiteness' is an abstract 
name, the name of an attribute, while 'white' is a concrete name, the name of all white 

4. Connotative and Non-Cbnnotative 

Summary: Mill next introduces what he claims is one of the most important distinc- 
tions: the difference between connotative and non-connotative names. "A non- 
connotative term is one which signifies a subject or an attribute only. A connotative 
term is one which denotes a subject and implies an attribute." Thus 'London' and 
'whiteness' are non-connotative terms. 'White' is a connotative term because it 
denotes all white things, and implies or connotes (con = with, notare = to mark) the 
attribute whiteness, which all these things possess. 

Consider the word 'man'. This term denotes Tom, Dick, Jane, and an indefinite 
number of other men, whether alive now or not. The term 'man' connotes a number of 
attributes, namely "corporeity, animal life, rationality, and a certain external form 
which, for distinction, we call the human." These attributes are essential to our calling 
a thing 'man'; anything which lacks even one of these attributes would not be called 
'man'. But it is not always easy to decide the connotation of a term: "In some cases it 
is not easy to decide precisely how much a particular word does or does not connote; 
that is, we do not know (the case not having arisen) what degree of difference in the 
object would occasion of difference in the name." 

An important problem for philosophers and scientists is to discover the proper con- 
notation of a term, which is essentially the process of definition. Mill claims that when 
a term in common use is defined, the connotation should be chosen in such a way that 


Mill's Logic: Of Names and Propositions 
it alters as little as possible the denotation of the term and contradicts as few as possi- 
ble of the propositions received as true about the things denoted. 

Comments: It is not clear what, exactly. Mill means by connotation. He frequently says 
that the connotation of a term comprises those attributes whose presence causes us to 
apply the term to a thing. This might suggest that the connotation of a name is the 
same as its definition, but Mill later (Chapter 5) states, "In defining a name, however, it 
is not usual to specify its entire connotation, but so much only as is sufficient to mark 
out the objects usually denoted by it from all other known objects." 

An alternate interpretation is that the connotation of a name is all of the attributes 
implied by the name. Thus the connotation of man would include rationality, animal- 
ity, his distinctive shape, the ability to use language, an opposable thumb, and so on, 
for an unlimited number of attributes. This is not Mill* view, however, since in Chapter 
7 he states, "Of all the innumerable properties known and unknown that are common 
to the class man, a portion only, and of course a very small portion, are connoted by its 
name; these few, however, will naturally have been thus distinguished from the rest 
either for their greater obviousness, or for their greater supposed importance." Also, 
he later distinguishes propositions that unfold the connotation of a term, such as 'Man 
is an animal', from those which express an "accidental" fact, such as 'Man is mortal'. 

How can we tell whether or not a given attribute is connoted by a name? In Killick 
(1909) we find, "The best mode of determining whether a name connotes a given attri- 
bute is to ask, Whether, if that attribute were removed, the name would still be applied 
to the subjects? Does 'man' connote mortality? The test is, should we apply the name 
'man' to beings exactly like men in other respects, but not mortal?" Although Killick 
does not answer the question in this particular case, presumably he would answer 
"Yes." On the other hand, if we ask whether we should apply the name 'man' to beings 
exactly like men in other respects, but not rational, presumably he would answer "No." 
I dare say that in common usage we would more likely apply the name 'man' to an irra- 


Of Names 
tional being otherwise like man, than to an immortal being otherwise like man. 

What, then, is th< -oot of Mill's notion of connotation? It appear to be that the ion- 
notation "of a general term or concept A [is] made up of all those general terms or 
concepts B for which 'All As are B' is a necessary truth ...." ['extension and intension' 
in Flew (1979)]. The presumption is that marine ss necessarily implies rationality, but 
only contingently (accidently) implies mortality. Thus Mill's notion of connotation 
seems to hinge on the distinction between necessary and contingent truth, a distinc- 
tion which, later, I will argue is fallacious 1 . For the time being the reader will be closer 
to the truth if he takes the connotation of a name to be all of the attributes that are 
truthfully predicable of everything denoted by the name. Thus, the connotation of man 
includes rationality, animality, an opposable thumb, the potential ability to cook food, 
the ability to fly to the moon, etc., etc. This is the interpretation of connotation that I 
will use throughout this work. 

Mill's discussion of the connotation of proper names is also unsatisfactory. Joseph 
(1906) says about it, "He confounded different distinctions, and raised a controversy 
about the connotation of proper names, to which there has been no satisfactory issue, 
because he never clearly realized to himself what he meant by connotation ..." The 
basis of Mill's belief that proper names have no connotation is presumably the idea that 
proper names do not imply anything in a necessary way; everything we know about 
them is contingent. Thus the name 'Caesar' does not imply necessarily that Caesar is a 
man. But observe that if we do not know the name 'Caesar', then for us it neither con- 
notes nor denotes. However, if we know the individual to whom t is name refers then 
we know both the i me's denotation and part of its connotation ^in the wider seise of 
connotation suggested above). The same of course applies to general names. Hence, 
with this wider notion of connotation one can see that a proper name has, in fact, a 
very rich connotation, since an individual has "more" attributes than any general 

1. The fallacy of the necessary /contingent dichotomy is discussed in PeikofT (1979), Quine (1953) and Hempel 


Mill's Logic: Of Names and Propositions 
class. (These issues are discussed at some length in Joseph (1906), Chapter 6.) 

Mill identifies a very important point: the denotation and connotation are both 
essential parts of a (connotative) term. There is a tendency to first define a term by 
identifying certain attributes as the connotation of the term, and then to replace the 
term by its definition. This ignores the fact that a definition is wrong if it substantially 
alters the denotation of the term (or, at very least, it is the definition of some other 
term). Such "denotation shifting" is in fact a logical fallacy. 

It can be argued that sometimes a definition must change the denotations of a term. 
For example, traditionally a whale might have been considered a fish because of its 
external form and its aquatic habitat. A better understanding of animal life later 
forced us to remove whales from the denotation of 'fish'. This is of course correct: we 
have altered the denotation of 'fish' on the basis of new evidence. More accurately, we 
have divided fish into true-fish and aquatic mammals. When we define these terms, we 
must preserve the denotation of each. If a definition were to substantially alter the 
denotation of a term, then a new term would be called for. 

We can take Shannon's Information Theory as an example of this fallacy. It is widely 
recognized among scientists that Information Theory has little to do with information 
in the colloquial sense, and that Shannon's measure of information does not measure 
informative ness (e.g., a book of random numbers has the most information). To one 
who recognizes the limitations of formal Information Theory, this theory can be a 
powerful tool. Nevertheless, there have been many misapplications of Information 
Theory resulting from an identification of Shannon's information with the usual infor- 
mation. Indeed, books mistakingly making this identification are still common in the 
popular press. 

The reader is also likely to encounter the terms intension and extension, which are 
often used as synonyms for connotation and denotation, although some writers make 
slight distinctions. We will adhere to Mill's terminology in this work. 


Of Names 
5. Positive and Negative 

Summary: Names can also be divided into positive and negative. Examples of positive 
names include man, tree, and good; examples of negative names include not-man, not- 
tree, and not-good. The distinction between positive and negative names is not a dis- 
tinction of form (such as the possession of a negative prefix like not- or un-), but a dis- 
tinction of meaning. For example, the word inconvenient , though negative in form, is 
positive in meaning, since it expresses a positive attribute, the presence of some cause 
of discomfort or annoyance. Similarly, the word innocent, although positive in form, is 
negative in meaning, since it expresses a negative attribute, the absence of an illegal or 
unethical act. 

Comments: The distinction between positive and negative names is crucial; unfor- 
tunately Mill's treatment of it is completely inadequate. In particular he barely treats 
the question of what makes a name positive or negative. 

Since the distinction between positive and negative terms is one of meaning rather 
than form, it has largely been lost from modern formal logic. We will see later that this 
notion is crucial to the understanding of inductive proof. 

In attempting to understand the difference between positive and negative names, we 
can begin with Aristotle, who said in On Interpretation (ii. 16a30-33): "The expression 
'not-man' is not a noun. There is indeed no recognized term by which we may denote 
such an expression, for it is not a sentence or a denial. Let it then be called an 
indefinite noun, since it refers to all kinds of things, non-existent as well as existent." 

Whately (1868) also calls a negative term indefinite "in respect of its not defining 
and marking out an object, in contradistinction to this, a positive term is called 
Definite ... because it does thus define or mark out." In other words, definite terms 
mark out or limit our view to one particular class of things, or one thing, while 
indefinite terms exclude such a class or individual, leaving undetermined the individu- 


Mill's Logic: Of Names and Propositions 
als of which we speak. 

Several logicians have noted that these purely negative terms have a limited value. 
De Morgan (1847) says, "There can be little effective meaning, and no use, in a 
classification which, because they are not men, includes in one word not-man, a planet 
and a pin, a rock and a featherbed, bodies and ideas, wishes and things wished for." 
Given the questionable value of negative terms, the reader might wonder why 1 dwell on 
them. The reason is that they are very common in modern symbolic logic, which con- 
siders not-man to be a predicate of essentially the same kind as man. 

Joseph (1906) explains both the reason for negative terms and the fallacy in their 
use. "Such negative terms as these do not really figure in our thought; they are 'mere 
figments of logic'; Aristotle long ago pointed out that [not-man] was not properly a 
name at all ...." These negative terms result from an "attempt to reduce negative and 
affirmative judgements to a common affirmative type, by throwing the negative into 
the predicate ...." This "is not really defensible for the negative term [not-man] does 
not signify the nature of anything, and so is not really a term; it should, if it were a gen- 
eral term covering everything except the corresponding positive, be predicable of all 
subjects except [men] in the same sense; but there is no common character in all of 
these which it is intended to signify .... [I]t is clear that we have not resolved the nega- 
tive into the affirmative form, when such affirmation can only be understood by res- 
toration to the negative." 

