- #1
cianfa72
- 2,566
- 274
- TL;DR Summary
- logical implication makes sense only within specific context/models.
I'm still confused about the use of material implication (material conditional) ##\to## vs logical implication ##\implies##.
From MSE the writing ##P \implies Q## makes the meta-logical assertion/statement that the logical statement ##Q## is logically implied by the logical statement ##P##.
First of all: saying for instance "logical statement ##Q##...." actually implicitly asserts that it is True, I guess.
Second: the meta-logical assertion as above should only make sense/apply within a specific context/model/interpretation. Consider for instance: $$x \gt 6 \implies x \gt 2$$ This logical implication actually makes sense within the context/model/interpretation of mathematical theory of natural numbers, however it makes no sense in other contexts in which there is no notion of ordering (how should one interpreter the symbol ##\gt## in such a case ?).
Therefore from the above logical implication follows (from a meta-logical theorem) that the material implication/conditional ##P \to Q##: "If ##x \gt 6## then ##x \gt 2##"
is true (i.e. it is a tautology) only within the specific context in which the logical implication applies.
What do you think about? Thanks.
Ps. note that I tend to use the english term "implication" for the logical implication ##\implies## and the form "If...then" for the material implication/conditional ##\to##.
From MSE the writing ##P \implies Q## makes the meta-logical assertion/statement that the logical statement ##Q## is logically implied by the logical statement ##P##.
First of all: saying for instance "logical statement ##Q##...." actually implicitly asserts that it is True, I guess.
Second: the meta-logical assertion as above should only make sense/apply within a specific context/model/interpretation. Consider for instance: $$x \gt 6 \implies x \gt 2$$ This logical implication actually makes sense within the context/model/interpretation of mathematical theory of natural numbers, however it makes no sense in other contexts in which there is no notion of ordering (how should one interpreter the symbol ##\gt## in such a case ?).
Therefore from the above logical implication follows (from a meta-logical theorem) that the material implication/conditional ##P \to Q##: "If ##x \gt 6## then ##x \gt 2##"
is true (i.e. it is a tautology) only within the specific context in which the logical implication applies.
What do you think about? Thanks.
Ps. note that I tend to use the english term "implication" for the logical implication ##\implies## and the form "If...then" for the material implication/conditional ##\to##.
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