Helmut, List: HR: I think, Jon A.S., you once gave me the following example: "Every unicorn is pink" is false, but "There is no unicorn that is not pink" is true.
Here is what I actually said a few months ago. JAS (https://list.iupui.edu/sympa/arc/peirce-l/2021-01/msg00075.html): For example, it is true that "there is not a unicorn that is not pink" because there are no actual unicorns, but we cannot infer from this that "every unicorn is pink." On the other hand, since by definition "every unicorn has a single horn," it does follow that "there is not a unicorn that does not have a single horn." The issue is not so much true vs. false as whether one proposition is a *valid inference* from the other. In classical logic, "there is not a unicorn that is not pink" is *equivalent *to "every unicorn is pink," so we can infer either one from the other. In intuitionistic logic, the former can be derived from the latter, but not vice-versa. HR: "Every unicorn is pink" is false, because it means "If it is a unicorn, then it is pink", and "If it is a unicorn" implies, that unicorns exist. No, "If it is a unicorn" *does not* imply the existence of unicorns, which is why this kind of proposition is sometimes called *hypothetical*. A more accurate statement of it is in the subjunctive mood, "If something *were *a unicorn, then it *would *be pink." This is false, not because no unicorns exist, but because being pink is not *essential* to being a unicorn. By contrast, having a single horn *is *essential to being a unicorn, so "Every unicorn has a single horn" and "There is not a unicorn that does not have a single horn" are both true--setting aside obvious exceptions such as a unicorn whose single horn has been removed. Nevertheless, intuitionistic logic still rejects the *inference *of the former from the latter, while allowing the *inference *of the latter from the former. HR: Is in classical logic "There is no unicorn that is not pink" equal with "Unicorns exist, and there is no unicorn that is not pink"? No, these are distinct propositions. HR: But this would mean, that the term "existential" in "Existential Graphs" means, that only existing things are allowed in them. Only things that exist *in the universe of discourse* are allowed in EG. So far we have been talking about the universe of *actual *existence, which contains no unicorns. If we were instead talking about the unicorns in an animated Disney movie, then those unicorns *would *exist in that universe--and "Every unicorn is pink" might be true in that universe! Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Wed, May 19, 2021 at 4:52 PM Helmut Raulien <h.raul...@gmx.de> wrote: > List, > > Does anybody know an example which justifies intuitionistic logic, so in > which classical logic fails? I think Jon, A.S., you once gave me the > following example: > > "Every unicorn is pink" is false, but "There is no unicorn that is not > pink" is true. > > "Every unicorn is pink" is false, because it means "If it is a unicorn, > then it is pink", and "If it is a unicorn" implies, that unicorns exist. So > it is equal with "Unicorns exist, and if it is a unicorn, it is pink". > Because unicorns donot exist, the proposition is false. > > "There is no unicorn that is not pink" sounds true, because there are no > unicorns at all, so there are no non-pink unicorns too. But if it would be > so, that this form of proposition too implied the existence-claim, it would > be false as well. Is that so? Is in classical logic "There is no unicorn > that is not pink" equal with "Unicorns exist, and there is no unicorn that > is not pink"? > > This might be so e.g. due to the fact alone, that the term "Unicorn" has > been mentioned. For EGs, it would mean, that every term written in any > place is a possible too in the blank sheet. Meaning, that it generally > exists. Otherwise it would not signify anything, it would e.g. be like "NOT > &/(", senseless. But this would mean, that the term "existential" in > "Existential Graphs" means, that only existing things are allowed in them. > > Another way to classically synchronize the two propositions might be to > say, that if a term signifies a nonexistent thing, it automatically > signifies its phantasy-concept instead. Then "Every unicorn is pink" is > false, because in some animated movie by Disney occurs a white unicorn. > "There is no unicorn that is not pink" then is false for the same reason. > This explanation is somewhat smoother than the first, but requires this > said automatism: If A does not physically exist, then A is the existing > concept of A. > > Best > Helmut >
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