Presupposition and Excluded Middle

If Socrates dies at time t, then Socrates was alive prior to t. If Socrates does not die at t, then Socrates was alive prior to t.  Since both 'Socrates dies at t' and 'Socrates does not die at t' entail 'Socrates was alive prior t,' we say that the latter is a semantic presupposition of 'Socrates dies at t.'

But wait a minute! Doesn't what I have written generate an inconsistent tetrad?

1) p entails q

2) Not-p entails q

3) Necessarily, for any p, either p or not-p (Law of Excluded Middle)

4) q is contingent.

The conjunction of the first three limbs entails the negation of the fourth. So something has to give.

It is a datum that q -- 'Socrates was alive prior to t' -- is contingent: true in some but not all possible worlds. So we either reject semantic presupposition (which requires the truth of both (1) and (2) ) or we reject Excluded Middle.

Why not reject Excluded Middle? Socrates dies at t and Socrates does not die at t are contradictories: each is the negation of the other.  There is no possible world in which both are true.  And yet there are possible worlds in which neither is true. Those are the worlds in which Socrates does not exist.

Comments

[The Bad Ostrich]

>>Why not reject Excluded Middle? Socrates dies at t and Socrates does not die at t are contradictories: each is the negation of the other. There is no possible world in which both are true. And yet there are possible worlds in which neither is true. Those are the worlds in which Socrates does not exist.
<<
Contraries, not contradictories, are such that they cannot be both true, but can both be false.

With contradictories, on the other hand, one must be true, the other false. Both cannot be true, both cannot be false.

‘Each is the negation of the other’. Is ‘A is B’ the negation of ‘A is non-B’? Not if wide scope negation is intended.

Clearly excluded middle fails for contraries. Does it fail for contradictories?

[BV]

'A is B' makes no sense from a Fregean perspective. Write:
'a is F.'

[The Bad Ostrich]

But 'a is F' is badly formed in Fregeanese. You mean 'Fa'. Frege has no copula.

We Aristotelians write 'A is B', where A and B are the two terms, and 'is' the copula.

You touch on an important point though. 'Fa' has only one negation, namely '~Fa'. These are contradictories, not contraries, moreover one implies the existence of something that is F, the other, something that is non-F.

Back to the Kepler problem. For Frege, both the singular proposition and its contradictory require/presuppose the existence of a referent for the singular term. But Excluded Middle says necessarily the proposition or its contradictory is true, ergo necessarily there is a referent for the singular term.

Ergo, Socrates necessarily exists.

[David Gudeman]

I'm not trying to nitpick here, but I do not get your point. In particular, I don't see how 'Socrates does not die at t' implies 'Socratess is alive prior to t'. Example:

A: Socrates died in the first century BC.
B: No, Socrates did not die in the first century BC (because he died in the fourth century BC).

If I try to reword this to get the required implication, I might try 'Socrates stopped living at t' and 'Socrates continued living at t'. Both of these imply 'Socrates was living before t', but these statements are not negations of each other so the excluded middle does not apply.

Another attempt would be to take 'Socrates is a man' and 'Socrates is not a man'. Both of these imply 'Socrates exists', from which one could conclude that 'Socrates exists' is tautological. However, in modern logic we would say that this is not a fact about the real world, it is a fact about the theory, that in all models of the theory, Socrates exists. This is true by definition: every name has to denote something in the model.

On the other hand, I view the excluded middle as part of the definition of what makes a proposition rather than a property of all propositions as such because there are many declarative sentences that we might want to call propositions for which excluded middle does not hold.

[David Gudeman]

In rereading the above comment, I see that my counter example does not work. Let me substitute instead:

A. Socrates died in the seventh century BC.
B. Socrates did not die in the seventh century BC (because he was not yet born).

[BV]

D. G.,

I'm flailing about, trying to understand the notion of presupposition and how it relates to better-behaved notions like that of entailment.

The above post is no good. I take your point.