tag:blogger.com,1999:blog-2631035637795172582.post8594610788222179275..comments2023-01-24T10:06:57.212-08:00Comments on (Blog&~Blog): What The Explosion Proof Isn'tBenhttp://www.blogger.com/profile/06702722560438833244noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-2631035637795172582.post-14181182749068964292009-11-08T22:01:23.763-08:002009-11-08T22:01:23.763-08:00Joel,
Thanks for the correction. Fixed it.
As fa...Joel,<br /><br />Thanks for the correction. Fixed it.<br /><br />As far as Modus Tollens, contraposition, etc., I'm not sure I see your point on that just yet. After all, in, e.g. Graham Priest's logic LP, those things *are* considered invalid, for the same reason.Benhttps://www.blogger.com/profile/06702722560438833244noreply@blogger.comtag:blogger.com,1999:blog-2631035637795172582.post-83481328939206071922009-11-08T21:59:23.231-08:002009-11-08T21:59:23.231-08:00Ben
"...once you seriously arrive at a contr...Ben<br /><br />"...once you seriously arrive at a contradiction, you must either reject some part of the offending argument, or reject logic itself."<br /><br />Why reject logic itself, as opposed to just rejecting classical logic?<br /><br />On the face of it, since (counterpossibly) if there were any true contradictions, then Disjunctive Syllogism wouldn't be truth-preserving, then abandoning classica logic in favor of paraconsistent logic seems like exactly the right move.<br /><br />The Hitler point is that "if Hitler won, B" is true for any B *on the assumption that Hitler didn't win*, but if we throw out that assumption by having "Hitler won the war" as the second premise, so we can derive triviality from Modus Ponens, we've illegitimately switched positions on Hitler winning halfway through the argument. We haven't shown, in any interesting sense, that "from Hitler winning, anything follows."<br /><br />By analogy, in the contradiction case, we've equally switched positions midway through the argument if we affirm the truth of a contradiction in the first premise, but then treat a rule that wouldn't be valid if contradictions were ever true as a valid rule of inference when reasoning about that contradiction, and we've equally failed to show that "from a contradiction, anything follows" in any particularly interesting sense.Benhttps://www.blogger.com/profile/06702722560438833244noreply@blogger.comtag:blogger.com,1999:blog-2631035637795172582.post-50487953796555245632009-11-07T13:49:51.109-08:002009-11-07T13:49:51.109-08:00You say "The falsehood of β is alone sufficie...You say "The falsehood of β is alone sufficient to guarantee the truth of that conditional in classical logic." This is not correct. The falsehood of (α ∧ ¬α) would be sufficient, but this doesn't make your point since we can't simply assume it is false in this case.<br /><br />I happen to think that the argument in question actually shows quite a lot. For example, it shows that there is an argument for the explosion principle that relies on apparently quite weak premises and not necessarily on something "purely definitional" like a model-based definition of entailment. <br /><br />You also point out that if there are any true contradictions, then assuming conjunction elimination is truth preserving, disjunctive syllogism isn't. I agree this is very close to begging the question, but it does show that explosion isn't just one separable thing and you can take it or leave it. Upon reflection, this argument easily transforms into others that show that modus tollens, contraposition, etc. are also invalid.<br /><br />Joel Velasco (can't seem to change the "Bob" name - not sure where that comes from)Anonymoushttps://www.blogger.com/profile/14122534020804478423noreply@blogger.comtag:blogger.com,1999:blog-2631035637795172582.post-67885797026606418362009-11-07T08:10:28.795-08:002009-11-07T08:10:28.795-08:00"Or, more weakly than this counterpossible cl..."Or, more weakly than this counterpossible claim, it could be seen as showing that anyone who believed that some such statement was true would be rationally compelled to believe that absolutely everything is true."<br /><br />I think a more direct informal way of making sense of it is by saying that once you seriously arrive at a contradiction, you must either reject some part of the offending argument, or reject logic itself. An explosion is the use of logic by rote after you've abandoned it for all meaningful purposes.<br /><br />"The obvious objection is that we have switched positions midway through the argument on the subject of whether Hitler won World War II. Quite so."<br /><br />I don't understand. You're confirming the antecedent in this example, not denying it, so you're not illustrating the point you wanted to make. If Hitler won, then β follows without event. Did you mean to say "Assume Hitler did not win"? Even so, though the whole if-then statement would be T, the consequent could still be either T or F, which is not a contradiction.<br /><br />But the next paragraph indicates that maybe you meant to say that we are putting an obvious contradiction in our premises, which is akin to changing our minds. If so, then a classical logician might just make the informal remark that we're not doing logic. It seems to me that the law of identity, etc., are not just formal tools, they're also informal demarcation criteria that tell us when we're just putting empty words to paper for purposes of personal amusement.BLS Nelsonhttps://www.blogger.com/profile/09221793753245953967noreply@blogger.com