The indefinite character of negative terms has led logicians to distinguish between 
contradictory and contrary terms (see, for example, De Morgan (1847), Joseph (1906), 
Whately (1868), Jevons (1919), and Creighton (1906)). Contradictory terms such as 
man and not-man divide the universe into two mutually exclusive, exhaustive classes; 
everything is either man or not-man. Contrary terms divide some class within the 
universe into mutually exclusive, exhaustive subclasses. For example, even-number 
and odaWatmber are contrary terms; some things are neither even numbers nor odd 


Of Names 
numbers, for example, Eucalyptus trees. 

Contrary terms can also be characterized as positive or negative. Consider the 
terms guilty and innocent; these are contraries rather than contradictories, since 
there are things, such as numbers, that are neither guilty nor innocent. How can we 
distinguish positive and negative terms? 

Positive terms are characterized by the presence of certain common attributes or 
qualities possessed by the things denoted by the term. Negative terms can only be 
characterized by the absence of such common attributes and qualities. For example, 
the positive term man is characterized by the conjunction of properties such as 
animality, rationality, and human form. Couldn't we similarly say that the negative 
term not-man is characterized by the disjunction of properties such as non-animality, 
non-rationality, and non-hurnan form? Yes, but notice that we have characterized the 
positive term man in terms of other positive terms, and the negative term not-man in 
terms of other negative terms. We will always find it to be the case that positives can 
be defined by positives, but that a negative requires at least one negative in its 

This seems to result in an infinite regression; to avoid it we must know by some 
other means (other than definition in terms of positives or negatives) whether a term is 
positive or negative. In fact we can know this, since any term which is defined osten- 
sively (by pointing) is by its nature positive. For example, to define the term blue we 
can pcint out a number of blue things. The hearer can then abstract out the common 
qualities of these things and understand what we mean by blue. It is not possible to 
communicate the term not-blue by pointing to things that aren't blue, because there 
are no common characteristics that the listener could abstract. 

One can define a positive term by pointing to things that have some quality; one 
can't define a negative term by pointing to things that don't have a quality. Thus the 
distinction between positive and negative terms ultimately rests on the presence or 


Mill's Logic: Of Names and Propositions 
absence of definite sensory and perceptual qualities. 

6. Relative and Mtsolute 

Summary: The fifth leading division of names is into relative and absolute (i.e., non- 
relative). Examples of relative names are 'father', 'son', 'longer', 'shorter', and 'equal'. 
Their characteristic property is that they are always given in pairs (e.g. 'father', 'son'), 
although in some cases [reflexive relations] the two elements of the pair are the same 
(e.g., 'equal', 'equal'). "Every relative name which is predicated of an object, supposes 
another object (or objects), of which we may predicate either that same name or 
another relative name which is said to be the correlative [or converse] of the former." 

The major reason for dwelling on relative names is that they provide insight into the 
nature of all attributes. "It is obvious, in fact, that if we take any two correlative 
names, father and son for instance, though the objects denoted by the names are 
different, they both, in a certain sense, connote the same thing. They don't connote 
the same attribute, but rather the same set of facts which we mean when we say A is 
the father of B and B is the son of A." "In this manner any fact, or series of facts, in 
which two different objects are implicated, and which is therefore predicable of both of 
them, may be either considered as constituting an attribute of the one, or an attribute 
of the other. This set of facts is what the schoolmen called the fundamentum rela- 
tionis, or foundation of the relation." 


Of the Things Denoted by Names 
3. Of the Things Denoted by Names 

1. Necessity of an enumeration of Name able Things 

Summary: Mill has discussed proofs, the constituents of proofs, which are propositions," 
and the constituents of propositions, which are names - "If, therefore, we knew what all 
names signify, we should know everything which, in the existing state of human 
knowledge, is capable either of being made a subject of affirmation or denial or of 
being itself either affirmed or denied of a subject." Thus we will attempt "an enumera- 
tion of all kinds of things which are capable of being made predicates or of having any 
thing predicated of them ..." 

Aristotle was the first to attempt an enumeration of "all things capable of being 
named; an enumeration by the summa genera, i.e., the most extensive classes into 
which things could be distributed ..." These highest predicates were called the 
categories. Table 1 is Aristotle's list of categories. 

TABLE 1. Aristotle's Categories 



, man or horse] 



, one foot long] 



, blue] 



, double] 



, to cut] 



, to be cut] 



, in the market-place] 



, yesterday] 



, sitting] 



, armed] 

Mill says that this list of categories is unphilosophical, superficial, redundant and 
defective. That is, m.ny of the distinctions are merely verbal, several of the categories 
overlap, and some things (such as states of consciousness) do not fall under any of the 

Comments: In Aristotle's defense it should be noted that he acknowledged that the 
categories overlap, and never claimed that the list was exhaustive. Also, Greek philoso- 
phy, like many modern philosophies, often confused linguistic distinctions with logical 


Mill's Logic: Of Names and Propositions 
distinctions. Finally, as I will discuss later, the notion of a summum genus is itself fal- 

2. Feelings, or states of Consciousness 

Summary: Mill uses feeling in a philosophical sense, i.e., to mean any state of cons- 
ciousness. "Feeling, in the proper sense of the term, is a genus, of which sensation, 
emotion, and thought, are subordinate species." Mill cautions us to carefully distin- 
guish objects from our ideas of them. "Even imaginary objects (which are said to exist 
only in our ideas) are to be distinguished from our ideas of them." 

Similarly, sensations (which are mental experiences) are distinguished from the 
objects which produce them and from the attributes of these objects which cause them 
to excite these sensations. Thus, the sensation of white must be carefully distinguished 
from the objects which produce this sensation, which we call white, and from the qual- 
ity, which we call -whiteness, that causes these objects to produce this sensation. 

3. Feelings must be distinguished from their physical antecedents 

Summary: Another distinction that must be carefully maintained is that "between ths 
sensation itself and the state of the bodily organs which precedes the sensation and 
which constitutes the physical agency by which it is produced." 

Comments: These distinctions - between sensations and the objects that product them, 
and between sensations and their physical antecedents - are often ignored by contem- 
porary philosophers. The important point is that sensations are primary; it is these 
that we have a direct awareness of, not objects or nerve impulses. Before we can be 
aware that there even are nerves, we must be able to see, so that we can look through a 
microscope or watch an oscilloscope screen. 

To reiterate, epistemologically sensations are primary: experience of them is prior 
to our experience of objects or nerve impulses. Causally (or ontologically) objects and 
nerve impulses may be prior to sensations, but that is a question for science to decide. 


Of the Things Denoted by Names 
To accomplish this, scientists will make use of the sensations they experience. 

4. Perceptions 

S ummar y: Mill comments on the notion of a perception as an intermediate link 
between the stimulation of our sense organs and the resulting sensation in our minds. 
He says that a perception "consists in the recognition of an external object as the 
exciting cause of the sensation." This notion is dismissed by Mill: "When a stone lies 
before me, I am conscious of certain sensations which 1 receive from it; but if 1 say that 
these sensations come to me from an external object which I perceive, the meaning of 
these words is that, receiving the sensations, 1 intuitively believe that an external cause 
of those sensations exists." Mill then says that the "laws of intuition and the conditions 
under which it is legitimate" fall within the field of psychology rather than logic. 

Comments: The notion of & perception is, in fact, central to logic, for it is perceptions, 
not sensations, that form the raw data of observations. Thus, contrary to the posi- 
tivists, we do not see patches of isolated colors. Rather, we see organized groups of 
sensations that are automatically integrated by our visual mechanism. It is by a pro- 
cess of abstraction that we can come to think of sensations (such as the color red) in 
isolation from the perceptions incorporating them. The above statements are vali- 
dated by experiments that each reader must perform individually in the laboratory of 
his own mind. That perceptions are primaries renders invalid any alternate attempt at 
their validation (say, by studying nerve impulses). 

In the previous paragraph, we have used perception to refer to an automatically 
integrated system of sensations. There is no reference in this definition to the 
existence of external causes. Mill is correct when he says that a perception does not 
give us immediate knowledge of an external object — we are all familiar with the 
phenomenon of hallucinations. In a wider sense, however, every perception is the 
result of an external condition: the relationship between our perceptual system 
(comprising, so far as we know, the brain and the sense organs) and the rest of the 


Mill's Logic: Of Names and Propositions 
external world. It is one of the tasks of science to sort out those characteristics of per- 
ception that are caused by our perceptual system (which is part of the external world) 
from those characteristics which are caused by the objects of our observation. 

Our perceptual system is the instrument through which we observe reality. Like any 
instrument, it has a definite identity and, hence, definite limitations. We do not con- 
sider measurement impossible because our instruments have limitations. Likewise, 
the impossibility of knowledge is not implied by the existence of limitations in our per- 
ceptual system. 

In summary, every perception has a cause in the external world. That cause is a 
complex of the observer and the observed. We will see later that a major task of sci- 
ence is to attribute characteristics of a perception to one or the other of these two 
external objects. 

5. Volitions and Actions 

Summary: Mill observes: "When we speak of sentient beings by relative names, a large 
portion of the connotation of the name usually consists of the actions of those beings; 
actions past, present, and possible or probable future." What is an action? It is "not 
one thing, but a series of two things: the state of mind called a volition, followed by an 
effect." For example, when I form the volition to move my arm, it moves, unless it is 
paralyzed or restrained. 

Comments: Mill alludes to, but does not identify, the fact that a volition is an irreduci- 
ble primary, just like a perception. That is, we do not see any "substeps" in an act of 
perception. That is, we do not see any "substeps" in an act of perception; it is literally 
immediate (no mediate steps). This epistemo logical primacy of perceptions does not 
contradict any causal primacy that might be identified by studying physiology. 

