FOL: What does the axiom schema A ⊃ (B ⊃ A) actually mean?

Asked Dec 22 '22 at 17:31

Active Dec 22 '22 at 17:31

Viewed 44 times

Of course, A ⊃ (B ⊃ A) works out to be true via truth function/table for material conditionals.

Does this mean that when you're doing proofs, if you had P written down as a true statement, then you could write down Q ⊃ P as a consequence?

And would that mean then P ENTAILS Q ⊃ P since (I think) a proof means logical consequence?

On one hand, this seems to fit the definition of entailment/logical consequence that says the premises can't be true while conclusion is false. (similar to the truth table for the material conditional)

On the other hand, that'd also mean that any true statement A ENTAILS that any statement B ⊃ A, regardless of any "relevance" between A and B... So, would the argument, "1 + 1 = 2, THEREFORE if triangles have 3 sides, then 1 + 1 = 2" be valid?

asked Dec 22 '22 at 17:31

csp

Here's a MathJax tutorial :) – Shaun Dec 22 '22 at 17:42
The only way $A\supset (B\supset A)$ is false is if $A$ is true while $B\supset A$ is false, which, in turn, is only false if $B$ is true while $A$ is false, a contradiction. – Shaun Dec 22 '22 at 17:47
Are the statements in the paragraphs correct? – csp Dec 22 '22 at 17:49
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It means that in a proof you can always add an unnecessary premise. See Weakening. – Mauro ALLEGRANZA Dec 22 '22 at 18:18
So does a true statement A entail B ⊃ A? – csp Dec 22 '22 at 18:37
Yup, thats what the axiom says – IsAdisplayName Dec 22 '22 at 19:39
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Regarding "relevance" between A and B, see Relevance Logic. – Mauro ALLEGRANZA Dec 23 '22 at 10:42

FOL: What does the axiom schema A ⊃ (B ⊃ A) actually mean?

0 Answers0