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Apologies ahead of time - I'm totally confused.

I'm trying to understand the difference between the Boolean truth function "->" and some sort of higher level "entailment" function "=>" (which I don't understand).

I understand the table for the "A -> B" conditional which you can evaluate when A and B are assigned (bound??? see below) Boolean values.

Let's try to define (again, apologies, because I'm out of my league here) a "function" $I(A, B)$ which takes as inputs two Booleans A and B, and returns the value (A -> B).

My understanding is that we say $$\mathscr{A} \implies \mathscr{B}$$ exactly when this vague condition holds:

"The conditional $\mathscr{A} \to\mathscr{B}$ is true if and only if $I(A, B)$ evaluates to true for all bindings of free variables involved" in the the left and right halves of the conditional, where they're bound in the same way.

This is the best definition that I've come up with, but I do not believe that it really makes sense.

I've also seen it phrased this way: "$\mathscr{A} \implies \mathscr{B}$ exactly when $\mathscr{A} \to \mathscr{B}$ is a tautology.". What does that mean? I know a tautology is a sentence which is always true (in some domain of discourse?)?

Question: How do I go from here to a better understanding, and also - how far off am I from some reasonable definition.

1 Answers1

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The '$\rightarrow$' has a very narrow meaning which can be captured by the following definition:

$$(\phi \rightarrow \psi) =_{df} (\lnot \phi \lor \psi)$$

That's basically all there is to the material conditional. The metalinguistic '$\Rightarrow$' is the 'if...then' of informal, but still rigorous, mathematical reasoning. The semantic entailment relation '$\models$' has different meanings according to different people. Tarski, for example, defined it as:

$$(\Gamma \models \phi) =_{df} \rm{all~models~of~\Gamma~are~models~of~\phi}$$

This is more or less the orthodox account. There are, however, logics (e.g. intuitionistic, relevantist, non-monotonic, paraconsistent, modal, and so on) that give different explications of the relation.

Hunan Rostomyan
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  • Thank you very much. I'm left with the uneasy feeling that "logical entailment"/"semantic entailment", in order to be defined properly, requires you (or me ;) ) to be working in some BIGGER logical system. That's weird, because I only want to do $\delta -\epsilon$ calculus proofs. –  Nov 01 '13 at 05:56
  • You're welcome. I'm glad you found it helpful. – Hunan Rostomyan Nov 01 '13 at 05:59
  • Again - thanks - but I'm not satisfied. It's the fancier meaning that we use when we're proving things like "functions which are differentiable on an open interval are continuous on that interval". –  Nov 01 '13 at 06:00
  • As regards your first comment. Yes, that follows from Tarski's undefinability theorem. – Hunan Rostomyan Nov 01 '13 at 06:04
  • Again - apologies for my confusion - I am asking ostensibly simple questions about logic, and you provide fancy Theorems? There should be an easier answer... –  Nov 01 '13 at 06:10
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    No need to apologize. I don't think you need to settle these questions in order to do calculus, but since you're interested, I think you should dig into the basics of model theory. Most mathematical logic texts introduce the elements, but if you're not able to find a suitable book, google "Moschovakis' Logic Notes" and go through chapters 1-2. I think once you take model theory for granted, logical consequence will not puzzle you as much anymore. – Hunan Rostomyan Nov 01 '13 at 06:18
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    Thanks. I appreciate it - now I have to assume my Calculus lives in this "model". –  Nov 01 '13 at 06:24
  • haha! I'm glad you're checking the notes out. I need to sleep now, but I think you should skip the examples of structures for now and go straight to the syntax (1B) and semantics of first-order languages (1C). – Hunan Rostomyan Nov 01 '13 at 07:09