What examples do we have of mathematicians who explicitly and publicly expressed their personal confidence that mainstream modern logic, as used in mathematics, either as object of study in itself or simply as a tool, was an appropriate representation of the sense of logic most of us have without having to study formal logic?
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2It is not easy to define what our natural "sense of logic" must be ... – Mauro ALLEGRANZA May 13 '18 at 09:58
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*Formal logic* started with Aristotle and since then has been characterized as the "scientific" way to represent/codify how reason is "implemented" into human language and discourse. – Mauro ALLEGRANZA May 13 '18 at 10:00
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Mathematical logic is twofold : (i) modern version of formal logic, using mathematical symbols and tools to formalize arguments; (ii) the application of formal mathematized logic to the study of a specific domain of reason and argument : mathematics. – Mauro ALLEGRANZA May 13 '18 at 10:02
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Maybe useful : Jean-Yves Béziau, [What is Formal Logic](http://www.jyb-logic.org/jyb-form-final-sp.pdf) and John MacFarlane, [WHAT DOES IT MEAN TO SAY THAT LOGIC IS FORMAL](https://johnmacfarlane.net/dissertation.pdf). – Mauro ALLEGRANZA May 13 '18 at 10:06
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This is a factual question about whether mathematicians as practitioners or even mere users of formal logic have or not expressed themselves publicly and explicitly as to the appropriateness of formal logic to our, or even their own, sense of logic. It's not asking for a definition of our sense of logic. – Speakpigeon May 13 '18 at 16:01
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"Mainstream modern logic as used in mathematics", i.e. classical logic, was only spelled out at the end of 19th century and is well known not to be a representation of our "sense of logic", for a number of independent reasons, see e.g. [Why are conditionals with false antecedents considered true?](https://philosophy.stackexchange.com/a/34084/9148) It is however a useful idealization for mathematical purposes. Moreover, as history shows "sense of logic most of us have without having to study" is a fiction, like mathematics itself logic is an acquired cultural artifact. – Conifold May 15 '18 at 03:16
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2@Conifold - I don't often disagree with you but this comment seems very wrong. We all have an inbuilt sense of logic or we wouldn't get through the day, and this would be true for stone-age man, horses and sheep. . . – May 15 '18 at 11:38
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@Conifold - II don't see how history could possibly show that my sense of logic is a cultural artifact. If it was, it wouldn't be a "sense" to begin with. You seem to be confusing "sense of logic" with logical systems. What is properly cultural is the actual practice around our sense of logic, like indeed our use of formal logic systems. Can you rephrase? – Speakpigeon May 15 '18 at 11:39
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1@Conifold - You yourself talk of the "intuition of implication" and of "natural reasoning" in the very piece you just linked in your comment! You seem to have contradicted yourself twice, at least according to my own sense of logic and my arithmetic expertise. – Speakpigeon May 15 '18 at 11:53
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1The "inbuilt sense" is not supported by modern cognitive psychology, not even more basic "sense" that Chomsky termed "universal grammar". I am not even sure what your basis for assuming it is, it is a sense because we call it "sense" is circular. As for intuitions, logical or linguistic, they are culturally accumulated and developmentally acquired, the time when Kant and others thought them a priori is long gone. – Conifold May 16 '18 at 19:37
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I'm no specialist in anything so I won't go into cognitive science! All I need, though, is the evidence of my own mental life. My brain once produced for me the value of a rather complex logical formula I was considering and it then took me several days to analyse the problem, understand the basic principle and find a convincing formal justification of this intuitive result. That has to be good enough for me. I'm not trying to convince you here, just saying you won't convince me. And so, we can agree we disagree on this. – Speakpigeon May 17 '18 at 13:32
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I also understand recent studies show that small children and some animals displays some basic logical capabilities. Further, if formal logic isn't ultimately justified on our logical intuitions, what could be its justification? Is it arbitrary, then? And the only justification I was able to find is itself a logical argument. It seems logic is ultimately justified on the intuition of at least some human being. Aristotle? Not even him as he basically looked around at what previous philosophers had said that had some logical relevance. – Speakpigeon Jan 14 '19 at 15:04
