Conditionality What does TFF mean?

The paradox of material implication p ⊨ q ⊃ p is valid (by the definition of the truth table of ⊃ ) but trivially wrong in application Two responses to the paradox The truth table for ⊃ is a good model for the truth values of conditionals. We then should attempt to explain why the paradoxes seem odd. The truth table for ⊃ is not a good model for the truth values of conditionals. The gap between ⊃ and ‘if’ then explains why the paradoxes seem so odd.

Truth and assertibility (Grice: Logic and conversation) A proposition is true if what it says is the way things are. A proposition is assertible if it is appropriate or reasonable to assert. – You ought to make assertions you believe to be true, and not make assertions you believe to be untrue.

Grice’s maxim of relevance I ought not say more than is necessary, and I ought not say less than is necessary. If what I say more or less than necessary then it is not assertible. Griceian answer to the paradox: The inference from p to q ⊃ p seems odd because we hardly ever assert q ⊃ p on the grounds that p.

Conditional Probability (Frank Jackson) Assertibility of a conditional p ⊃ q varies with the conditional probability Pr(q|p) – Pr(q|p) the probability that q obtains, given that p obtains Given the special role that conditionals play in our own reasoning (such as in dealing with hypothetical situations, and their use in the context of limited information), conditionals ought to be robust in the context of increasing information, and that this motivates the definition of its assertibility in terms of conditional probability. Robust: it survives into the new context in the discourse

Another paradoxical example: tautology (p ⊃ q) v (q ⊃ r) is a tautology but wrong Either, if the Pope is a Catholic, Queensland wins the Sheffield Shield, or if Queensland wins the Sheffield Shield, the moon is made of green cheese.

Another approach: material conditional is not conditional (Bertrand Russell) Truth table of p ⊃ q = the truth table of ~(p&~q) (draw them) I shall now write ‘if p then q’ as ‘p→ q’, to make clear that we are not necessarily assuming the material conditional, but arguing that the conditional we use is the material conditional. The argument: If p→q is true then if p is true, q must be true (that is what a conditional says, after all) so you do not have p true and q false, so p&~q is not true, and consequently ~(p&~q) is true. Conversely, if ~(p&~q) is true, consider what happens if p is true. In that case, you don’t have q false, since p&~q is not true (that is what we have assumed: ~(p&~q) is true.) So, if you don’t have q false, q must be true. So, if p is true, so is q. In other words, p→q is true.

The problematic part: – Conversely, if ~(p&~q) is true, consider what happens if p is true. In that case, you don’t have q false, since p&~q is not true (that is what we have assumed: ~(p&~q) is true.) So, if you don’t have q false, q must be true. So, if p is true, so is q. In other words, p→q is true. P: I’m dead Q: I’m alive Then – ~(p&~q) is true – p→q is false

The problem If ~(p&~q) is true, then what happens if p is true. – when p is true is that I’m dead. In this case, you don’t have q false, since p&~q is not true. – This seems wrong: In the situations in which I am dead, we do have q false. The reasoning presumes that we can still call on the truth of our original assumption ~(p&~q) in a situation in which we have assumed that p is true. – If I am dead, it is no longer true that I’m not both dead and not alive.

Introducing necessity □~(p&~q) □: one-place connective of necessity □ is not truth-functional □ is a modal operator □A does not depend merely on the truth value of A Some propositions p might be true, while □p is false (p might be true by accident) Other propositions p might be true with □p true