Wikipedia:Reference desk/Archives/Mathematics/2024 March 18

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March 18

Tautology logical reasoning

Does every tautology have a verbal logical reasoning not requiring any use of truth tables or logical connectives, just like combinatorial identities/formulae (at least many) have combinatorial reasonings?
For example, this tautology here: ${\bigl (}a\to (b\to c){\bigr )}\to {\bigl (}b\to (a\to c){\bigr )}$. יהודה שמחה ולדמן (talk) 18:44, 18 March 2024 (UTC)

Well one can always read out the maths! But it would become very tedious and you might lose attention for anything much more complicated than the above. That though can be transformed by saying: A implies B means A is false or B is true. Then A implies B implies C means the same as A is false or B is false or C is true and that is the same as B is false or A is false or C is true. I mean what are you looking for beyond that? To start using the ancient business of syllogisms with names like Baroco? NadVolum (talk) 19:30, 18 March 2024 (UTC)

I mean literally, how do we deduce with logical reasoning that the initial part $a\to (b\to c)$ implies the final part $b\to (a\to c)$?

Of course I understand that we use symbols to express arguments, such as $a\to b=(\neg a\land \neg b)\oplus (\neg a\land b)\oplus (a\land b)=\neg a\lor b$.

But I am also asking if this is a definition apriori (${\stackrel {\rm {def}}{=}},{\stackrel {\triangle }{=}}$) or a theorem. יהודה שמחה ולדמן (talk) 20:53, 18 March 2024 (UTC)

If I tell you that $(a\,\natural \,(b\,\natural \,c))\,\natural \,(b\,\natural \,(a\,\natural \,c))$ is a tautology without supplying an interpretation of the connective "$\natural$" and ask you to provide verbal reasoning for this tautology, you will not be able to make any inroads. You can't do anything if you don't know the meaning of "$\natural$". So it is not clear to me what kind of "verbal reasoning" might correspond to this tautology without referencing the interpretation of the material conditional connective "$\rightarrow$". To me, this would be like requiring a verbal proof of $a+b=b+a$ without any use of addition. The tautology can be verbalized: "If we know that if A is true, then if B is true then C is true, it follows that if B is true, then if A is true then C is true." If you can wrap your head around it, this should be self-evident, but a proof by natural deduction of the tautology can also be verbalized. ("Assume we know that if A is true, then if B is true then C is true. We want to show that it follows that if B is true, then if A is true then C is true. So assume that B is true. ...") However, unless one has an uncanny ability, one will get lost in keeping track of which assumptions hold at various steps and which have been discharged.  --Lambiam 08:49, 19 March 2024 (UTC)