I’ve been reading Ayer’s Language, Truth and Logic, and been struck by how hard it is define verifiability, which is crucial since Ayer thinks that statements are meaningful if and only if they are verifiable or analytic (or, I suppose, denials of analytic statements, though oddly he doesn’t mention that possibility). Ayer rightly notes that we don’t want verifiability to mean conclusive verifiability. In the body of the text, he offers three criteria for the (weak) verifiability of p, apparently without realizing they are quite different:
it is possible for an experience to “make [p] probable”
a possible observation would be “relevant to the determination of [p’s] truth or falsehood”
together with some additional premises it entails an observation statement that cannot be derived from the premises alone.
In the 1946 introduction, he realizes that (3) makes every proposition verifiable (just make the additional premise be “if p, then o” for any observation statement o) and offers this:
- a statement is directly verifiable if it is an observation statement or if conjoined with one or more observation statements it entails an observation statement that is not entailed by the latter observation statements,
then gives a complicated account of indirectly verifiable statements, and finally defines verifiable statements as ones that are directly or indirectly verifiable.
Here’s a curious thing: (1), (3), (4) as well as the complex account of indirect verifiability can apply to a statement p without applying to its negation. For instance, consider any crazy “metaphysical” non-verifiable statement q such as his example “the Absolute is lazy” and let o be any observation statement. Then q ∧ o satisfies (3) and (4), since it entails o by itself (and hence as conjoined with some other independent observation statement). On the other hand, the negation of q ∧ o is equivalent to ∼q ∨ ∼o and does not satisfy (3), or (4), or the indirect verifiability criterion. Similarly, q ∨ o satisfies (1), since the experience reported in o would make q ∨ o conclusively probable, but the negation does not seem to satisfy (1).
But we would expect that if a statement is meaningful so is its negation. Certainly, Ayer thinks so: non-meaningful statement are neither true nor false on his view.
Account (2) is nicely symmetric between truth and falsity, and hence escapes this worry. But it has the consequence that the conjunction of two meaningful statements can be meaningless. For again let q be any crazy non-verifiable statement, and let o be the statement “It’s bright here”. Then q ∨ o is verifiable by (1), since observing brightness is clearly relevant to the truth of the disjunction, and so is q ∨ ∼o, since observing darkness is clearly relevant. Thus, both q ∨ o and q ∨ ∼o are meaningful. But (q ∨ o)∧(q ∨ ∼o) is logically equivalent to q. Thus, either the conjunction of two meaningful statements can be meaningless, or every statement is meaningful.
Maybe the best move for the positivist is to allow that meaningfulness is not closed under negation?
7 comments:
Could Ayer plausibly restrict these criteria to atomic sentences. And then add that A is verifiable, then so is ~A; if A and B are verifiable, so is AvB and A&B etc?
That's an option, but we will have trouble with quantified sentences, since they aren't built out of atomic sentences. We also will run into trouble with gruesome predicates.
Have you seen this interview?
https://www.youtube.com/watch?v=nG0EWNezFl4
According to the older Ayer, the main flaw with his book is that nearly everything in it was false.
It's worth noting that in your example q does not need to be a metaphysical nonsense statement. It's enough for q to just be the proposition "false", or any proposition that is tautologically false. Then (q and o) implies o, satisfying (3) and (4). But (not (q and o)) is tautologically true, so doesn't imply anything (except other true tautologies).
In general, the definitions (1)-(4) call any tautologies meaningless. This may be intended, but of course has the consequence of making the set of meaningful statements not closed under certain logical operations, because those operations let us construct tautologies. I'm not that familiar with positivism, but however positivists deal with that problem is likely how they will deal with this one.
One possibility(?): a positivist could use is to use richer logical operators, on analogy with mathematical intuitionism or type theory, so that "p or not-p" actually demands evidence. For instance, a positivist could identify the semantic content of a statement with the set of sufficient packages of evidence for it. Then they'd define "and" to mean Cartesian product and "or" to mean union. So then "p or not-p" is a statement demanding either evidence of p or evidence of its negation. It is distinct from the tautology "true" in that it is unsupported if we have no evidence either way. I have no idea how this idea would pan out but it might fix some of these issues.
I think you missed the first sentence: "Ayer thinks that statements are meaningful if and only if they are verifiable or analytic (or, I suppose, denials of analytic statements, though oddly he doesn’t mention that possibility)."
Tautologies are analytic.
Yep; my mistake. Sorry about that.
On further reflection, it seems your suggestion of not having meanginfulness closed under negation would be pretty palatable to people who want truth to have to do with science, because exactly of examples of this type.
For instance, "electrons exist" is meaningful because it entails certain predictions about the electrostatic force, etc. But "electrons do not exist" formally allows for the possibility that instead of electrons there are pairs of "schmelectrons", and each pair of "schmelectrons", working together, generates exactly the same observations as one "electron" would have.
That is, "electrons do not exist" could explain anything, including the same observations that "electrons exist" predicts. So scientists would not count it as a workable hypothesis unless it also explicitly denied that the electron theory's observational predictions are true. But that would make it more than just the formal negation of "electrons exist".
(The above example is roughly your example, with "electrons exist" = (q and o) and q the "metaphysical part" of that statement while o is the "empirical part." I'm not sure if this can quite be expressed as a conjunction formally in this way, but that was the intent here.)
So it seems a positivist should be totally fine with meaningfulness not being closed under negation. Or perhaps they think the correct sense of negation should be something other than negation as usually defined (e.g. by claiming that (not x) must also assert the existence of some observational entailment of x that is false).
It's a standard Bayesian observation that, given regularity (all probabilities strictly between 0 and 1), if P makes Q more likely, then ~P makes ~Q more likely. Thus, if the hypothesis that electrons makes some observation more likely, then the denial of that hypothesis makes the non-occurrence of that observation more likely. And typically that's something observational.
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