Jon, List,
Thank you! I need examples. This one is tricky, I have to read it some more times, as now I don´t see the asymmetry. And I don´t understand
"Nevertheless, these are different propositions that signify different states of things, which is reflected by their different existential graphs (see attached from R 490:22).
, because one of them, the one with the "or" should be an entitive graph, due to the "or", or not? But as I said, I must read it again and again, maybe I will understand, what about classical logic is not sufficient.
Best, Helmut
27. Dezember 2020 um 02:22 Uhr
"Jon Alan Schmidt" <jonalanschm...@gmail.com>
"Jon Alan Schmidt" <jonalanschm...@gmail.com>
Helmut, List:
HR: I have tried the reversion including possibility with an example:
Those examples are not appropriate since they include "possibly" within the statements themselves, which makes them modal propositions. Moreover, what Peirce finally recognized in accordance with his pragmatism is that there are real possibilities, which have nothing to do with our knowledge or ignorance.
HR: Is that what intuitionism is about? Negating a weak affirmation, not noticing, that this negation is weak too, and applying nevertheless the NOT-operator?
No, it is simply about omitting excluded middle such that not-(not-X) is not equivalent to X, thus requiring constructive proofs for formal theorems rather than accepting proofs by contradiction (reductio ad absurdum) as valid. In classical logic, given that X is true, not-(not-X) is also necessarily true; and given that X is false, not-X is necessarily true. In intuitionistic logic, given that X is true, not-(not-X) is not necessarily true; and given that X is false, not-X is not necessarily true.
HR: Or can somebody give a better example?
Peirce's own example and explanation of two propositions that are equivalent in classical logic but not so upon acknowledging the reality of some possibilities is a bit lengthy and complicated. It comes right before the passage that I quoted in my previous post (retained below), although CP 4.580 leaves out some of the pertinent text.
CSP: Let us assert that there is a man A and a man B, who may or may not be the same man, and if A becomes bankrupt then B will suicide. Then, if we add that A and B are the same man ... the strange rule to which I refer is that ... it does not affect the interpretation in the least. It seems monstrous to say that these two come to the same thing, that, on the one hand, there is a man, B, who will commit suicide if a certain man, A, not necessarily a different man, becomes bankrupt, and, on the other hand, to say that there is a man who will commit suicide if he becomes bankrupt. But here is the reasoning. A conditional proposition is false only if the condition of it is satisfied, while the consequent is falsified. ,,, But a proposition that is not false is true. So, then, this proposition about A and B will be false only in case, whatever two men or whatever one man, be chosen to be called A and B, A will go bankrupt while B will not suicide. That is, it will be false only in case every man goes bankrupt, and no man suicides. By the same reasoning, the proposition that there is a man who if he goes bankrupt will commit suicide is false only in case, taking any man you please, he will go bankrupt, and will not suicide. That is, it is falsified only if every man goes bankrupt without suiciding. But this is the same as the state of things under which the other proposition is false; namely, that every man goes broke while no man suicides. This reasoning is irrefragable as long as a mere possibility is treated as an absolute nullity. (R 490:22-25, 1906)
In classical logic, the conditional proposition "if X then Y" is evaluated as true unless X is true and Y is false, making it equivalent to both "not-(X and not-Y)" and "not-X or Y." As a result, "if some man goes bankrupt then some man commits suicide" is equivalent to "either some man does not go bankrupt or some man commits suicide." That is why, as Peirce says, "it will be false only in case every man goes bankrupt, and no man suicides." Likewise, "if some man goes bankrupt then he commits suicide" is equivalent to "some man either does not go bankrupt or he commits suicide." Again, "it is falsified only if every man goes bankrupt without suiciding." Nevertheless, these are different propositions that signify different states of things, which is reflected by their different existential graphs (see attached from R 490:22).
As Peirce says while discussing a similar example in a contemporaneous passage, "The equivalence of these two propositions is the absurd result of admitting no reality but existence" (CP 4.546, 1906). Again, the problem is what he calls "the strange rule" of classical logic that produces this outcome; namely, "that every conditional proposition whose antecedent does not happen to be realized is true." In a world where it is really possible for a man to go bankrupt but it just so happens that no man ever actually does so, this alone does not make it true that some such incident would have led another man, or even the same man, to commit suicide. In the latter case, the fact that this particular man never actually goes bankrupt, even while others do so, does not make it true that he would have committed suicide had he done so.