Similarly, a volition is immediate; I observe no steps between my volition to move 
my arm and the motion of my arm. There is of course nothing mystical or superna- 


Of the Things Denoted by Names 
tural about the irreducibility of a volition; it is an epistemological irreducibility, not a 
causal (or onLjlogical) irreducibility. Epistemological irreducibility uoes not contrad- 
ict any explanation by science of volition in terms of nerve impulses or other physical 

Thus, we can call a perception an afferent primary and a volition an efferent pri- 

6. Substance and Attribute 

Summary: Having dealt with feelings (sensations, thoughts, emotions, and volitions), 
Mill proceeds to the two remaining classes of namable things, substances and attri- 
butes. Previous logicians, Mill observes, have usually drawn this distinction on the 
basis of a word's grammatical function (e.g., noun or adjective) rather than on the 
basis of distinctions among the things the words name. In answer to the classical ques- 
tion, whether substances can exist without attributes, or attributes without substance, 
Mill says, "we can no more imagine a substance without attributes than we can imagine 
attributes without a substance." "Whiteness, without any white thing, is a contradiction 
in terms." 

7. Body 

Summary: Mill defines a body as "the external cause to which we ascribe our sensa- 
tions." Are we justified in ascribing our sensations to an external cause? Certainly "a 
part of our notion of a body consists of the notion of a number of sensations of our own, 
or of other sentient beings, habitually occurring simultaneously." Is there any reason 
to presume a substratum underlying these recurring groups of sensations? After 
reviewing the theories of Locke, Hartley, Hamilton, and Berkeley, Mill concludes that 
"of the outward world, we know and can know absolutely nothing except the sensations 
which we experience from it." 

Comments: There are many hypotheses that can explain our sensations: external 


Mill's Logic: Of Names and Propositions 
reality, hallucinations, perceptual errors, etc. In most cases the simplest hypothesis is 
that external reality is the cause of our sensations. Why we choose the simplest 
hypothesis, and how we judge simplicity, are topics we take up in Book III, Of Induc- 

8. Mind 

Summary: Mill says that, just "as our conception of a body is that of an unknown excit- 
ing cause of sensations, so our conception of a mind is that of an unknown recipient or 
percipient of them; and not of them alone, but of all our other feelings." "As bodies 
manifest themselves to me only through the sensations of which I regard them as the 
causes, so the thinking principle, or mind, in my own nature makes itself known to me 
only by the feelings of which it is conscious." 

Comments: Recall (Section 3) that perceptions, not raw sense data, are primary, and 
that perceptions are automatically integrated sensations (both external and internal). 
It seems that our notion of self is a perception automatically integrated from internal 
sensations of our own mental processes. Thus, our notion of 'self is an epistemological 
primary. (See Nozick (1981), Chapter 1, for a good discussion of borderline cases in 
the notion of 'self'.) 

9. Qualities 

Summary: Mill turns from substances to attributes, which he says are of three kinds; 
qualities, quantities and relations. He says that "if we know not and cannot know any- 
thing of bodies but the sensations which they excite in us or in others, those sensations 
must be all that we can, at bottom, mean by their attributes ..." 

Mill asks what it is that we mean when we ascribe a quality, such as whiteness, to 
some object, such as snow. Do we mean only that when snow is presented our sense 
organs a certain sensation (that we call white) is experienced? Or do we mean that the 
object possess some inherent "potency," the attribute whiteness, which produces in us 


Of the Things Denoted by Names 
that sensation? In short, is there any difference between a quality and a sensation? 

Although Mill says that this distinction is not important to logic, he does claim that, 
"when we say that snow is white because it has the quality of whiteness, we are only re- 
asserting in more technical language the fact that it excites in us the sensation of 
white. If it be said that the sensation must have some cause, I answer, its cause is the 
presence of the assemblage of phenomena which is termed the object." Mill says that 
there is no point in interpolating a "potency" between the object and the sensation. 

10. Relations 

Summary: "The qualities of a body, we have said, are the attributes grounded on the 
sensations which the presence of that particular body to our organs excites in our 
minds. But when we ascribe to any object the kind of attribute called a Relation, the 
foundation of the attribute must be something in which other objects are concerned 
besides itself and the percipient." Two things can be said to be related when "there 
exists or occurs, or has existed or occurred, or may be expected to exist or occur, 
some fact or phenomenon, into which the two things ... both enter as parties con- 
cerned. This fact, or phenomenon, is what the Aristotelian logicians called the fun- 
damentum relationis. " 

Relations can be based on complicated series of facts (as are legal relations), or can 
be based on very simple facts. An example of the latter is our experience of two events 
as either simultaneous or successive. Mill claims that these latter relations are per- 
ceptual primaries that canno be analyzed further. 

11. Resemblance 

Summary: Mill claims that two other sorts of relations, likeness and unlikeness, are 
also primaries. "Resemblance is evidently a feeling, or state of the consciousness of 
the observer." These relations are not capable of [epistemological] analysis because 
they are presupposed in every analysis "Likeness and unlikeness, therefore, as well as 


Mill's Logic: Of Names and Propositions 
antecedence, sequence, and simultaneousness, must stand apart among relations, as 
things sui generis. " Certainly, however, complex cases of likeness and unlike ness can 
be resolved into simpler ones. "All likeness or unlikeness of which we have any cog- 
nizance resolve themselves into likeness and unlikeness between states of our own, or 
some other, mind." 

Comments: A fundamental requirement of good science is to recognize epistemological 
primaries. To attempt to analyze a primary is an error, specifically, a category error. 
This is because it is an error to attempt to base an epistemological primary upon 
something that is not a primary. 

12. Quantity 

Summary: Mill asks us to consider two comparisons: the first, between a gallon of water 
and ten gallons of water; the second, between a gallon of water and a gallon of wine. In 
the first case we say that they different in quantity, in the second in quality. Both of 
these assertions are grounded on differences in the sensations they excite. But what is 
the distinction between a quantitative difference and a qualitative difference? Mill 
says, "This likeness and unlikeness 1 do not pretend to explain, no more than any other 
kind of likeness or unlikeness." 

Comments: Certainly, at base, our notion of quantity is an epistemological primary, 
just like our notions of color and shape. Depending on circumstances (such as arrange- 
ment) we can directly perceive numbers up to about a dozen. That is, our perceptual 
system tells us automatically whether some number of objects is equal, greater, or less 
than some other number of objects. This automatic perceptual integration is the basis 
for our recognition of quantity, as it is for our recognition of color, shape, etc. 

Of course, we extend our notion of quantity beyond those quantities that are 
immediately perceivable, just as we extend our notions of color and shape beyond the 
immediately perceivable. (For example, certain colors and shapes are indistinguish- 


Of the Things Denoted by Names 
able on a direct perceptual basis.) This extension of ideas from the directly perceivable 
primaries to non-primaries is the basis for measurement. But, to understand this 
extension process, we must investigate more carefully what distinguishes quantity from 
other attributes, which we will do in the comments accompanying Book II, Chapter 6 
(On the Science of Number). 

13. Attributes Concluded 

Summary: "Thus, then, all the attributes of bodies which are classed under quality or 
quantity are grounded on the sensations which we receive from those bodies ...." The 
same applies to every attribute of mind, which "consists either in being itself affected 
in a certain way or affecting other minds in a certain way." This provides the key to 
the analysis of attributes such as 'beauty': "As we thus ascribe attributes to minds on 
the ground of ideas and emotions, so may we to bodies on similar grounds, and not 
solely on the ground of sensations: As in speaking of the beauty of a statue, since this 
attribute is grounded on the peculiar feeling of pleasure which the statue produces in 
our minds, which is not a sensation, but an emotion." Thus, all attributes are ulti- 
mately grounded on feelings, i.e., states of consciousness. 

14. Recapitulation 

Summary: In summary, the things which can be named are in three categories. First, 
there are feelings (states of consciousness), which are of four sorts: sensations, 
thoughts, emotions, and volitions. Second, there are substances, which are of two 
sorts: bodies and minds. Finally there are attributes, which are of three sorts: quali- 
ties, relations, and quantities. But, Mill has argued that all attributes are reducible to 
sensations or states of consciousness. These considerations lead Mill to the following 
enumeration of the categories of all namable things: 

1. Feelings, or states of consciousness. 

2. Minds, which experience those feelings. 


Mill's Logic: Of Names and Propositions 

3. Bodies, or external objects which excite certain of those feelings. 

4. Successions and coexistences, likeness and unlikeness, between feelings or states 
of consciousness. 

Mill concludes this section by distinguishing psychological or subjective facts, which 
are composed solely of feelings or states of consciousness, from objective facts, which 
also incorporate substances and attributes. "We may say, then, that every objective 
fact is grounded on a corresponding subjective one, and has no meaning to us (apart 
from the subjective fact which corresponds to it), except as a name for the unknown 
and inscrutable process by which that subjective or psychological fact is brought to 

Comments: This latter view, that objective facts are based on subjective facts, is essen- 
tially the view of British empiricism, in the tradition of Locke, Berkeley, and Hume. 
The intended readers of this work, scientists, generally don't need to be convinced that 
the real world exists. Nevertheless, it may be worthwhile to spend a few sentences to 
discuss Mill's subjective empiricism. Certainly, if one doubts the existence of the real 
world, then there is very little reason to engage in science, since the purpose of sci- 
ence is to give us knowledge about the real world. That is, the existence of the real 
world is a presupposition of the study of logic or the scientific method. To put this 
another way, the purpose of science is to establish true propositions, i.e., statements 
that correspond with the state of affairs in the real world (the so-called correspon- 
dence theory of truth). Without the real world, there is no significance to truth. 
Hence, we will assume that any reader interested in Mill's scientific method does not 
question the existence of reality. 

