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Can you think of any possibility of logical deductive argument without the use of logical symbols (negation, material conditional, all, some, etc.), or equality?

I was at first thinking perhaps the "element of" or "subset of" relations could be used for a deductive entailment, but it would still require axioms of set theory which require logical symbols.

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    of course, although the answer is trivial. for any prop letter, P implies P
    – emesupap
    Commented Jun 26, 2023 at 14:42
  • Do you mean without any truth functional operators at all or just without the usual ones typically used in logic or maybe a set that cannot be used to define the ones usually used in logic? Commented Jun 26, 2023 at 16:14
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    You could have entailment relations that are not formal. E.g. John has a brother, therefore John has a sibling. Some logicians call this 'material validity' or 'material consequence' and distinguish it from formal consequence. Others would treat it as enthymematic with the hidden premise that anyone who has a brother has a sibling, and others would regard it as an example of analytic consequence.
    – Bumble
    Commented Jun 26, 2023 at 16:23
  • So the answer is YES (see first comment above) but the conclusion is: without "logical terms" no logic at all. Commented Jun 28, 2023 at 7:22

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Can you think of any possibility of logical deductive argument without the use of logical symbols (negation, material conditional, all, some, etc.), or equality?

Arguments are linguistic in nature and therefore formal and so we use whatever formal logical operators as part of the language we happen to know. It seems all natural languages rely on the same set of logical operators--which, by the way, does not include the so-called material "implication", which by the way is not even an implication, since there is just one implication, and it is not "material".

There are 16 basic potential logic operations, and we only use the negation, the conjunction, the disjunction, something akin to the exclusive disjunction but not quite, logical equality (i.e. both true or both false), and a few more exotic things but we don't use all 16 of them (in particular, we don't use the horseshoe or its variants), which suggests the logical operators we use reflect some basic cognitive processes of the human brain, somewhat like Boolean logic reflects the electromagnetic processes taking place in computers. This also means that logic is not a formal discipline. Logic is the logic of these basic cognitive processes in our brain, and we cannot pick and choose what they are. They are the result of natural selection and evolution of species, and we just inherit our logic through our DNA.

Would it be possible to do otherwise? Probably, yes, but we would still need logical operators. They are necessary. Logic is fundamentally analytical, which requires to combine ideas, which requires logical connections between ideas, which requires logical operators to compute what follows from what. We can implement the logical operators we don't normally use on computers, but we still won't use them ourselves when we think, because the logic of our thinking is predefined by our DNA.

I was at first thinking perhaps the "element of" or "subset of" relations could be used for a deductive entailment, but it would still require axioms of set theory which require logical symbols.

We can always define new concepts and their logical relations. This is what mathematics does and this is what we do when we use a natural language, but this cannot be any substitute for the basic logical operations we know. We can always define analytical concepts like "man", "unmarried" and "bachelor", or like x ∈ S₁ and y ∈ S₂, but we still need logical operators to express our assumptions about the connection between the truth values of "man", "unmarried" and "bachelor", or between the truth values of x ∈ S₁ and y ∈ S₂.

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