But that is not the same as the question of whether the capacity is part of human nature.Originally Posted by Speakpigeon
For example, it is apparent that human nature restricts the sorts of grammars that can arise in natural languages (i.e., languages that are formed as communication means, and informally, as opposed to formal languages). It might be that logic in natural languages is even more restricted, either to a few, or even to only one single logic. The capacity to tell whether a conclusion follows from premises would still not be a part of human nature as long as language isn't, since (if language isn't) there could be a human community that has no language (of the relevant sort; see my previous post on the matter), so they would not have the capacity to tell whether a conclusion follows from premises (indeed, they would not even understand the idea of a premise), but that does not mean that there isn't a single object of study.
While I do not know whether language (and thus, the capacity you mention) is part of human nature (and I think probably neither do you or anyone else; there is insufficient information about human psychology as far as I can tell), I think there is a high probability that there is a single human logic for naturally arising languages, and if there is not a single one, there are a few and their common points can still be studied. If there is one, I think that classical mathematical logic does capture it (at least, better than Aristotle's logic), but that is a matter for one of the other threads, so I will leave it for later.
Could you give an example of those contradictions?Originally Posted by Speakpigeon
You would have to consider how the word 'logic' is used in each context, and then analyze the logics in question.Originally Posted by Speakpigeon
Some are (in some contexts), some aren't. For example, when some mathematicians argue in support of intuitionistic logic and argue against ¬¬P->P, whereas others disagree and argue for classical mathematical logic (which accepts ¬¬P->P), then you have a disagreement about the correct logic. By the way, both accept P&¬P->Q for any Q, so you disagree with both of them. But that is a disagreement. There are also philosophers and mathematicians who support other logics. Perhaps, you would like relevance logicOriginally Posted by Speakpigeon
On the other hand, a mathematician might choose to restrict their work to only finitistic arguments because that sort of proof might be useful for some applications, but they're not claiming infinite arguments are not part of human logic. So, it depends on the case. One should take a careful look at what people are saying if one wants to understand them.
Sure. For example, "when some mathematicians argue in support of intuitionistic logic and argue against ¬¬P->P, whereas others disagree and argue for classical mathematical logic (which accepts ¬¬P->P), then you have a disagreement about the correct logic" (Angra Mainyu, 05-31-2019, 04:07 AM).
Why should I do that. My point is that mathematicians either don't use the word "logic" to mean logic or they do but they are wrong.
I don't want or need to understand them. The topic of this thread is whether you think humans have an inherent capacity to decide that a conclusion follows necessarily from premises. Follows necessarily from premises. I am in no doubt that this is the case, and that it is pretty obvious, too.
I also think that this is the view shared by most people outside mathematical logic (and possibly outside computer sciences). And this poll seems to confirm.
Mathematical logic has made itself irrelevant here since contrary to what it says, the conjunction "It's true Jack is blue and it's false Jack is blue" doesn't imply anything since nothing follows necessarily from it.
EB
That is not a contradiction between different areas of math. That is a philosophical disagreement between mathematicians. It is not the same. Moreover, those who argue against ¬¬P->P do not argue that it is false. They argue that it is not proper to use it, for philosophical reasons. Depending on the person, they might say it's neither true nor false.Originally Posted by Speakpigeon
Still, now that I see what you have in mind when you say that " The practice of Mathematical logic today suggests on the contrary that logic is arbitrary. Mathematical logic itself is a branch of mathematics, not a method or a theory of logic. As a branch of mathematics, it brings together a very large number of theories and methods (calculus) which are all different from each other and in effect mutually contradictory.", I would say that philosophical disagreement and debate about what the correct logic is is more common among philosophers than it is among mathematicians. Would you agree also that the practice of philosophy today suggests that logic is arbitrary?
You should do that because you said " this makes it impossible to decide whether anyone of these theories or methods is really about the logic of valid arguments as used by humans and as first described by Aristotle.", but it does not make it impossible. Even if the word "logic" is used in more than one sense, it is possible to listen to what people say, in order to ascertain what sense that is, etc. There is no impossibility to decide (or, more precisely, ascertain) whether any of those theories or methods is about the logic of valid arguments, in general. It depends on the available information.Originally Posted by Speakpigeon
By the way, when mathematicians do debate about the correct logic (e.g., intuitionists vs. classical logic mathematicians), then that is about the logic of valid arguments. But it's not about the logic as applied in languages such as English, French or Spanish, but about as logic applied to mathematical language. Mathematical language is much more restrictive than colloquial languages (to give them a name), and not all usages of the words and connectives in colloquial language are in use in mathematics. As a result, the correct logic for mathematics may well be (and I think it is) a proper part of the full correct logic for colloquial languages. This is not to say that it's in any way in conflict with it, but rather, that it is a part of it - not all.
To give you an example, there is a use of the conditional in colloquial languages that involves causality, and which cannot be properly modeled by truth-tables, whereas there is also a use that is like that of math, and is properly modeled by truth tables. In mathematical language, I would say that the former use is not present. Of course, trying to apply truth-tables to the causal use of the conditional will give the wrong results, but that is not a problem with truth-tables in mathematics.