Although it is certain that reality is, it is not certain what it is. The establishment of 
the nature of reality (as opposed to its existence) is, of course, the purpose of science. 
However, the only way we know reality is through our senses (including our introspec- 
tive awareness of our own mental states). This is the sense in which Mill is correct: 


Of the Things Denoted by Names 
epistemologicalLy, that is, in the order in which we gain knowledge, subjective facts are 
prior to objective facts. Indeed, this is the essence of objectivity: we only believe wh^ 
we ultimately can see, hear, touch, etc., either directly or indirectly. The goal of 
scientific methodology is the elaboration and refinement of the previous sentence. 


Mill's Logic: Of Names and Propositions 
4 . Of Propositions 

1. Nature and office of the copula 

Summary: A proposition is a sentence in which a predicate is affirmed or denied of a 
subject. To accomplish this affirmation or denial, a copula is necessary (usually the 
verb to be). Hence, the first division of propositions is into affirmative and negative. 
An affirmative proposition is one in which the predicate is affirmed of the subject; a 
negative proposition is one in which the predicate is denied of the subject. 

2. Affirmative and Negative Propositions 

Summary: The distinction between affirmative and negative propositions is real. Some 
writers, such as Hobbes, have claimed that negative propositions are just disguised 
affirmative propositions. For example, 'Caesar is not alive' is really a disguised form of 
'Caesar is non-alive'. But this analysis has just replaced one proposition denying a posi- 
tive predicate with another affirming a negative predicate. This has accomplished 
nothing. "The distinction between affirming and denying is real and is not to be got rid 
of by a verbal juggle." Thus, "when we affirm a negative name, we really affirm the 
absence, not the presence, of anything; not that something is, but that it is not. " 

Comments: The distinction between affirmative and negative propositions is an essen- 
tial one that has been abandoned in symbolic logic. The value of this distinction will be 
more apparent when 1 discuss induction and scientific method in Book III. As Mill 
notes, the distinction between affirmative propositions is grounded on the distinction 
between positive and negative terms, which I discussed in Chapter 2. 

3. Simple and Complex 

Summary: "A simple proposition is that in which one predicate is affirmed or denied of 
one subject. A [complex] proposition is that in which there is more than one predicate, 
or more than one subject, or both." Complex propositions in turn can be divided into 


Of Propositions 
categorical and hypothetical propositions. 2 

Categorical propositions are those "in which the assertion is not dependent on a 
condition." Categorical complex propositions are equivalent in meaning to two or more 
simple propositions. For example, 'Caesar is dead, but Brutus is alive' is equivalent to: 

1. Caesar is dead. 

2. Brutus is alive. 

3. Propositions 1 and 2 should be thought of together. 

4. There is a contrast between propositions 1 and 2. 

The function of the particle 'but' is to abbreviate propositions 3 and 4. 

Unlike the categorical complex propositions just considered, a hypothetical proposi- 
tion is not "a mere aggregation of simple propositions." Although a hypothetical pro- 
position may contain several subjects and several predicates, it makes only one asser- 
tion. Take as an example one kind of hypothetical proposition, the conditional proposi- 
tion. The conditional proposition 'If A is B, C is D' is just an abbreviation for 'The propo- 
sition C is D, is a legitimate inference from the proposition A is B'. The latter is a sim- 
ple proposition whose subject is 'the proposition C is D' and whose predicate is 'a legiti- 
mate inference from A is B'. Thus, a conditional proposition is a simple proposition 
whose terms are the names of propositions. 

"Like other things, a proposition has attributes which may be predicated of it." For 
example, ' /hat th whole is greater than its parts, is an axiom in mathematics' is a ] re- 
position whose subject is a proposition. 

Another common kind of hypothetical proposition is the disjunctive, for example, 
either A is B or C is D. Whately (1868) and others have shown that this is resolvable into 
the two conditionals: 

2. AH simple propositions are categorical. 


Mill's Logic: Of Names and Propositions 
. If A is not B, C is D. 

- If C is not D, A is B 

Comments: Mill's and Whately's explication of the inclusive or connective is easy to see 
in the prepositional calculus: 

(~P -» Q) & (~Q -> P) 
(~~P V Q) & (~~Q V P) 
(P V Q) & (P V Q) 
P V Q 

On the other hand, Mill's definition of the conditional proposition cannot be expressed 
in symbolic logic because it mixes the object language and metalanguage levels. That 
is, it is a proposition about the inferability of propositions. To express such a notion, 
we have to use techniques like Godel's for embedding metalanguage propositions in the 
object language. Traditional logic avoids the object language /metalanguage distinc- 

4. Universal, Particular, and Singular 

Summary: Propositions may be divided into three major classes on the basis of the 
degree of generality which the subject is understood to have. These classes are: 3 

Universal, for example, Ml men are mortal. 

Particular, for example Some men are mortal. 

• Singular, for example, Julius Caesar is mortal. 

A proposition is singular when its subject is an individual name. "When the name which 
is the subject of the proposition is a general name, we may intend to affirm or deny the 
predicate, either of all the things the subject denotes, or only of some." In the former 

case the proposition is universal, in the latter particular. Since in a singular 

3. Following Mill, we have used the traditional terms, although Bain's (Logic) terms total and partial are 
perhaps more descriptive. 


Of Propositions 
proposition the predicate is affirmed or denied of the entire subject, a singular propo- 
sition is usually considered a universal proposition. 

Traditionally, an occurrence of a name is said to be distributed when it stands for 
each and every individual the name denotes and is said to be undistributed otherwise. 
Thus, in a universal proposition the subject is distributed, but in a particular proposi- 
tion the subject is undistributed. Clearly, the subject is distributed in a singular pro- 

Comments: It is worth adding that in a negative proposition the predicate is distri- 
buted and in an affirmative proposition the predicate is undistributed. Thus we have 
the classification shown in Table 2. 

TABLE 2. Classification of Propositions 











All S and P 





Some S are P 





No S are P 





Some S are not P 

TABLE 3. Set Expressions Equivalent to Types of Propositions 




Set Expression 




S £P 








s c~p 





Since the extension of a general term is a class, we can express these propositional 
forms in the algebra of classes. This is shown in Table 2. Here we have used '5" } P' as 
an abbreviation for S n P* 0. Notice that the forms containing 'C distribute the sub- 
ject and those containing '~' distribute the predicate. Furthermore, note that if we 
always write a negative term with a '~' sign and a positive term without one, then the 
form of the proposition will be obvious regardless of "verbal juggling." Thus, whether 
we say 'some men aren't mortal' or 'some men are non-mortal', the form is 'men \ 
~mortal'. The symbolic notation also simplifies transforming propositions into 


Mill's Logic: Of Names and Propositions 
equivalent forms. For example, since S c ~P is equivalent toPc ~S (as we can con- 
vince ourselves with Venn Diagrams), we know that 'no man is a fish' is equivalent to 'No 
fish is a man'. Of course, the early development of the algebra of classes by De Morgan 
and Boole took place after Mill's Logic had been written, so he didn't have the benefit of 
notations such as this. 


Of the Import of Propositions 
5. Of the Import of Propositions 

1. Is Proposition a Relation Between Two Ideas? 

Summary: Mill asks, "What is that which is expressed by the form of discourse called a 
proposition, and the conformity of which to fact constitutes the truth of the proposi- 
tion?" Since a proposition consists of two terms (a subject and a predicate) connected 
by a copula, we must first understand what the terms represent, and then ask what 
kind of connection between them the copula asserts. 

Philosophers, "from Descartes downward, and especially from the era of Leibnitz 
and Locke," have taken a proposition "to consist in affirming one idea of another." will 
disagrees with this view. "When I say that fire causes heat, do I mean that my idea of 
fire causes my idea of heat? No, I mean that the natural phenomenon, fire, causes the 
natural phenomenon, heat." In fact he claims that focusing on ideas of things rather 
than the thin, s themselves was the cause of much of the sterility of pre-scientific rea- 
soning. The scientific view is that propositions "are not assertions respecting our ideas 
of things, but assertions respecting the things themselves." 

2. Is a Proposition a Relation Between the Meanings of Two Names? 

Summary: Mill next addresses Hobbes' view that a proposition signifies "the belief of 
the speaker that the predicate is a name of the same thing of which the subject is a 
name." When we assert that all oxen ruminate it is certainly true that all the individu- 
als denoted by the name 'ox' are asserted to be among all the individuals denoted by 
the name 'ruminating'. Thus Hobbes' analysis is true of all propositions. And in fact it 
is the entire meaning of some propositions, which "only shows what an extremely 
minute fragment of meaning it is quite possible to include within the logical formula of 
a proposition." 

Mill says that this account of meaning could be considered adequate only because 
Hobbes, in common with the nominalists, sought the meaning of terms entirely in their 


Mill's Logic: Of Names and Propositions 
denotation and ignored their connotation. What is wrong with this view? As explained 
earlier, "the meaning of all names, except proper names and that portion of the class 
of abstract names which are not connotative, resides in the connotation." Otherwise 
we could not explain the meaning of a proposition such as 'diamonds are combustible'. 
Certainly when mankind fixed the meaning of the word 'combustible' they did not know 
that the individuals denoted by 'diamond'. The names happen to fit the same objects 
because of a certain fact, which fact was not known when the names were invented. 
Thus "the objects are brought under the name by possessing the attributes connoted 
by it: but their possession of the attributes is the real condition on which the truth of 
the proposition depends; not their being called by the same name. Connotative names 
do not precede, but follow, the attributes which they connote." 

