Speakpigeon said:
Sure. I could have said most of this.
Please explain to AM. Explain in particular that the various definitions of validity used in mathematical logic have absolutely zero value and zero authority when it comes to any human being considering or discussing the validity of logical arguments. This is a point he still doesn't get.
Stop grossly misrepresenting my position.
First, I never said or suggested that the various definitions of validity used in mathematical logic had value when it comes to the colloquial meanings of 'valid'. In fact, I said that in natural languages, like English, usually people do not need a fine-grained distinction that separates arguments with false premises with arguments in which the conclusion does not follow from the premises, so a colloquial use of "invalid" may include arguments of the former kind.
Second, I was never discussing the different usages of 'valid' in different logics in mathematics. Rather, I was talking about
the specific definition that you called "the definition of logical validity used in mathematical logic since the beginning of the 20th century" (not "a", but "the"), and which is
as follows:
Validity
A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.
Internet Encyclopedia of Philosophy -
https://www.iep.utm.edu/val-snd/
Note that this definition is
not limited to mathematics. It's a standard definition used in philosophy (indeed, you quoted from philosophy Encyclopedia), and it is in that context used in the study of human logic. I would say it as a proper definition of an argument in which the conclusion follows from the premises (whether it is called it 'valid' or not), intuitively. But you have different intuitions. It does not matter, because that is not one of the points I intended to make.
Third, I did not claim or suggested that the definition in question matched common usage. But I showed that
under assumptions you made, that definition
provided our best method for finding mathematical truth. I showed that in several posts, for example
this one.
Fourth, I showed that under your own assumptions, the rational assessment is that
human logic is a disaster, as it fails to preserve truth, so we should ditch it and choose another, better definition of validity, such as what you call "the" definition of mathematical logic. Of course, I am not suggesting either that the definition in question fails to match human logic, or that human logic fails to preserve truth. I just
debunked your position once again
Fifth, I did further debunking on your position, showing its untenability in several threads, such as
this one,
this one,
this one,
Speakpigeon said:
Further, there is the question of the use of the theories developed in mathematical logic. Mathematicians are human beings and they reason logically when they want to, and presumably they have to to prove theorems. As human beings, their logic must be the same as that of any other human being. So, either mathematical logic has no role whatsoever in how mathematicians prove theorems or it does. If no role, mathematical logic is literally useless, which would contradict what you say here. So, I will assume that you think it has a role. Now, i asked several times, without result, AM to provide examples of important mathematical theorems whose demonstration relied on mathematical logic. Can you yourself provide examples?
First, again by "mathematical logic", you meant what you defined as such, namely the definition of validity that says "A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false."
I already told you:
that is our intuitive sense of logic. We reason that way all the time. But I went further. I
posted the relevant link to the SEP article in which it is explained that "Unfortunately, as Meyer and Friedman have shown, relevant arithmetic does not contain all of the theorems of classical Peano arithmetic." (I also provided more information about what happens when you limit logic in different ways).
Second, you misunderstand Aristotelian logic, and as a result, you say things follow in Aristotelian logic when they do not, which makes it hard to give examples, since you can simply say that that too follows in Aristotelian logic. In fact, you contradict of course your own position. For example, you
claim
One derives a contradiction.
The right word is indeed "inconsistent". The premises are inconsistent, i.e. one premise implies the negation of the other premise.
Contradictory premises would be p and not p and that's not what we have here.
The premises here are not contradictory.
You need to make sure you know the basics before posting silly arguments.
EB
So, you claim that it is not a problem when "one premise implies the negation of the other premise.", but rather, the problem is with premises like "p and not p". Of course, that already debunks your own claim that the
"Improved Squid Argument" (and several similar arguments you asks about) is invalid, since that is precisely a case in which no premise contradicts itself, but one or more premises imply the negation of another.
Remember: I do not want to talk to you anymore, but whenever you reply to a thread and misrepresent what I said (in that thread or any other(s)), I will reply by debunking some of what you claim. Either you will stop misrepresenting what I said, or I will keep debunking your position. (of course, that will not motivate you to stop if you fail to realize I'm debunking your position, but no matter, the debunking will continue in that case).