Excluded Middle, Bivalence, and Disquotation

LEM: For every  p, p v ~p.

BV: Every proposition is either true or false.

These principles are obviously not identical.  Excluded Middle is syntactic principle, a law of logic, whereas Bivalence is a semantic principle. The first says nothing about truth or falsity. The second does. (See Michael Dummett, Truth and Other Enigmas, Harvard UP, 2nd ed. , 1980, p. xix; Paul Horwich, Truth, Oxford UP, 2nd ed., 1998, p. 79) Though not identical they might nonetheless be logically equivalent.  Two propositions are logically equivalent iff each entails the other.  Entailment is the necessitation of material implication. Can it be shown that (LEM) and (BV)  entail each other? Let's see.

The logical equivalence of the two principles can be demonstrated if we assume the disquotational schema:

DS: p is true iff p.

For example, snow is white is true iff snow is white. Or, if you insist, 'snow is white' is true iff snow is white. In the latter forrmulation, which does not involve reference to propositions, the truth predicate  -- 'is true' -- is merely a device of disquotation or of semantic descent. On either formulation, 'is true' adds no sentential/propositional content:  the sentential/propositional content is the same on both sides of the biconditional.  The content of my assertion is exactly the same whether I assert that snow is white or I assert that snow is white is true.  But if (DS) is granted, then so is:

DS-F: p is false iff ~p.

For example, snow is white is false iff ~(snow is white).    

Now if the disquotational schemata exhaust what it is to be true and what it is to be false, then (LEM) and (BV) are logically equivalent.

Given (DS) and (DS-F), we can rewrite (LEM) as

LEM-T: For every p, p is true v p is false.

Now (LEM-T) is simply a restatement of (BV). The principles are therefore logically equivalent given the disquotational schemata. 

But this works only if falsehood can be adequately explained in terms of the merely logical operation of negation.  This will NOT work if negation can only be explained in terms of falsehood.  For then we would enter  an explanatory circle of embarrassingly short diameter. 

Ask yourself: when is one proposition the negation of another? The negation of p is the proposition that is true iff p is false and false iff p is true.  To explain the logico-syntactic notion of negation we have to reach for the semantic notions of truth and falsehood.  But then falsehood cannot be exhaustively understood or reduced to negation.

It is telling that to explain negation and the other logical connectives we use TRUTH tables.  Such explanation is satisfactory.  But it would not be if the redundancy or disappearance or disquotational schemata gave the whole meaning of 'true' and 'false.'  (The point is made by M. Dummett, Truth and Other Enigmas, p. 7)

I take this explanatory circle to show that there is more to truth and falsehood than is captured in the above disquotational schemata.

Conclusion: if one's reason for accepting the logical equivalence of (LEM) and (BV) is (DS) then that is a bad reason.

Are there counterexamples to (DS)?  It seems to fail right-to-left if 'Sherlock Holmes is a detective' is plugged in for 'p' on the RHS of (DS).  Arguably, Holmes is a detective, but it is not true that Holmes is a detective.  For it to be true that Holmes is a detective, 'Holmes' would have to refer to something that exists.  But this requirement is not satisfied in the case of purely fictional items.  I am assuming that veritas sequitur esse, that truth 'follows' or supervenes upon being (existence):

VSE:  There are no true predications about what does not exist.

Since Holmes does not exist, 'Holmes is a detective' appears to express a proposition that is neither true nor false. Likewise for its negation, 'Holmes is not a detective.'  (LEM) is not violated since either Holmes is a detective or Holmes is not a detective. But (BV) is violated since the two Holmes propositions are neither true nor false.

It is worth noting that from 'Only propositions have truth-values' one cannot validly infer 'All propositions have truth-values.'  

Comments

[Horace Jeffery Hodges]

I thought BV meant Bill Vallicella!

Sorry, Bill, bad joke . . .

Jeffery Hodges

* * *

[BV]

Hi Jeff,

Great to hear from you. Are you on Facebook?

Just finished listening to Trump's State of the Union address. Impressive. He really rose to the occasion.

Best to you and yours,

Bill

[The Bad Ostrich]

A few points. First, the schemata I set out in my previous comments were not disquotational.

TRUE: S says that p and p
FALSE: S says that p and not-p

I used a ‘that’ clause to specify what S says. Of course there is a schema, but that is because ‘p’ is a placeholder for any meaningful sentence.

Second, I note you have only two schemata, whereas Aristotle gives 4. I suppose two are sufficient on the assumption that ‘p’ could stand in for a negated sentence, e.g. ‘grass is not green’. So if S says that grass is not green and it is not the case that grass is not green, then S speaks falsely.

Third, your schema is

DS: p is true iff p.

This is not quite the same as my schema, as is obvious from above. Let’s assume it doesn’t matter for the moment.

>> This will NOT work if negation can only be explained in terms of falsehood. For then we would enter an explanatory circle of embarrassingly short diameter. The negation of p is the proposition that is true iff p is false and false iff p is true. To explain the logico-syntactic notion of negation we have to reach for the semantic notions of truth and falsehood. But then falsehood cannot be exhaustively understood or reduced to negation.<<

But as I have argued elsewhere, we can define negation without reaching for the notions of truth and falsity. Assume that the notion of ‘all possible situations’ is coherent, and suppose it is coherent for any proposition ‘p’ to map onto a subset of that set. Then ‘not p’ maps onto the complement. The question is whether the very idea of a complement or of a subset covertly appeals to the concept of negation. But then that suggests that negation is a primitive indefinable concept, rather than what you are claiming (namely that it is truth and falsity which are primitive).

[The Bad Ostrich]

>> Are there counterexamples to (DS)? It seems to fail right-to-left if 'Sherlock Holmes is a detective' is plugged in for 'p' on the RHS of (DS). Arguably, Holmes is a detective, but it is not true that Holmes is a detective.<<

No. It is true that Conan Doyle said that Sherlock Holmes is a detective. It is not true, understood strictly and literally, that Sherlock Holmes is a detective.

[David Gudeman]

Bill, can you elaborate on a couple of things? First, how is DS-F a consequence of DS?

Second, you say "But this works only if falsehood can be adequately explained in terms of the merely logical operation of negation." I don't see where this comes from unless you are thinking that DS-F is to be taken as a definition of falsehood, but DS-F is symmetrical, so it could just as easily be taken as a definition of negation, or for that matter it could be taken as an axiom that is not a definition at all.