3. Is a Proposition an Expression of Class Membership? 

Summary: "The most generally received notion of a predication decidedly is that it 
consists in referring something to a class, that is, either placing an individual under a 
class, or placing one class under another class. ... If the proposition is negative, then ... 
it is said to exclude something from a class." This is essentially the same as Hobbes' 
theory, since "a class is absolutely nothing but an indefinite number of individuals 
denoted by a general name." 

Mill believes that this theory is an example of a common logical error, hysteron pra- 
teron 4 (last first), or "explaining a thing by something which presupposes it." This is 
because, only after having judged that snow is white and several other objects are white 
do I gradually begin to think of white objects as forming a class. ""We place the indivi- 
dual in the class because the proposition is true; the proposition is not true because 
the object is placed in the class." Mill claims that this view seems to treat classes as 
preexisting, as though having once been laid down by the framers of language. Furth- 
ermore, we must think of the meaning as constantly changing as individuals become or 

4. ixTTspov irpoTtpov. 


Of the Import of Propositions 
cease to be members of the class. Thus, a name would have no definite meaning. "The 
only mode in which any general name has a definite meaning is by being a nam of an 
indefinite variety of things, namely, of all things, known or unknown, past present, or 
future, which possess certain definite attributes." Thus, attributes are prior to classes. 

Mill concludes by noting the dominance of these two erroneous views: "Since the 
revolution which dislodged Aristotle from the schools, logicians may almost be divided 
into those who have looked upon reasoning as essentially an affair of ideas and those 
who have looked upon it as essentially an affair of names." 

Comments: This trend has certainly continued through the 20th century, although the 
idea-oriented viewpoint has tended to be displaced by the class-oriented view begun by 
Boole, Russell and Whitehead, and the formal view that descended from symbolic logic. 
Although these views capture most of deductive reasoning, they are inadequate for 
explaining induction. The following sections present Mill's attempt to solve these prob- 

4. What it Really Is 

Summary: Consider a singular proposition, such as 'Socrates is wise'. The meaning of a 
proposition such as this is that 'the individual thing denoted by the subject has the 
attributes connoted by the predicate'. Next, consider a proposition whose subject is 
connotative, such as 'All men are mortal'. In this case we are also asserting that the 
objects denoted by the subject possess the attributes connoted by the predicate. How- 
ever, in this case the objects are not individually pointed out— they are idenufied by 
some of their attributes, namely those connoted by the name 'man'. Thus, a proposi- 
tion of this form means that "whatever has the attributes connoted by the predicate; 
that the latter set of attributes constantly accompany the former set." 

Mill carries this analysis one step further by recalling that "every attribute is 
grounded on some fact or phenomenon, either of outward sense or of inward conscious- 


Mill's Logic: Of Names and Propositions 
ness, and that to possess an attribute is another phrase for being the cause of, or form- 
ing a part of, the fact or phenomenon upon which the attribute is grounded. ... The 
proposition which asserts that one attribute always accompanies another attribute 
really asserts ... that one phenomenon always accompanies another phenomenon ..." 

Comments: There are two ways to interpret Mill's explanation, based on the exact 
notion of connotation we adopt. If by the connotation of a term we mean all the attri- 
butes shared by the individuals denoted by that term, then his analysis is correct. For 
example, among the many attributes connoted by man we find the attributes connoted 
by mortal. This is what we mean when we say 'all men are mortal'. It is of course one 
of the tasks of science to determine whether the attributes connoted by mortal are 
among those connoted by man. 

Another interpretation is that the connotation of a term includes only certain essen- 
tial attributes. For example, the connotation of man might by rational animal, or 
featherless biped. In this case the connotation of man does not include the connota- 
tion of mortal. By this view 'man' can be considered to be just an abbreviation for 
'rational animal' or 'featherless biped'. This analysis can easily be seen to be inade- 
quate, since we can easily imagine discovering featherless bipeds that we would not be 
willing to call men. The The reason is that they would not share the other attributes of 
men, such as rationality. Thus the connotation must include all the attributes. 

We can see in the above interpretation a different class-oriented analysis of proposi- 
tions. 'All men are mortal' does not mean that the denotation of man is a subclass of 
the denotation of mortal, but that the connotation of man is a superclass of the conno- 
tation of mortal. 

5. What it Is that Propositions Assert or Deny 

Summary: "The object of belief in a proposition, when it asserts anything more than 
the meaning of words, is generally ... either the co-existence or the sequence of two 


Of the Import of Propositions 
phenomena." For example, when we say, "A generous person is worthy of honor," we 
"affirm that wherever and whenever the inward feelings and outward facts implied in 
the word generosity have place, then and there the existence and manifestation of an 
inward feeling, honor, would be followed in or minds by another inward feeling, appro- 

Although propositions asserting sequences and co-existences among phenomena are 
the most common, "we make propositions also respecting those hidden causes of 
phenomena, which are named substances and attributes." As noted previously, 
though, "no assertion can be made, at least with a meaning, concerning these unknown 
and knowable entities, except in virtue of the phenomena by which alone they manifest 
themselves to or faculties." Thus, propositions asserting the co-existence or the 
sequence of substances and attributes reduce to propositions concerning the co- 
existence and sequence of phenomena. [This view is known as phenomenalism]. 

Besides propositions that assert co-existence and sequence, there are those that 
assert simple existence and causation. Some logicians, such as Bain, have asserted 
that the concept of simple existence is empty. His Law of Relativity says that tilings 
can be perceived or apprehended only by contrasting them with other things. But, 
since "we have no other class to oppose to Being, or fact to contrast with Existence," 
these words are merely "fictitious and unmeaning language." Mill disagrees. The 
meanings of existence and being lie in the fact that to exist is to excite, or be capable 
of exciting, any sensations or states of consciousness ...." A thing can't be without 
being something . Causation and existence are discussed further in Book III. 

"To these four kinds of matter-of-fact or assertion must be added a fifth, resem- 
blance. This was a species of attribute which we found it impossible to analyze; for 
which no fundamentum distinct from the objects themselves could be assigned." Thus, 
a statement, such as "This color is like that color," cannot be analyzed [epistemologi- 
cally] into any more basic propositions. 


Mill's Logic: Of Names and Propositions 
"It is sometimes said that all propositions whatever of which the predicate is a gen- 
eral name do, in point of fact, affirm or deny resemblance." There is only a slight 
degree of foundation for this remark, for although the arrangement of things into 
classes in based on resemblance, it is not a mere general resemblance, but rather a 
resemblance that "consists in the possession by all those things [in the class] of cer- 
tain common peculiarities." It is those peculiarities "which the terms connote, and 
which the propositions consequently assert, not the resemblance." 

There are some exceptional classes that are founded on general unanalyzable 
resemblance. "The classes in question are those into which our simple sensations, or 
other simple feelings, are divided." Thus when I classify things as white, the basis for 
this classification is an unanalyzable [i.e., epistemologically primary] sensation of 

"Existence, co-existence, sequence, causation, resemblance: one or other of these 
is asserted (or denied) in every proposition which is not merely verbal. This five-fold 
division is a exhaustive classification of matters-of-fact, of all things that can be 
believed or tendered for belief, of all questions that can be propounded, and all 
answers that can be returned to them." 

Comments: The position of phenomenalism, that all proposition are ultimately proposi- 
tions about sensory phenomena, has developed into one of the dominant modern 
schools of the philosophy of science, the logical positivism of the Vienna Circle ori- 
ginated by Mach, Schlick and others. Against this position we can use Mill's own argu- 
ments (Section 1). When we say that an eclipse is caused by the moon coming between 
us and the sun, we clearly intend to say something about these objects in reality (i.e., 
the sun, moon and ourselves), not our sensations of these things. We use different ver- 
bal forms to talk about sensations. For example, "Our sensation of the sun is caused 
by its light falling on our retina, which causes impulses to travel down the optic nerve 
...," and so forth. Mill would be more correct in saying that propositions concerning 


Of the Import of Propositions 
substances and attributes are in fact intended as statements about the real world, but 
that the evidence for these propositions are propositions about the phenomena we 
experience directly. 

6. Propositions with Abstract Terms 

Summary: In the previous analysis Mill has addressed only propositions whose terms 
are concrete. However, since the meaning of such a proposition is based on the attri- 
butes which its terms connote, it is easy to extend the analysis to propositions whose 
terms are abstract, since abstract terms directly denote attributes. For example, the 
proposition, 'Prudence is a virtue', in which the terms are abstract, may be rendered 
'All prudent persons, in so far as prudent, are virtuous', in which the terms are con- 

In the previous section Mill showed that whenever a proposition has a concrete 
predicate, what we are predicating is an existence, co-existence, or resemblance. The 
interconvertibility of abstract and concrete terms leads Mill to conclude that an attri- 
bute is necessarily either an existence, co-existence, causation, sequence or resem- 

Comments: Although in common discourse concrete propositions are the more com- 
mon, the above analysis suggests that semantically abstract propositions are more fun- 
damental. This is because abstract terms directly denote the characteristic attri- 
butes, whereas concrete terms only connote them, and the meaning of a proposition is 
based on these attributes. 


Mill's Logic: Of Names and Propositions 
6. Of Propositions Merely Verbal 

1. Essential and Accidental Propositions 

Summary: Mill has already refuted the Conceptualists' claim, that propositions state 
relations between ideas, and the Nominalists' claim, that propositions express the 
agreement or disagreement between the meanings of names. He has claimed that pro- 
positions "assert five different kinds of matters of fact, namely, Existence, Order in 
Place, Order in Time, Causation, and Resemblance. ..." 

