Sunday, 30 January 2011

A Problem You Can't Define

There is some disagreement, particularly pertinent in the recent debates about whether science conflicts with religion, about what kind of things constitute, or determine, a truth. Leaving aside the debate about the existence of intangibles, most people still accept at least two kinds of truth: empirical truths - that is, the true statements about the facts of our world - and truths by definition - that is, statements which are true purely by virtue of what they mean. It is the latter notion which I wish to discuss.

Most people agree that logical truths, such as

(1) all cats are cats

are true by definition. It does not matter what goes on in the world around us, they say; "all cats are cats" is true under all possible circumstances, and we know this just so long as we know the meanings of the words.

There is some problem with ambiguous terms which may be misleading; suppose that, in the first instance of the word "cats" we are just saying it the way Keith Richards says it, and in the second use we are talking about felines, then our statement will be false. Obviously if a word can have more than one meaning, then we need to know which meaning it has in the context.

This lead some theorizers to suggest that it is not the words (as in the sounds or symbols), but the concepts which they represent, which are to be the constituents of truths by definition. This seems more intuitive but it isn't particularly explanatory, as "concept" is just used the same way as "unit of meaning", and we need some notion of what meaning actually is. We'll come back to that later.

So how can a definition generate a truth such as in statement (1)? Gottfried Wilhelm Liebniz proposed a fundamental cleavage of "truths of reason" and "truths of fact"; that the former class were true of all possible worlds, and the latter only of our own. Logical truths, therefore, were necessary, whereas factual ones were just contingent. But how are we to test which things are true of all possible worlds, when we can only observe this one we are in? Saul Kripke's notion of necessity suffers the same paucity of information; and it is this question that the idea of "analytic" truths - that is, truths by definition - was supposed to answer. [1] [2]

Immanuel Kant proposed that an analytic truth was one in which the predicates are contained within the subject. So for instance, in

(2) all bachelors are unmarried

"unmarried" is contained within "bachelor". W.V.O. Quine pointed out that this suffers two major flaws. Firstly, it is not particularly explanatory, because we don't have a way of finding out whether a subject contains a predicate; worse, the word "contain" is surely metaphorical in this instance. In short: what does it mean for one concept to contain another, and how are we to find out whether it does in any particular instance? [3] [4]

The second problem is that we cannot apply this to statements such as

(3) either it is raining or it is not raining

in which we do not simply have the form "[subject] is [predicate]"; we rather have the form "([subject] is [predicate]) or ([subject] is not [predicate])". If such statements are analytic, then Kant's explanation will not do; and if they are not, then analyticity fails as an explanation of logical truth, at least within non-Intuitionistic logics. Furthermore, surely nobody claims that (3) was not true before the invention of language? If it had been false factually, would it still have become true analytically upon the invention of languge? If so, what is the use (or even the sense) of linguistically generated truth?

It has been suggested, alternatively, that a truth-by-definition is a statement the denial of which entails a contradiction. This is no better, however, because a contradiction is surely just a falsehood-by-definition. How do we test whether two concepts contradict each other?* And what does that mean exactly?

How do we know that a predicate is implicit within a subject? Statement (2) is of the form "(all A are B)", so its analyticity is not formally explicit. What we need is definition, because this gives us a means of transforming non-logical truths into logical truths* (and vice versa). For instance, if we can define "bachelor" as unmarried man" then we can show (by substitution) that "all bachelors are unmarried men" is synonymous with

(4) all bachelors are bachelors

and so we have a logical truth. We need, therefore, to do two things: (a) to show that logical truths such as "all A are A" are analytic, and (b) to show that definition can function without presupposing the analyticity of logical truth.

(a) Definition: The first problem is that not all terms can be explained in terms of definition, because the last term would have nothing in terms of which to be defined. This is known as the problem of the status of the primitives: if the last term is empirically understood then its meaning is uncertain, and so must be everything which is explained in terms of it. In natural English there are a vast plethora of such terms; within an artificial language, Rudolf Carnap managed to get it down to the word "is". Nonetheless, there must always be one such term, and so, the rest of the class of statements belonging to the language can be on no firmer ground than this undefinable term. [5]

(b) Definition prior to logical truth: The second problem is that we have some sort of circularity. Consider Carnap's notorious example of how to stipulate a "truth by definition" (or in his word, "convention"):

(i) For every x, y and z, if z is the result of putting x for "p" and y for "q" in "if p then q", and x and z are true, then y is true.

