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I am a mathematics major student and interested in logic. I have some questions, in math(both pure and applied aspects) study and research, do we clearly distinguish between logical equivalence and material equivalence as what we do in the pure logic study, or do we only discuss and require material equivalence (and roughly think that the two equivalences are the same meaning), because I find that in my daily math study we only discuss and require the material equivalence while ignoring logical equivalence. I want to know the answer and why, I am very grateful!

Guang Yang
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    Perhaps give examples of what you mean by "material equivalence" and by "logical equivalence"? For example, is my description here consistent with your understanding, or not quite? – ryang Mar 14 '22 at 11:57
  • How good do you know mathematical logic and, for example , Goedel's incompleteness theorems ? You should know that they do not apply in pure/applied math , let alone as a decision of abstract questions like whether God exists (Trials to prove God with Goedel were actually made). To model the reality, we have to make assumptions anyway and the math we finally need is secure of unexpected contradictions or incompletenesses. – Peter Mar 14 '22 at 12:18
  • But usually, mathematicians asserts that when they prove a theorem in a theory T, this theorem is a consequence of the axioms of the theory. Thus, "logical" and not "material" seems involved here. – Mauro ALLEGRANZA Mar 14 '22 at 14:08

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Two observations from the propositional perspective:

  • If $A$ and $B$ are already established propositions and their connection is already well understood, one usually does not have to view this as a deduction each time, $A\vdash B$, and may carry on with material implication $A\rightarrow B$ instead.
  • More generally, the deduction theorem allows one to pass from one view of matters to another (from the logical point of view): $$\Gamma,A\vdash B\Longleftrightarrow\Gamma\vdash A\rightarrow B$$

The same goes for material equivalence, $\leftrightarrow$, and logical equivalence, $\dashv\vdash$.

It should be remarked that logic is not a description of working mathematicians' cognitive processes, nor of communicative practices. Such logical distinctions exhibit themselves when mathematical arguments are needed on some purpose to be regimented and rearranged into a logical form.

Tankut Beygu
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