Mill reminds us, however, that there is a class of propositions that "do not relate to 
any matter of fact ... at all, but to the meaning of names." These would not be worth 
spending much time on, but for the fact that they occupy "a conspicuous place in phi- 
losophy" and that some philosophers regard them as expressing the most essential 

Comments: There are two senses in which we could say that proposition is about the 
meaning of names. One expresses a matter of fact about language, namely, that a 
given configuration of sounds or letters refers to a given concept. For example, when I 
say '6Lw$pomo<; is the Greek word for man', I assert a matter of fact about the Greek 
language. However, when I assert 'man is the rational animal', I am not stating a 
matter of fact about English. Rather, I am stating a fact of fundamental importance 
about the entities which we group together under the concept man. Such a definition 
is more than simply an assertion about the way English speakers use the word 'man'. 

2. Essential Propositions are Identical Propositions 

Summary: The Schoolmen divided the attributes as anything into two classes: the 
essential attributes and the accidental attributes. The essential attributes were con- 
sidered to be part of the essence of a thing, and thus go deeper than the accidents. 
For example, rationality was considered part of the essence of man, whereas that he 
cooks his food was considered an accident (i.e., not essential). This theory was based 


Of Propositions Merely Verbal 
on the Schoolmen's view that an object borrowed some of its properties form a univer- 
sal substance (an essence) and that the rest belonged to it individually (were acciden- 
tal). Although this view is often attributed to Aristotle, Mill notes that in the Categories 
Aristotle expressly denies that general properties inhere in a subject; they are merely 
predicated of it. 

Following Locke, Mill claims that an essential proposition only unfolds the meaning 
of a term. For example, the word man connotes all of man's attributes, including cor- 
poreity, rationality, and being living. Thus, when we say 'Man is rational' we have 
merely singled out one of the attribute connoted by man. Thus, since the essences of 
classes are merely the signification of their names, and the signification of a name is 
just its connotation, we can see that all propositions that have been called essential are 
in fact identical. 

Is there any value, then, to an essential proposition? Since it states part of the 
meaning of a term, it can only be informative to someone who does not already know 
the full meaning of the term. Thus, Mill says that the only really useful kinds of essen- 
tial propositions are definitions. Even here he notes, "In defining a name, however, it is 
not usual to specify its entire connotation, but so much only as is sufficient to mark 
out the objects usually denoted by it from all other known objects. And sometimes, a 
merely accidental property, not involved in the meaning of the name, answers this pur- 
pose equally well." 

Comments: What, ther , can we make of this distinction between essence and accident, 
between verbal and real propositions? There certainly seems to be a distinction 
between a proposition such as 'Man is rational' and propositions such as 'Man can cook 
his food,' 'Man has five fingers,' and 'Man domesticates animals.' What is this 
difference? They all state properties of men, but some seem more fundamental, 
(more essential), while others seem less fundamental. What makes some properties 
seem more fundamental? 


Mill's Logic: Of Names and Propositions 
As noted before, one aspect of this distinction is explanatory power. The rationality 
and animality of man implies that he can cook food, provide shelter for himself, com- 
pose symphonies, design computers, etc. Thus, we feel we have got to man's essence, 
when we realize he is a rational animal. Notice, however, that in this sense it is no 
accident that man builds houses; it is a consequence of his rationality and animality. 

Some properties of man do seem to be accidents: that man has five fingers on each 
hand does not seem to be a consequence of him being a rational animal. Further, this 
property doesn't seem to explain many others: it gives us no hint why he cooks or 
designs computers. Yet having five fingers is no less a property of man than is having a 
rational faculty. In this sense there is nothing accidental about it. 

Thus, if we take connotation in the wide sense, to mean all the properties, known 
and unknown, shared by the members of the denotation of a term, then we can see that 
any true, universal affirmative proposition only "unfolds" what is already contained in 
the connotation of the subject. Yet we can hardly call such propositions "merely ver- 

Should we then discard the notion of essential properties? No, for it expresses a 
useful epistemological fact: that at given point in our understanding of a subject some 
propositions have greater explanatory value than others. Peikoff (1979) says, "To 
designate a certain characteristic as 'essential' or 'defining' is to select, from the total 
content of the concept, the characteristic that best condenses and differentiates that 
content in a specific cognitive context." Note that properties are not inherently essen- 
tial: whether they are essential or not depends on their explanatory value relative to 
our knowledge. "The characteristic(s) which most fundamentally distinguishes a cer- 
tain type of entity from all other existents known at that time, may not do so within a 
wider field of knowledge, when more existents become known and /or more of the 
entity's characteristics are discovered" (Peikoff, 1979). 


Of Propositions Merely Verbal 
3. Individuals Have No Essences 

Summary: Since the essence of a term is its connotation, and, as described earlier, Mill 
claims that individual terms have no connotation, it follows that "individuals have no 

Comments: This conclusion results from taking the narrower notion of connotation. If 
we take connotation to refer to all the properties of a thing, then clearly individual 
terms have a connotation. But do they have essences? The essence of an individual 
would be those characteristics that explain and make possible the largest number of 
the rest. In many cases we would find ourselves in agreement with the Schoolmen on 
the essence of an individual. For example, the essence of Julius Caesar is the same as 
the essence of man: rationality and animality. 

4. Real Propositions, How Distinguished from Verbal 

Summary: "An essential proposition, then, is one which purely verbal" and therefore 
"either gives no information or gives it respecting the name, not the thing. Non- 
essential, or accidental propositions, on the contrary, may be called real propositions, 
in opposition to verbal." This is because they predicate of a thing "some attribute not 
connoted by [its] name." Thus, they tell us a new fact that we didn't know before. 

Comments: Although Mill's distinction between verbal and real propositions is falla- 
cious, he has hinted at an important point. When we are told something we already 
know, we consider the proposition "merely verbal" since it tells us something that was 
already part of our understanding of the meaning of the subject. In this sense, it only 
unfolds the meaning of that term. Thus, a proposition is verbal to me if it expresses a 
part of the connotation of the term with which I was already familiar. In contrast, a 
real proposition is one that tells me something new, something that adds to my 
knowledge of the connotation of the subject. 


Mill's Logic: Of Names and Propositions 
In this sense 'Man has five fingers' is a verbal proposition to most people, since most 
people have known that men have five fingers almost as long as they have grasped the 
concept man. On the other hand 'Man is a rational animal' is for many people a real 
proposition the first time they hear it, since it tells them a matter of fact that they 
might not have previously considered: that the characteristic most distinctive of man 
and that explains the largest number of his other characteristics, is that he is rational. 

5. Two Modes of Representing the Import of a Beat Proposition 

S ummar y: There are two different aspects in which a real, universal proposition can be 
considered, One "is best adopted to express the import of a proposition as a portion of 
our theoretical knowledge"; the other is useful "when the proposition is considered as 
a memorandum for practical use." In the first case the proposition 'All men are mor- 
tal' "means that the attributes of man are always accompanied by the attribute mor- 
tality." In the second case it "means that the attributes of man are evidence of, are a 
mark of, mortality. ..." The latter is usually the most useful view when we are studying 
the reasoning process. 

Comments: We can express these notions symbolically as follows. That the attributes 
of man are always accompanied by the attribute mortal, means that mortal is one of 
the attributes of man, mortality e attributes(man). Alternately we can say that the 
connotation of man includes the connotation of mortal, connotation(man) D 
connotation(mortal). The second interpretation says that when ever we find something 
that we can identify as man, we will know that that thing has the attribute mortal, 
man(x) -* mortal(x). 


On Classification and the Predicables 
7. On Classification and the Predicables 

1. Classification, How Connected with Naming 

Summary: Although the ideas of a class and classification play an important role in the 
work of many logicians, Mill does not stress these ideas. This is because he takes "gen- 
eral names as having a meaning, quite independently of their being the names of 
classes." Usually classes owe their existence to general names, since "[a]s soon as we 
employ a name to connote attributes, the things, be they more or fewer, which happen 
to possess those attributes, are constituted ipso facto a class. But in predicating the 
name we predicate only the attributes; and the fact of belonging to a class does not, in 
many cases, come into view at all." Killick (1909) illustrates this as follows: "Suppose 1 
take two attributes, 'perfect molecular mobility' and 'inelasticity' and devise a name 
'liquid', which shall connote or mean those properties, a class is ipso facto formed con- 
taining all objects possessing those two attributes." 

Occasionally the opposite process takes place: classification precedes the formation 
of general name. This occurs when "we have thought it useful for the regulation of our 
mental operations, that a certain group of objects should be thought of together." Mill 
cites as an example the classes, orders, etc. of Cuvier's classification of plants and 
animals. Yet, even in these cases the resulting classes are, "as much as any other 
classes, constituted by certain common attributes, and their names are significant of 
those attributes, and nothing else." 

Comments: In fact, classification usually precedes the definition of a general name in 
terms of essential attributes. For example, the meaning of 'liquid' is more than the 
two properties, 'perfect molecular mobility' and 'inelasticity.' Indeed we knew what 
liquids were long before we recognized that they are characterized by these two pro- 
perties. And it is possible that in the future we might discover a substance having 
these two properties that we are not willing to classify as a liquid, because it is 
different from liquids in too many other respects. 


Mill's Logic: Of Names and Propositions 
We initially form a class on the basis of certain perceived similarities between the 
members of the class, that is, on the basis of certain common properties. However, the 
resulting general name connotes more than the properties that originally motivated 
the formation of the class. The general term denotes all of the individuals possessing 
the originally identified common attributes, and connotes all of the common properties 
of these individuals. 

For example, the class of liquids is probably formed initially on the basis of certain 
unanalyzable resemblances. The denotation of 'liquid' is all liquids, and its connotation 
(in the wider sense) is all the properties of liquids. Among these, we may later dis- 
cover, are perfect molecular mobility and inelasticity. If, within the context of our 
knowledge, these two properties seem the most characteristic of liquids, then we are 
justified, for now, in defining a liquid as an inelastic substance with perfect molecular 
mobility. We could then say that, within the context of our knowledge, these attributes 
are the essential attributes of liquids. 