This tells us that if we have a true conditional statement with a true antecedent, then the consequent of the conditional is true. Suppose we already know:

(ii) z is the result of putting x for "p" and y for "q" in "if p then q", and x and z are true

then we can infer

(iii) y is true

but only if we use the logic of "if-then". The fourth English word of (i) is "if" and the fifth from the end is "then"; we know that given (i) and (ii) we can infer (iii), because we understand the English expression "if_then_". But this understanding is not provided by (i); rather it must be presupposed by it, in the sense that we can only understand the import of (i) if we already understand the notion if-then. More generally, the statement of definitions cannot be what determines logical truths or logical relationships, because it is only by virtue of logical relationships that logical truths or relationships are derivable from them.

Beyond this, we have another, simpler problem with definition. The fact that we are able to stipulate a definition (such as "every mare is a horse") by no means demonstrates that we have created a truth. What would stop Darwin from stipulating that

(5) species arise by means of evolution

is a definition? In other words: when somebody makes an assertion, what distinguishes it as the creation of a truth by definition, rather than merely as the (true or false) expression of a fact?

We are no closer, then, to finding some way of determining whether a predicate contains a subject, or whether two terms are interchangeable within the right context. What we need is some clear notion of meaning itself.

a) John Stuart Mill suggested that the meaning of a proper name is its reference. Consider:

(6) Morning Star
(7) Evening Star

These terms are co-referential, but do they mean the same thing? Gottlob Frege notes that the discovery that they are one and the same was one of astronomy, not analysis of the meanings of the words; so we cannot accept reference as an explanation of meaning. [6] [7]

b) It is not extension, because

(8) centaur
(9) unicorn

are alike in extension but differ in meaning, and

(10) creature with a heart
(11) creature with a kidney

may also turn out to be co-extensional, even though they differ in meaning.

c) It is not nomination because

(12) "8"
(13) "the number of planets"

name the same abstract entity, but are not synonymous; and conversely because, as noted in the case of (1), a word can have more than one meaning.

d) Meaning is not a phenomenological state, because we may be in different phenomenological states and yet still understand each other. Consider

(14) My uncle became a lawyer yesterday

Two speakers may picture entirely different things when they think of "uncle" or "lawyer" or "yesterday", and yet still altogether understand each other. Some predicates such as "clever" seem to have no corresponding imagery whatsoever. There may be associated imagery, but this is unimportant. There is imagery associated with nonsense-syllables.

e) It is not interchangeability salva veritate. The statements

(15) 'Mare' has four letters
(16) 'Female horse' has four letters

have different truth-conditions. Even excluding the special case of quotation: in an extensional language, the truth of a statement will always depend on the extension of the referents, and not merely on their meaning; in an non-extensional language, we have the same problem with (8) and (9). We might, in a non-extensional language, get a sufficient criterion by prefixing the assertions with some modal term. Consider:

(17) Necessarily, every creature with a heart is a creature with a heart
(18) Necessarily, every creature with a kidney is a creature with a heart
(19) Necessarily, every bachelor is an unmarried man
(20) Necessarily, every unmarried man is an unmarried man

This would seem to give us a criterion of meaning, because (17) and (18) differ in meaning and are not interchangeable; (19) and (20) are alike in meaning and are interchangeable. This might seem to complete the project, if only we could get an explanation of the term "necessarily", but as we saw earlier, no such explanation which would allow us to assert the interchangeability of (19) and (20) has yet been forthcoming.

It may be the case that two terms are synonymous if neither they nor any pair of compounds respectively including them differ in meaning. For instance, (8) and (9) are co-extensional while

(21) Picture of a centaur
(22) Picture of a unicorn

are not, so we get a difference in meaning. However,

(23) Female horse
(24) Mare

are co-extensional and so are

(25) Picture of a female horse
(26) Picture of a mare

so we get a synonymy. If this serves as a criterion for synonymy, then we can build around it a notion of analyticity and necessity.


(27) description of a female horse
(28) description of a mare

differ in extension.

