CHAPTER IV OF LANGUAGE, TRUTH, AND LOGIC

CONTENTS OF LECTURE

Here we discuss: (1) the positivist dismissal of the synthetic a priori, and (2) the positivist account of analyticicity.

1  THE SYNTHETIC A PRIORI
1.1  Cartoon version of some of Kant's views

Kant notoriously thought that a lot of sentences were both synthetic and a-priori knowable.

 A-priori A-posteriori Analytic Logical Truths  (i.e. instances of ``P or not-P'')  Conceptual Necessities (i.e.  ``All bachelors are unmarried males.'') Synthetic Scientific Laws (i.e. ``force = mass times acceleration'')  Mathematical Truths (i.e. ``2 + 2 = 4'')  Geometrical Truths  (i.e. The sum of the interior angles of  a triangle equals 180 degrees)  General Moral Truths  (i.e.  It is wrong to lie.) Specific factual claims (i.e. ``The cat's on the mat.'')  General non-lawlike claims  (i.e. ``Everyone in this room has seen Star-Trek.'')

Without belaboring Kant's specific characterization of analyticity, etcetera, such that he grouped things this way we should see why one might want to have this list of synthetic a-priori truths.  Intuitively all of  Kant's synthetic claims are true in virtue of meaning and the world because they describe the way the world is in some sense.  When we say that ``2 + 2 = 4'' this claim is intuitively made true by the plus than and equality relation on the natural numbers.  That is, the natural numbers and the properties they have make the claim true.  (Kant's interesting story of how this works needn't detain us).  Thus, while the test mentioned above fails for Kant's synthetic/a-priori truths, we can see why Kant didn't want these truths to be analytic.

1.2  Empiricist rejection of the synthetic a priori

Empiricism is the view that all of our contentful factual knowledge comes through the senses.  That is, all of our evidence for the truth of a claim is sensory evidence.

Ayer takes himself to be an empiricist.  He takes Hume's arguments to show that correctness of  empiricism to entail that there are no synthetic a-priori truths.  The arguments would go like this:

1. A synthetic a-priori truth would be necessary, since it is a-priori.
2. A synthetic a-priori truth would refer to a matter of fact, since it is synthetic.
3. But truths that refer to matters of fact are by empiricism verified or falsified by sensory evidence.
4. But Hume showed us that no sentence verified or falsified by sensory evidence can ever be necessary, since no matter how many times a sentence is verified in practice, there always remains the possibility that it will be confuted on some future occasion.
5. Therefore, since a synthetic a-priori truth would refer to a matter of fact, it is not necessary.
6. But then, a synthetic a-priori truth would be both necessary and not necessary, which is incoherent.
7. Therefore, there are no synthetic a-priori truths.

It should be noted that Ayer really didn't need to appeal to empiricism (and open himself up to the accusation of question begging) in this argument. He should have argued along the following lines.

1. A synthetic a-priori truth would be necessary, since it is a-priori.
2. A synthetic a-priori truth would refer to a matter of fact, since it is synthetic.
3. But truths that refer to matters of fact are not necessary truths; for the facts could always have been different.
4. Therefore, since a synthetic a-priori truth would refer to a matter of fact, it is not necessary.
5. But then, a synthetic a-priori truth would be both necessary and not necessary, which is incoherent.
6. Therefore, there are no synthetic a-priori truths.

It should be noted that the basic way we made sense of a synthetic truth was in terms of whether we could imagine a situation where the sentence was false, without changing the meaning of the sentence. If this is central to our notion of a claim being synthetic, then all synthetic truths are contingent (such that they could have been false), and hence not necessary truths. So Ayer seems to be on good ground here.

1.3  Cartoon version of some of Mill's views

Ayer presents Mill as holding, in virtue of his (Mill's) empiricism that all sentences are synthetic/a-posteriori.  In the case of mathematical truths, Mill thinks that they are simply high-level inductive generalizations.   For Mill we experience adding 2 things to 2 things and get 4 things enough times that we end up making this generalization, for all things if you add 2 of them to 2 of them, you get 4 of them, and this is what ``2 + 2 = 4'' means.  It's as if we see enough white swans and then infer that all swans are white.

Ayer argues that Mill's position that Mathematical and Logical truths are synthetic/a-posteriori is itself absurd.  Ayer's argument goes like this

1. Synthetic a-posteriori truths are contingent (that is, non-necessary)  since they are a-posterioiri.
2. Contingent truths are such that, if they are true, then it is possible that they are false.
3. Mathematical and Logical truths are such that, if they are true, then it is not possible that they are false.
4. Therefore Mathematical and Logical truths are not synthetic a-posteriori.

1.4  Some of Ayer's views

At this point Ayer has to face the music.  How can an empiricist account for the necessity of mathematics and logic?  Part of the genius of Logical Positivism was to attempt to do this by reshuffling Kant's box this way, and characterizing Analyticity etcetera such that this reshuffling made sense.

 A-priori A-posteriori Analytic Logical Truths (i.e. ``P or not-P'')  Conceptual Necessities (i.e.  ``All bachelors are unmarried males.'')  Mathematical Truths (i.e. ``2 + 2 = 4'')  Geometrical Truths (i.e. The Pythagorean Theorem) Synthetic Scientific Laws  (i.e. ``f = m(a)'')  Specific factual claims (i.e. ``Frank's on the mat.'')  General non-lawlike claims  (i.e. ``Everyone in this room has seen Star-Trek.'')
2  ANALYTICITY

Ayer characterizes analytic truths in this manner,

A proposition is analytic if and only if it follows logically from a meaning characterizing definition.

If you want a good idea of what Ayer had in mind, think of your high-school geometry.  You start with some very general definitions concerning point and line, etcetera, and then by following strictly logical reasoning you prove any geometric statement you want to prove.  In Ayer's time, massive world-historical advancements in logic were beginning to be wrought such that  something like this method was beginning to be fruitfully applied to all areas of mathematics (though many of its limitations of this method yet to be discovered), and the idea caught hold in other areas of thought.

Once Ayer gives the above characterization of analyticity, he needs to characterize the meaning-characterizing definitions and the logical inference rules.  His (and other positivists') idea is that the logical deductive inference rules are guaranteed as valid in virtue of the meanings of words such as ``and,'' ``or,'' ``not,'' ``if - then,'' ``all,'' and ``some.''

Then, the initial definitions that corespond to (say in the case of Euclidean Geometry) the postulates of Euclidean geometry for example, are considered to be true in virtue of linguistic convention.  The initial definitions are in some sense considered to be stipulative;  they stipulate what we take the meaning of the terms occuring in the definitions to amount to.

Finally, Ayer has the task of trying to show how analytic truths such as these can be useful in our general theorizing about the world.  For example, a lot of mathematics is part of our physical theories.  If this mathematics is just the result of stipulating meaning of the mathematical terms by picking definitions and then following the logical consequence relation, it is very unclear why, say,  calculus is such an intergral part in describing the world with physics.

Note that ethical judgments occur nowhere in Ayer's boxes.  In Chapter 6 he argues that ethical statements are not even meaningful.

Also notice that scientific laws are now considered to be synthetic/a-posteriori by Ayer.

For Ayer the synthetic truths will be those which are not analytic, but are still meaningful by the principle of meaningfulness given above.  This brings us to Chapter 5 of Ayer's book.