2. The Predicables 

Summary: "The predicables are a fivefold division of General Names, not grounded as 
usual on a difference in their meaning, that is in the attribute which they connote, but 
on a difference in the kind of class which they denote." The predicables, handed down 
from Aristotle, are: 

• Genus 

• Species 

• Differentia 

• Property 

• Accident 

These distinctions do not apply to general names in isolation; they refer to all the ways 
that a name cai be a predicate in a proposition. "The words genus, species, &c, are 


On Classification and the Predicables 
therefore relative terms; they are names applied to certain predicates, to express the 
relation between them and some given subject ...." 

3. Genus and Species 

Summary: In popular usage genus and species are relative terms. Thus, "any two 
classes, one of which includes the whole of the other and more, may be called a Genus 
and Species." For example, man is a species of the genus animal, and mathematician 
is a species of the genus man. 

The Aristotelian logicians used these terms in a more restricted sense. "It was 
requisite, according to their theory, that genus and species should be of the essence of 
the subject." Thus, man and brute could be considered coordinate species under the 
genus animal, but biped would not be considered a genus with respect to man; it would 
be considered a property or accident only. Does this distinction make any sense? The 
distinction is important, but the recourse to essence confuses the issue. 

Comments: Full comments follow the next section. 
4. Kinds Have a Real Existence in Nature 

Summary: When we consider the classes denoted by general names, they seem to be of 
two kinds. "There are some classes, the things contained in which differ from other 
things only in certain particulars which may be numbered, while others differ in more 
than can be numbered, more even than we need ever expect to know." For example, 
the class of white things is distinguished by no common properties other than white- 
ness. "But a hundred generations have not exhausted the common properties of 
animals and of plants, of sulphur or of phosphorus; nor do we suppose them to be 
exhaustible ..." 

"The differences [on which classification is based] are made by nature, in both 
cases, while the recognition of those differences as grounds of classification and of 
naming is, equally in both cases, the act of man ..." However, in the first case (e.g., 


Mill's Logic: Of Names and Propositions 
white) the act of classification is motivated by convenience; in the second (e.g., sulfur) 
nature requires us to form the class: "the ends of language and of classification would 
be subverted if no notice were taken of the difference." Hence the latter classes are 
called real kinds, and the former not^real kinds. 

The Aristotelian logicians considered only real kinds to be genera and species. In 
particular, the lowest real kind to which an individual is referable is called its species: 
the species of Isaac Newton is man. Although Newton belongs to many other classes 
(e.g., mathematician and Englishman) these are not species. The sexes, races, etc. 
would be considered not-real kinds if their differences turn out to be reducible to a few 
primary differences; if not, then they must be considered real kinds (i.e., species). 

In summary, a real kind is a class "which is distinguished from all other classes by 
an indeterminate multitude of properties not derivable from one another." A real kind 
is a genus or species or both. A genus is a real kind which is divisible into other real 

Comments: The crux of the problem with Mill's distinction between real and not-real 
kinds can be seen in the class sulfur. Mill calls this a real kind, since we will never 
exhaust the list of properties shared by all pieces of sulfur. On the other hand, it is 
quite possible that we may find that one property, say, its atomic structure, is the 
cause and explanation of all the rest. Then we would no longer be able to consider sul- 
fur a real kind. Thus, if we can find a finite number of properties that explain all the 
rest (i.e., if we can define the class) then it is a not-real kind. A kind remains real only 
in so far as we are ignorant of it. 

A class is initially demarcated on the basis of a finite number of properties. As we 
learn more about the class we continue to be aware of only a finite number of its pro- 
perties. If all the known properties are explainable in terms of just a few, then we con- 
sider it a not-real kind. If this is not the case, and we suspect there is an "inexhausti- 
ble supply" of more unexplainable properties, then it is a real kind. By this reasoning 


On Classification and the Predicables 
electron is not a real kind, because we believe all an electron's properties are explain- 
able by a few quantum numbers. This illustrates the fact that the real/not-real distinc- 
tion is at best an inessential one that is a reflection of our current knowledge of the 
class. It would probably be best to discard the distinction altogether. 

5. Differentia 

Summary: The word differentia "is correlative with the words genus and species, and 
... signifies the attribute which distinguishes a given species from every other species 
of the same genus." Can we use any attribute that will distinguish the given species? 
The Aristotelian logicians say "No;" the differentia must be of the essence of the sub- 
ject. Thus rationality could be considered the differentia of man; that he cooks his 
food could not be, since it is only an accidental property. Mill rejects this notion of 
differentia since it is based on the fallacious idea of essence. What then distinguishes 
the differentia from other properties? Mill observes that the connotation of a species 
includes the connotation of its genus. Therefore, since he has said that the essence of 
a name is just its connotation, he concludes that "the Differentia is that which must be 
added to the connotation of the genus, to complete the connotation of the species." 

Comments: With regard to the notion that the differentia is added to the genus to yield 
the species, Joseph (1906) says, "Provided it is not supposed that the differentia is 
added to the common character of the 'larger class' in the same extraneous way that 
sugar is added to tea, there is no fresh harm in this mode of expressing oneself." But 
if the differentia is not an extraneous property added to the genus, then what is it? We 
are inclined to say, with the Aristotelians, that the differentia should in some way be 

If we take essence not in the medieval sense of an essential substance, but in the 
sense of those properties which cause or explain most of the rest, then the essence can 
be the source of the differentia. Joseph (1906) suggests the following criteria for the 
characteristics which form the differentia: "these characteristics should be (a) of the 


Mill's Logic: Of Names and Propositions 
same general kind for each type within one genus, or ... variations upon the same 
theme, in order to exhibit the mutual relations of agreement and divergence among 
the various types; (b) important, or ... pervasive: that is, they should connect them- 
selves in as many ways as possible with the other characters of the species." These 
issues will be considered further when we discuss definition, in the next chapter. 

6. Property 

Summary: In the Aristotelian theory genus and differentia are of the essence of the 
subject. In contrast, properties and accidents are non-essential. The difference 
between properties and accidents is that properties follow necessarily from the 
essence, whereas accidents do not. Thus, that man uses language follows necessarily 
from his having a rational faculty, and is thus a property. That man has five fingers on 
each hand, follows neither from his rationality nor his animality, and is thus an 

"One attribute may follow from another in two ways; and there are consequently two 
kinds of [property]. It may follow as a conclusion follows premises, or it may follow as 
an effect follows a cause." An example of the former is the property of triangles that 
their angles sum to 180 degrees; an example of the latter is the property of man that 
he understands language. 

Comments: Since the distinction between essential and nonessential attributes is a 
contextual distinction, the distinction between properties and accidents is also contex- 
tual. If we take the essence of a name to be those attributes which explain or cause 
most of the rest, then the properties are those attributes that the essence does 
explain, and the accidents are those attributes that i.t doesn't. Frequently an accident 
is an unexplained property. An accident may also be an attribute which, for reasons of 
cognitive economy, we have chosen not to include in the essence. 


On Classification and the Predicables 
7. Accident 

Summary: Accidents are divided into two classes, separable and inseparable. Separ- 
able accidents are attributes that are universal to the species, but not necessary to it. 
For example, so far as we know, all crows are black, but this does not follow necessarily 
from crowness. And, if we discovered birds otherwise like crows but being white, we 
would call them 'white crows'; we would not say that they're not crows. 

"Separable Accidents are those which are found, in point of fact, to be sometimes 
absent from the species; which are not only not necessary, but not even universal." 
For example being red-haired is a separable accident of man, since some, but not all, 
men are red-haired. Similarly, those attributes that are not even constant in an indivi- 
dual, such as to be sitting or walking, must be considered separable accidents. 


Mill's Logic: Of Names and Propositions 
a Of Definition 

1 . A Definition, What 

Summary: "The simplest and most correct notion of a definition is, a proposition 
declaratory of the meaning of a word, namely, either the meaning which it bears in 
common acceptation, or that which the speaker or writer, for the particular purposes 
of his discourse, intends to annex to it." 

Notice that this means that words with no meaning, such as proper names, are not 
susceptible to definition. "In the case of connotative names, the meaning ... is the con- 
notation, and the definition of a connotative name is the proposition which declares its 
connotation." This is most commonly done by predicating of the name intended to be 
defined "another name or names of known signification, which connote the same aggre- 
gation of attributes." Since analysis means the resolution of a "complex whole into the 
elements of which it is compounded," it can be seen that a definition is an analysis. 

Comments: It is certainly counter-intuitive to say that proper names have no meaning, 
yet this follows from Mill's notion of connotation. As discussed earlier (Chapter 2, Sec- 
tion 4 and Chapter 6, Section 3), if we take connotation in the wider sense then we 
must conclude that individuals have a very rich connotation. Thus, as the connotation 
of a name grows, its extension narrows until the name becomes individual. Thus, as will 
be discussed below, proper names can be defined in the same ways as general names: 
by ostension (pointing) or by genus and differentia. 

If we take connotation in the wide sense, as referring to all the properties possessed 
in common by the members of the denotation, then it is clear that a definition cannot 
cite all of the connotation. This would not be a definition. Rather, it would be a catalog 
of everything known about the subject, and it would still be incomplete. For a 
definition to serve the goals of cognitive economy it must condense our knowledge of 
the subject. Thus a definition should be in terms of the essential characteristics of the 


Of Definition 
thing defined. One way to accomplish this is to formulate the definition in terms of 
genus and differentia. 

