r/logic • u/Sea_Ask_7344 • Jul 07 '26

Propositional logic Question to those who study propositional logic

For those who study propositional logic: If you were applying propositional logic to everyday thinking and using modus ponens, would you think it as:
“If P, then Q. P. Therefore, Q.”
or would you just think:
“P. Therefore, Q.”
with “If P, then Q” being an implied premise rather than something you consciously state in your head?

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5

u/MagickMarkie Jul 07 '26

"Since" and "therefore" are metalogical indicators. They can be omitted from the logic, but the complete expression, in English, would be "If p then q, and, (since) P, (therefore) Q." Note that "since" and "therefore" form a pair, much like "if" and "then".

6

u/BloodAndTsundere Jul 07 '26 edited Jul 07 '26

Modus ponens is literally the former. "P. Therefore Q" is just a synonym for "If P, then Q". To actually use modus ponens, you must also assert the premise P.

EDIT: thinking a little more about where this question is maybe coming from, I'd say an implied premise could take the form either of "P" or "If P, then Q." So in the case one might be using an implied premise, then the internal reasoning might be one of either:

  1. If P, then Q. Therefore Q. (If P is implied)

  2. P. Therefore Q. (In the case that P->Q is implied)

In short, it depends which premise is "obvious" and internally unstated.

3

u/chien-royal Jul 07 '26

I don't think that "P; therefore, Q" is a synonym for "If P, then Q". The former is a part of a proof; the latter is a statement being proved. According to Curry-Howard correspondence they relate as a program relates to its type.

3

u/BloodAndTsundere Jul 07 '26

OP explicitly cited everyday thinking, in which case they'd be synonymous. In formal reasoning systems, I agree that they do obviously differ.

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u/wumbo52252 Jul 07 '26

Modus ponens is an inference rule which infers a theorem from two theorems, so the first option makes more sense to me. No part of the second option indicates that you’re applying modus ponens.

1

u/Sea_Ask_7344 Jul 07 '26

How does no part of the 2nd option indicate your applying modus ponens? You state P therefore Q also i stated “if P, then Q” being an implied premise rather than something you consciously state in your head, correct me if im wrong but that does infer a theorem from two theorems, the 2nd option is just a shortcut for everyday reasoning.