The disconnect comes from neglecting the fundamental asymmetry of inference, which is captured by the unsymmetrical relation of consequence but not by the symmetrical relation of negation. There must be "a real movement of thought in the mind" from antecedent to consequent, since it would be "absurd to say that a real change of A into a sequent B consists in a state of things that should consist in there not being an A without a B. For in such a state of things there would be no change at all" (R 300:49[48], 1908). In other words, the conditional proposition is only true if the reason why some man would commit suicide is because some man--perhaps the same man himself--goes bankrupt. As Frederik Stjernfelt puts it, "There is a link, a 'real possibility' connecting bankruptcy and suicide that is not addressed if you adopt the 'strange rule'" (http://frederikstjernfelt.dk/Peirce/Optimal%20and%20operational%20iconicity.%202011%3A2014.pdf, p. 23).
This also serves as a good illustration of the point about abduction/retroduction that first prompted me to resume this thread a couple of weeks ago (https://list.iupui.edu/sympa/arc/peirce-l/2020-12/msg00004.html). A man committing suicide is a surprising fact that calls for an explanation. Another man going bankrupt seems unlikely to fit the bill except under relatively unusual circumstances, while the same man going bankrupt is a plausible hypothesis from which the surprising fact could be expected to follow as a matter of course. By deduction we ascertain that the man's financial records would reveal his net worth, and by induction we then investigate accordingly. If the evidence indicates solvency, then the hypothesis is falsified; but if the man did go broke, then it is rendered much more plausible--perhaps even probable--though still not certain since there might be other circumstances such that his bankruptcy is a secondary factor or even coincidental.
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Structural Engineer, Synechist Philosopher, Lutheran Christian
On Sat, Dec 26, 2020 at 12:13 PM Helmut Raulien <h.raul...@gmx.de> wrote:
Supplement: The attachment has arrived, with Johns post, sorry.I am afraid, that in my example a weak negation is blended in, so the example doesn´t count. You cannot say "NOT possibly female", but have to say instead "possibly not female". The adverb, I think, indicates the weak negation, which does not include a NOT-operator, I think. The weak negation is the only way to negate a weak affirmation like "possibly is female". The strong affirmation (resp. its negation) "is a possible female" is absurd, it would mean, that the dodo has the choice of its gender, or its gender changes from time to time (resp. NOT).Is that what intuitionism is about? Negating a weak affirmation, not noticing, that this negation is weak too, and applying nevertheless the NOT-operator?Or can somebody give a better example?Jon, List,The attachment has not arrived. I have tried the reversion including possibility with an example:Let a dodo be a animal close to extinction, of which we only know about its extinction status, that maybe all still living dodos are male, maybe females still exist too. We have a blurred photograph (sex not visible) of a dodo:"If this is a dodo, THEN it is possibly female."-- is still correct, because it merely is a possibility, that the last ones are all male.Reversion:"If it is NOT possibly female, THEN this is NOT a dodo"-- is false.BUT: If we do not subsume "not-knowing" under "possibility", as in fact it is something quite different from the possibility of something being one case of existing cases, then the reversion works:"If this is a dodo, THEN we donot know, if it possibly is female"-- is correctReversion:"If it is NOT so, that we donot know, if it possibly is female, THEN this is NOT a dodo"-- is correct, though in this case not helpful.Has anybody got other examples?Best, HelmutCSP: I soon discovered, upon a critical analysis, that it was absolutely necessary to insist upon and bring to the front, the truth that a mere possibility may be quite real. That admitted, it can no longer be granted that every conditional proposition whose antecedent does not happen to be realized is true ...[T]he verso of the sheet of Existential Graphs represents a universe of possibilities. This, taken in connection with other premisses, led me back to the same conclusion to which my studies of Pragmatism had already brought me, the reality of some possibilities. This is a striking proof of the superiority of the System of Existential Graphs to either of my algebras of logic. For in both of them the incongruity of this strange rule is completely hidden behind the superfluous machinery which is introduced in order to give an appearance of symmetry to logical law, and in order to facilitate the working of these algebras considered as reasoning machines. I cannot let this remark pass without protesting, however, that in the construction of no algebra was the idea of making a calculus which would turn out conclusions by a regular routine other than a very secondary purpose. (CP 4.580-581, 1906)
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