Several differences should be noted between this notion of definitions and that most 
common among modern logicians. The modern view is that a definition is merely a dev- 
ice for specifying the meaning of a word or phrase; it provides a sort of shorthand for a 
longer phrase. As a result, the word defined and its definition are completely inter- 

The view here is that the meaning of the name is probably already known; in any 
case the function of the definition is not to inform us of the meaning of the name. If we 
don't know what a liquid is, then being told that it is an incompressible substance with 
perfect molecular mobility, will not help much. Similarly, if we don't know the meaning 
of man, then being told the definition 'rational animal' won't be very useful; we still 
won't even know what a man looks like. Since a definition does not come close to 
exhausting the meaning of a name, it can be seen that a name and its definition are not 

A correct definition results from a scientific analysis of the things named. The 50a! 
of this analysis is to determine the essential attributes of these things. Of course, as 
our scientific knowledge expands, we may find that in a wider context the attributes we 
thought were essential can no longer be considered so. We will then have to find attri- 
butes that are essential in this wider context. Thus, a definition is a kind of scientific 
law and must be validated like other scientific law. 

2. What Names can be Defined? 

Summary: "A name, ... whether concrete or abstract, admits of definition, provided we 
are able to analyze, that is, to distinguish into parts, the attribute or set of attributes 
which constitute the meaning both of the concrete name and of the corresponding 
abstract: if a set of attributes, by enumerating them; if a single attribute, by dissect- 
ing the fact or phenomenon (whether of perception or internal consciousness) which is 


Mill's Logic: Of Names and Propositions 
the foundation of the attribute." "The only names which are unsusceptible of 
definition, because their meaning is unsusceptible of analysis, are the names of the 
simple feelings themselves." For these "we are obliged to make a direct appeal to the 
personal experience of the individual whom we address." 

3. Complete versus Incomplete Definitions 

Summary: The only scientific definition of a name is one which declares the whole of 
the facts that the name involves in it signification. Most people do not want such a 
scientific definition however; they are seeking a guide to the correct use of a term. 
There are two sorts of these unscientific definitions. The first is an essential but incom- 
plete definition, which defines a name on the basis of some but not all of the connota- 
tion of the name. An example is the definition, 'Man is the rational animal'. "Such 
definitions ... are always liable to be overthrown by the discovery of new objects in 
nature." For example, if we discovered a species of rational fish, we might have to 
revise our definition to 'Man is the rational mammal'. 

Incomplete definitions are what the logicians had in mind when they said a definition 
should be by genus and differentia. The differentia is seldom all that is peculiar about 
the species; it is usually just one of many peculiar attributes. 

Comments: In fact, almost all practical definitions are incomplete. As discussed previ- 
ously; (Section 1) a definition does not simply define an abbreviation for a set of attri- 
butes; it specifies the essential attributes — those distinguishing attributes that best 
explain the other distinguishing attributes. Guidelines for the choice of a differentia 
have been discussed in the commentary accompanying Chapter 7, Section 5. 

4. Complete Definitions versus Descriptions 

Summary: The second kind of unscientific definition is the accidental definition, or 
description, which bases the definition of a name on something which is not part of the 
connotation at all, but still "enables us to discriminate the things denoted by the name 


Of Definition 
from all other things." "What would otherwise be a mere description may be raised to 
the rank of a real definition by the peculiar purpose which the speaker or writer has in 
view." In this case the author is in fact giving to some general name, "without altering 
its denotation, a special connotation," and then defining the name in terms of this spe- 
cial connotation. An example is Cuvier's definition of Man as "a mammiferous animal 
having two hands." "Scientific definitions, whether they are definitions of scientific 
terms, of of common terms used in a scientific sense, are almost always of [this] kind;" 
"their main purpose is to serve as landmarks of scientific classification. And since the 
classifications in any science are continually modified as scientific knowledge advances, 
the definitions in the sciences are also constantly varying." 

Comments: This discussion, although based on Mill's incorrect idea of connotation, 
correctly points out the contextual nature of definitions: the correct definition of a 
term depends on the context in which that definition is to be used. This does not mean 
that definitions are arbitrary; rather, it means that the correctness of a definition is 
relative to a context. Within a given context of knowledge and purpose, a correct 
definition cites those properties that explain and make possible the greatest number of 
other attributes. 

5. Real Definitions versus Nominal Definitions 

Summary: Mill next turns to an ancient doctrine, that there are two kinds of 
definitions: definitions of names and definitions of things. "The former are intended to 
explain the meaning of a term; the latter, the nature of a thing, the last being incom- 
parably the most important." Mill notes that the nominalist trend in his time was tend- 
ing to discard the notion of real definitions (definitions of things) in favor of nominal 
definitions (definitions of names). [This trend has continued to the present.] 

Mill takes all definitions to be nominal. "We apprehend that no definition is ever 
intended to 'explain and unfold the nature of a thing.' " His basis for this is that no one 
has ever been able to distinguish a definition of a tiling from a proposition about that 


Mill's Logic: Of Names and Propositions 
thing. Hence he concludes that "All definitions are of names, and of names only ...." 
But in some definitions, "besides explaining the meaning of the word, it is intended to 
be implied that there exists a thing corresponding to the word." For example, the 
definitions of geometrical figures both give names to these figures and assert that they 
exist. If they didn't do the latter, it would be impossible to deduce true theorems from 
them. For example, when we say 'a triangle is a figure bounded by three straight lines', 
we are really making two statements in one package: 

1. There exists a figure bounded by three straight lines. 

2. We will call such a figure by the name 'triangle'. 

Comments: I disagree with Mill on this point. If we consider a definition such as 'Man is 
the rational animal' it becomes apparent that if we hadn't previously known the mean- 
ing of 'man' then this definition wouldn't have been very useful. We still wouldn't know 
the most rudimentary things about men, such as their appearance. Rather, when we 
assert 'Man is the rational animal' we are generally assuming that the hearer knows the 
meaning of 'man'. This means that he is aware of a substantial part of its denotation 
and connotation. That is, he can recognize men when he encounters them, and he 
knows a number of the properties of men. 

What, then, is the purpose of a definition such as 'Man is the rational animal'? Since 
it is not to inform us of the meaning of the word 'man', we can only conclude that it is 
to inform us of some fundamental fact about men. Thus against Mill I claim that, with a 
few trivial exceptions, there are no nominal definitions; the only definitions of interest 
in science and logic are real definitions. 

Mill's objection to real definitions, that there is no way to distinguish a real definition 
from any other proposition asserting a property of the thing, can be answered as fol- 
lows. He is right in the following sense: there is no way, once and forever, to pick out 
one or a few of an object's properties as the defining properties. This would imply 


Of Definition 
omniscience about the thing. As discussed previously, a real definition should be in 
terms of essential properties. But which properties are essential and which are not is a 
contextual issue: it must be settled relative to the available knowledge and the 
intended use of the definition. 

6. Mathematical Definitions 

Summary: What is the meaning of a mathematical definition? Consider a definition 
such as 'A circle is a plane figure bounded by a line all the points of which are at an 
equal distance from a given point within it'. This cannot be considered an assertion 
about real circles, since in no real circle are the radii exactly equal. Some people have 
said "that the subject-matter of mathematics, and of every other demonstrative sci- 
ence, is not things as they really exist, but abstractions of the mind. A geometrical 
line is a line without breadth; but no such line exists in nature; it is a notion merely 
suggested to the mind by its experiences of nature." 

Mill disagrees with this view, since he claims the mind "cannot conceive length 
without breadth; it can only, in contemplating objects, attend to their length, 
exclusively of their other sensible qualities, and so determine what properties may be 
predicated of them in virtue of their length alone." Thus, "the postulate involved in the 
geometrical definition of a line is the real existence, not of length without breadth, but 
merely of length, that is, of long objects." 

Comments: Another way of saying this is that a line is a thing whose breadth is negligi- 
ble. When is a thing's breadth negligible? That depends on circumstances; for some 
purposes a thing's breadth will be negligible, while for others it won't. Thus, whether a 
real thing is considered a line or not is a contextual issue. This is the case with most 
mathematical concepts. All mathematical concepts are models of the real world that 
attend to the significant aspects, and ignore the negligible aspects, of some real 
phenomena. Indeed, mathematics can be defined as the systematic ignoring of the 
negligible . 


Mill's Logic: Of Names and Propositions 
7. Definitions Grounded on Knowledge of Corresponding Things 

Summary: "Although, according to the opinion here, definitions are properly of names 
only, and not of things, it does not follow from this that definitions are arbitrary. How 
to define a name, may not only be an inquiry of considerable difficulty and intricacy, 
but may involve considerations going deep into the nature of the things which are 
denoted by the name." These inquiries are not "so much to determine what is, as what 
should be, the meaning of a name ...." 

It often happens that names are applied to things solely on the basis of resem- 
blance; the names have no distinct connotations in the minds of their users. "This, as 
we have seen, is the law which even the mind of the philosopher must follow, in giving 
names to the simple elementary feelings of our nature; but, where the things to be 
named are complex wholes, a philosopher is not content with noticing a general resem- 
blance; he examines what the resemblance consists in[,] and he only gives the same 
name to things which resemble one another in the same definite particulars. The philo- 
sopher, therefore, habitually employs his general names with a definite connotation." 
To accomplish this requires an inquiry into matters of fact. 

"In giving a distinct connotation to the general name, the philosopher will endeavor 
to fix upon such attributes as, while they are common to all the things usually denoted 
by the name, are also of greatest importance in themselves; either directly, or from 
the number, the conspicuousness, or the interesting character, of the consequences to 
which they lead." He will select those differentiae that lead to the greatest number of 
interesting properties. But this may be "one of the most difficult of scientific prob- 
lems" — and one of the most important. "[S]ome of the most profound and most valu- 
able investigations" in philosophy have "offered themselves under the guise of ••• 
inquiries into the definition of a name." 



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