I do not see a way around making some kind of distinction here. Either only mathematics (and logic) gets knowledge and deduction ― and everything else gets rational belief and probability ― or there are two kinds of knowledge, and two kinds of deduction. Pick your poison.
Mathematical knowledge and empirical knowledge differ so greatly they barely deserve the same name. Obviously the history of philosophy includes almost every conceivable way of either affirming or denying that claim. — Srap Tasmaner
I am not sure what you mean by saying "If I am American then I am the President" is true in propositional logic. — NotAristotle
And yet here we are. Turns out folk do speak like that.But no one speaks like that and no one would make such an inference. — NotAristotle
I would prefer there not to be equivocal definitions of validity, it appears that there are, one formal, the other informal. — NotAristotle
Okay, but I can actually see how the edited conditional could be true. For instance, if Michael is a really great citizen, then maybe he would end up being President were he American, if so, then in the ordinary sense, the sentence can be "true" based on what it means. — NotAristotle
If I uttered: "If it is raining then it is not raining." ... If formal logic is "mappable" onto ordinary language, then you should be able to infer "oh okay, it's not raining." But no one speaks like that and no one would make such an inference. At least, no one would consider such an "argument" "valid." That being so, while I would prefer there not to be equivocal definitions of validity, it appears that there are, one formal, the other informal. — NotAristotle
One isn't inferring Not A in such cases — sime
What is the edited conditional? — TonesInDeepFreeze
So you would say that a reductio ad absurdum is not an inference in the proper sense? — Leontiskos
What would be the implications if we would say for any given argument under all values of the antecedent the conclusion may not result in a logical contradiction or the argument will be deemed invalid? — Benkei
Still, it also appears that the conclusion is an unwarranted logical leap from the premises, so that is why I think there might be room to argue that the argument is not valid according to some informal definition of logical validity. That is to say, the conclusion doesn't follow or doesn't lead to the conclusion. I understand that this is not the definition of validity formally speaking. — NotAristotle
I am referring to the "it is raining" example; the conclusion in that argument appears to be a logical leap. I get that the argument is formally valid, that's the entire point - while formally valid, the conclusion does not appear to "follow." — NotAristotle
Michael, the argument is simply this:
If it is raining then it is not raining.
Therefore, it is not raining.
Who in there right mind would conclude the conclusion from the premises in a conversational setting? — NotAristotle
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