I am curious about vacuous truth in logic. I searched Google for various resources, including here. What I could learn from the paper was that I could not know the history or background of vacuous truth, but I could see that vacuous truth was developed by philosophers and mathematicians such as Quine. However, I am not sure whether the content in question is about mathematical vacuous truth, and I would like to know about the history, background, and origin of the beginning of mathematical vacuous truth.
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$\begingroup$ I'm not sure that we can find the "first usage" of this term... $\endgroup$– Mauro ALLEGRANZACommented Apr 10 at 7:09
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$\begingroup$ The "origin" of the concept are clearly due to the truth table for material conditiona. $\endgroup$– Mauro ALLEGRANZACommented Apr 10 at 7:24
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$\begingroup$ @MauroALLEGRANZA Hello, thank you for your comment. But is it true that the Quine in the file I posted in the question above is related to vacuous truth in a mathematical sense? $\endgroup$– user1274233Commented Apr 10 at 8:25
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$\begingroup$ I think not; see page 4; when the Author says that "statements that spell out meaning relations [...] are true purely in virtue of their meaning: they are vacuously or degenerately true", he is referring to statement like "all bachelors are unmarried" that are true because thay are expressing a definition. $\endgroup$– Mauro ALLEGRANZACommented Apr 10 at 8:38
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1$\begingroup$ For an early occurrence of "vacuous truth" in the sense of universal conditional with empty antecedent, see e.g. Roderick M. Chisholm, The Contrary-to-Fact Conditional (Mind, 1946), page 298: "the paradoxes of 'vacuous truth' which we have considered in the cases of the material and universal conditionals". $\endgroup$– Mauro ALLEGRANZACommented Apr 10 at 8:39
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I think that the content on p.66 of this book can be a fundamental starting point, even if it is not mathematical, for vacuous truth.
Kant produced an elaborate transcendental idealism to solve his own problem for mathematics and for purportedly apriori cognition in the natural sciences. Kant's theory is philosophically profound and a continuing source of philosophical insight. But, quite apart from its appeal to a putative capacity for pure intuition, its resort to the view that spatial and temporal structures are at bottom mind-dependent seems to me to disqualify it from serious candi- all in dacy They any The for genuine held way being that positivists made pure true. knowledge true responded by, of mathematics a and to are subject the and not logic true matter problem are of, a by must only subject be trying to vacuously guided matter. protect and true They and justified held empiricism. are that not by perceptual-causal relations to the subject matter. Kant anticipated this view by holding that logic, though not mathematics, is vacuously true. These appeals to vacuity were, I think, decisively overthrown by Quine in the middle of the twentieth century
As above, I believe that Kant's thoughts are the starting point of vacuous truth.
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1$\begingroup$ But this is not the vacuous truth of universally quantified conditional statement. This refers to the same meaning referenced in the OP's post: "statements that spell out meaning relations [...] are true purely in virtue of their meaning: they are vacuously or degenerately true". $\endgroup$ Commented Apr 12 at 10:32
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$\begingroup$ @MauroALLEGRANZA Thank you for your comment. I was saying, didn't the ideological starting point of vacuous truth begin with Kant? $\endgroup$ Commented Apr 12 at 11:04
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$\begingroup$ @MauroALLEGRANZA I am writing a comment because I am not sure. Where did you look at the part I quoted in the book and how did you judge that it was not a universally quantified conditional statement? $\endgroup$ Commented Apr 12 at 18:35