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I am struggling to understand the basics of logic. I don't understand the following.

I know that in propositional logic the statement "A and (not B)" is true under certain interpretations of the symbols and false under others. It is also unprovable since it is not a tautology.

This seems to me to be analogous to true unprovable statements of number theory. These are also true only in some models becasue if they were true in all models they would be provable by the completeness theorem.

Why don't we consider propositional logic incomplete, then?

amWhy
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Adam
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    The point of been complete is not that every statement can be proved, but that every true statement can be proved. – jjagmath Nov 09 '24 at 10:48
  • But if A is true and B is false then the statement is true. – Adam Nov 09 '24 at 11:05
  • Compare with the number theory statement $n>2$. It is true under certain interpretations of $n$. – user8268 Nov 09 '24 at 11:20
  • Thanks guys, I think I am starting to get it. – Adam Nov 09 '24 at 11:49
  • You said "But if A is true and B is false then the statement is true." which means that under the hypothesis of A been true and B been false, we have A and (not B) is true. In symbols $(A \wedge \lnot B) \implies (A \wedge \lnot B)$ which is indeed always true. – jjagmath Nov 09 '24 at 12:11
  • Focus on validity. $A\to B$, with A as true and B as false is false, @Adam. – amWhy Nov 09 '24 at 20:15
  • Voting to reopen because the duplicate target isn't a genuine duplicate, whereas Bram's answer below actually addresses the OP's question. – ryang Nov 10 '24 at 02:35

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There is a big difference between mathematical truth and logical truth.

A logical truth is something like $A \lor \neg A$: a statement that is true in all possible worlds. The statement $A \land \neg B$ may be true in some worlds, but it is not a logical truth, since it is not true in all possible worlds.

The statement $1+1=2$ is also not a logical truth: it is true in the world of numbers, but it is not true in all possible worlds.

The notion of ‘completeness’ is likewise different depending on whether we talk about logic or some specific mathematical domain like number theory.

The completeness of propositional logic is that every logical truth can be proven, not that all things that are true for some particular world can be proven.

But it is the latter that we would mean by the completeness of number theory: that all statements that are true in the world of numbers can be proven.

Bram28
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  • To elaborate: neither the completeness of number theory nor the completeness of propositional logic implies the other. And this answer explains a bit more the difference between logical truth and mathematical truth. – ryang Nov 10 '24 at 06:02