Realism and antirealism

4. Logical and semantic versions

It is often said that realism and antirealism can be distinguished by their attitude towards the law of excluded middle (the logical principle that, given two propositions one of which is the negation of the other, one of them must be true): the realist accepts it, the antirealist does not. Again, we can understand this if we think back to the original characterization of the distinction in terms of what is independently there and what we ‘construct’, what is the case ‘in itself’ and what is so because of our ways of experiencing (see Intuitionistic logic and antirealism).

For explanatory purposes we may consider the world of literary fiction. Most people will be happy enough with the idea that, in so far as anything can be said to be true of the world of Macbeth, just those things are true which Shakespeare wrote into it. But in that case neither ‘Lady Macbeth had two children’ nor its negation ‘Lady Macbeth did not have two children’ is true, since Shakespeare’s text (we may suppose) does not touch on that question; the law of excluded middle fails in this ‘constructed’ world.

Passing now to a genuinely disputed case, there are those who think that whether a mathematical statement is true is one thing, whether it can be proved quite another; and there are those who think that truth in mathematics can only mean provability. For the latter the law of excluded middle is unsafe. From the fact that not-p cannot be proved, it does not follow that p can be proved; perhaps neither is provable and hence, on this view of mathematical truth, perhaps neither is true. And anyone who equates truth, in whatever sphere, with verifiability-in-principle by us will be liable to the parallel conclusion: only for those propositions p where failure to refute p is ipso facto to verify p may we rely on the law of excluded middle. Where verifying p and verifying not-p are distinct procedures, excluded middle fails. (It is because they are characteristically distinct when the proposition in question makes some claim about an infinite totality that we hear so much about infinite totalities and the rejection of excluded middle.) This explains why some writers (in particular Dummett) often say that the difference between realist and antirealist lies in the difference between their conceptions of truth (see Antirealism in the philosophy of mathematics; Realism in the philosophy of mathematics).

It can also be seen why it should have become common to express the realism–antirealism opposition as an opposition between theories of meaning, and why philosophers should be found speaking of realist and antirealist semantics. Any theory which ties meaning to verification, which equates the understanding of a sentence with a knowledge of those conditions that would verify it or would justify us in asserting it, promotes the view that we have no other idea of what it is for it to be true than for these conditions to be satisfied. Hence the realism–antirealism debate often exhibits neo-verificationist features; sometimes (especially by Dummett) antirealism is presented as the outcome of Wittgensteinian ideas about meaning, sometimes (especially by Putnam) of the alleged impossibility of explaining how our language could ever come to refer to the mind-independent items that realism posits (see Meaning and verification).