That is a difficulty, because you are raising disparaging accusations against "them", in the context of your argumentation.Originally Posted by Speakpigeon
That is the thread title. However, I have only addressed points you made about mathematicians and their use of the words.Originally Posted by Speakpigeon
I already addressed that point. I do not believe you know that. I understand "inherent capacity" as a question about human nature, because you explained that that is what you meant. On the basis of that, I say that I do not have sufficient information to tell whether humans have such inherent capacity. Very probably, no one has sufficient information.Originally Posted by Speakpigeon
It would be unwarranted, though. We do not know whether there can be a human society without language. Is language like, say, morality? (so, the anwswer is negative). Or is it a technology, like fire? (so, it is positive).Originally Posted by Speakpigeon
I gave more details in my reply here.
Still, you are not asking the right question (see here).
If you were right about the lack of implication, mathematical logic would not be irrelevant. It would fail to capture human logic in that particular case (though it was meant to capture logic as used in math, but anyway). But it would still remain a (much) better tool for finding mathematical truths than Aristotelian logic (see my post in the other thread), or (if human logic truly lacks the Principle of Explosion) than human logic.Originally Posted by Speakpigeon
By contradiction, I mean just that. One theory will say something is true, the other will say something is false.
So here, Wiki says intuitionistic logic will say formula A is not a valid inference whereas classical logic will say formula A is a valid inference. Contradiction.Intuitionistic logic
Intuitionistic logic can be understood as a weakening of classical logic, meaning that it is more conservative in what it allows a reasoner to infer, while not permitting any new inferences that could not be made under classical logic. Each theorem of intuitionistic logic is a theorem in classical logic, but not conversely. Many tautologies in classical logic are not theorems in intuitionistic logic (...)
https://en.wikipedia.org/wiki/Intuitionistic_logic
EB
How could they decide which logic is correct?! I have asked you for a justification that the definition of validity used in mathematical logic was correct, to no avail. There is no such justification. So, they may have arguments about which is correct, but none of them can support their respective claim.
There is just one deductive logic. We all use the same. Whether in our linguistic utterances, our thoughts, in writing, colloquial or formal. What is not deductive logic is mathematical logic. What is funny is that mathematicians use deductive logic like everybody else and somewhat with more rigour, yet they keep up the fiction that mathematical logic is "correct". It's not. They know it's not but they keep pretending. One is reminded of the tremendous capability of human beings for dissembling. That's toeing the party line and nothing else.
No it's not. Human logic and mathematical logic are mutually contradictory.
Causality, if it exists at all, is a fixture of natural world. As such, there can't be any logical problem with causality.
EB
Your point is irrelevant. Logic does not depend on any formal system and does not depend on language. You are confusing the communicating of the message with the meaning conveyed. Logic is what makes us say what we say when we speak logically but 99.999% of the time we don't even verbalise the logical inferences we make. Indeed, we are not even aware we are making them because, essentially, they remain unconscious. Aristotle pointed at logic, and like most people you're still looking at his finger.
Mathematical theorems are understood by mathematicians using their intuition mostly. Proofs are sketchy indications of the validity of the theorem meant for other mathematicians. They understand each other mostly without using formal logic. Mathematical logic is irrelevant except probably in the few cases where it is misleading.
EB
However, that is not what happens between mathematical theories. It is what happens between philosophical theories about mathematics. Some philosophers and some mathematicians adhere to some of those theories; others, adhere to other such theories.Originally Posted by Speakpigeon
In any case, if you think that this suggests logic is arbitrary, it seems it's not so much the practice of mathematical logic today (no contradiction there), but the philosophical practices of some mathematicians and some philosophers. However, by the same disagreement criterion, we could say the practice of, say, physics today suggest that physics is arbitrary; the practice of ethical philosophy or ethical debates today suggests ethics is arbitrary, and so on. I think it's an extremely weak case.
Hold on. There are different ways in which mathematicians use intuitionistic logic. When they are doing mathematics, they use it to infer things from premises. Now, when they're doing that, they are making no claims whatsoever about whether all classical inferences are valid. They might be using intuitionistic logic because they think that some classical inferences are not valid but all intuitionistic inferences are. Or they may have no objection to the validity of classical inferences, but instead they use intuitionistic logic because the sort of constructive proof they're much more likely to get is also useful for some purposes they may have, whereas classical proofs would not give them the practical tools they need in that particular case. Or they might have a different motivation. But regardless of their motivation, they're not saying (even if they believe it) anything about the validity of classical inferences.Originally Posted by Speakpigeon
Now, when some mathematicians and/or philosophers claim and/or argue that some classical inferences are not valid, they are engaging in philosophical argumentation. Nothing wrong with that of course, but it's something else. Similarly, for that matter, different physicists and philosophers propose competing (and mutually incompatible) theories of, say, the laws of nature, or even about time or space (sometimes, the disagreement is scientific; sometimes, it seems it's philosophic, and sometimes, it's a mixture). Different philosophers (and posters on the internet) defend competing (and mutually incompatible) metaethical theories, or first-order ethical theories, or epistemic theories, etc.
My point is that what you describe as "contradictions" are just examples of philosophical disagreement. It's not a particular feature of mathematics, but it's all over the place - and it was so in the time of Aristotle as well, even if the matters about which there was disagreement were not the same as they are in the present.