(29) "female horse that is not a mare"

is an example of (27) but not of (28). By this argument, it seems no two terms will ever be synonymous.

Equally, cases (29) and

(30) "mare that is not a female horse"

show that Kant's notion of containment cannot be maintained.

Perhaps we can salvage some notion of meaning from the more specific notion of synonymy (or sameness-of-meaning). So how do we test statements for synonymy?

Peter Strawson and Paul Grice suggested that "two statements are synonymous iff any experience which, on certain assumptions about the truth values of other statements, confirm or disconfirm one of the pair, also, on the same assumptions, confirm or disconfirm the other to the same degree". But this suffers a similar setback to Liebniz' attempt; how can we test a proposition against all experiences? [8]

A suggestion is that two terms of synonymous if they stand for the same Essence or Platonic Idea. This is not much help, as we do not know how to discover whether they do so.

Another suggestion is that two terms are synonymous if they stand for the same mental image. Nelson Goodman notes that "It is not clear what we cannot and cannot imagine. Can we imagine a man 1,001 feet tall? Can we imagine a tone we have never heard?" [9]

A further suggestion is that two predicates P & Q differ in meaning iff we can conceive of something which satisfies P but not Q. What exactly does it mean to conceive something? We can define a five-dimensional body, although we cannot imagine it. But by this criterion we can conceive of a square circle. It might be contested that "square circle" is inconsistent; this leads us around in an obvious circle, however, for the reasons explained near the start of this essay.

Synonymy might be explained in terms of possibility. Two predicates P and Q are synonymous iff there is nothing possible that satisfies P but not Q. However, if we know that all things satisfy (10) and (11), we no longer regard it as possible that there exists something which satisfies (10) but not (11). Proponents of the possibility criterion may protest that there is a non-actual possible which satisfies (10) and not (11). However, it is difficult to accept that there is an entity which we know cannot be actual (because (10) and (11) are co-extensional) yet still retains the status of being possible. Furthermore, this seems to confound the meaning of "is" or "exists" altogether; if an entity is non-actual, then in what sense does it exist?

How are we to determine when there is a non-actual possible for some collection of predicates? It cannot be by testing whether "is P and ~Q" is consistent; because as long as P and Q are different predicates it will be logically self-consistent, and we have no means of determining whether it is otherwise consistent. In fact this latter question amounts to asking whether P and Q are synonymous, so we have come full circle. Hilary Putnam made an attempt to salvage the notion by suggesting a "one-criterion" notion of synonymy, but Jerry Fodor points out that "criterion" is on no better a footing. Furthermore. if a criterion is merely a condition for verification, then we have the same problem which confronts Strawson and Grice. [10] [11]

In conclusion, unless we can get some proper explanation of "meaning" which is not circular and does not rest upon equally dubious terms, let alone of "truth purely by virtue of meaning" then we will not have an adequate basis for supposing that any of our commonly accepted laws of logic, such as "1=1" are analytic. We will simply have to accept that they are facts about our world, and as such they should be treated as scientific theorems.


* Alonzo Church proved that there is no algorithmic means for determining, in arithmetic, whether "x = y", where x and y are arithmetical expressions. Quine claims that it follows that there can be no general means of testing for contradictoriness. [12] [13]


[1] Gottfried Wilhelm Liebniz (1714) - "Monadology"
[2] Saul Kripke (1980) - "Naming and Necessity"
[3] Immanuel Kant (1781) - "Critique of Pure Reason"
[4] W.V.O. Quine (1951) - "Two Dogmas of Empiricism"
[5] Rudolf Carnap (1928) - "Der Logische Aufbau der Welt"
[6] John Stuart Mill (1843) - "A System of Logic"
[7] Gottlob Frege (1982) - "The Basic Laws of Arithemetic"
[8] Peter Strawson and Paul Grice (1956) - "In Defense of a Dogma"
[9] Nelson Goodman (1949) - "On Likeness of Meaning"
[10] Hilary Putnam (1983) - Two Dogmas Revisited"
[11] Jerry Fodor (1987) - "Psychosemantics"
[12] Alonzo Church (1936) - "An Unsolvable Problem of Elementary Number Theory"
[13] W.V.O. Quine (1953) - "On What There Is"

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