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@Conifold "As for intuitions, logical or linguistic, they are culturally accumulated..." Do you have any references for this? Also, would "developmentally acquired" still support your position if it happens pre-linguistically? - Speakpigeon Do you have some reference for: "studies show that small children and some animals..."? - Also, having shortly prior finished classes in Logic and critical thinking, I had an opportunity to witness my daughter produce a textbook example modus ponens argument, when she was not yet three years old and barely able to construct a full sentence... – christo183 Jan 31 '19 at 08:49
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@christo183 There is now a general field of studying intuitions called [Experimental Philosophy](https://plato.stanford.edu/entries/experimental-philosophy/). On linguistic intuitions see e.g. [Beebe-Undercoffer survey Individual and Cross-Cultural Differences in Semantic Intuitions](https://brill.com/abstract/journals/jocc/16/3-4/article-p322_8.xml). On logic see e.g. [King's study of medieval debates](http://individual.utoronto.ca/pking/articles/Consequence_as_Inference.pdf). – Conifold Jan 31 '19 at 09:12
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@christo183 Sorry, no reference. The science of our sense of logic seems in its infancy. I'm aware of a study done I think in Soviet Russia, possibly in the 1930's, repeated more recently, which seems to have concluded that adults without any formal education don't seem interested in reasoning in a logical way. I take this to be inconclusive since mammals confined in a dark room from birth don't develop a visual sense. The method was to observe the subjects' reaction to sentences, which as such are essentially formal logic. What needs to be tested is our capability to have logical intuitions. – Speakpigeon Jan 31 '19 at 09:13
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Nobody is going to claim that things like 'Ex Falso Quodlibet' or the Zermelo-Frankel constructions represent natural human logic. They are formal dodges that avoid confusing aspects of naive logic on purpose.
One important example: The idea that you cannot have a set of all sets is not reasonable to most humans, naively. It has to be motivated by a need to evade paradoxes, and they eventually accept it, but it clearly contradicts a very natural impulse.
We go so far as to have different set-theories (e.g. Zermelo-Frankel and Godel-Bernays-von-Neumann) that do or do not allow for a universal set, because not having one seems too counter-intuitive to some mathematicians. In the latter, you can have sets that include all the sets, but you still can't have a set of all sets, because Russel's paradox still can't be permitted.
So the already artificial notion of 'too big a collection to be contained', the closest intuition we can usually impart for why there should not be such a set, actually fails to capture what is going on. There is a real gap here between the formalized solution and our vocabulary that humans don't actually seem to be able to accommodate fully.
But in the end, the purpose of formalization is to improve the system in some way. If it captured all the confusing parts and all the potential paradoxes, it would not actually achieve anything.
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It does not seem that difficult to come up with a concept of set that is immune to Russell's paradox. What may be problematic is using such a concept in the context of the usual mathematical formalism. My guess is that the current situation merely results from the contingencies of historical convenience. Same thing with formal logic. – Speakpigeon Jan 14 '19 at 15:16
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1@Speakpigeon We certainly have a system that is immune to Russell's paradox, and that is easy to use. The point is that it is not a model of intuition, it is not intuitive itself, and the way of identifying exceptions is not any more intuitive, either (because the intuition itself is not ultimately logical -- human beings are flawed). That is what the question actually asked. – Jan 14 '19 at 15:19
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Sorry, I should have been more explicit. I meant a concept of set reflecting our ordinary intuitive notion and immune to Russell's paradox. Humans are flawed, but their brain is the end result of several hundred million years of natural selection of neurobiological systems evolved over an enormous biomass. On the face of it, evolution is a much better guarantor than a few mathematicians spread over two millennia in terms of producing good logic. Formal logic is still very young. So, I don't see any good reason to think there's likely nothing better than current logic and current set theory. – Speakpigeon Jan 14 '19 at 16:10
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1@Speakpigeon No. Our intuition has a contradiction built into it. And another intuition of ours rejects contradiction. You cannot axiomatize naive set theory. Period. You can relax our natural intuition about contradiction, the way Intuitionism does, as it just seems wise to be less arrogant. But that is not the same thing as solving the problem, which really is impossible. If there is a future solution facilitated by evolution it will have to come because inborn human intuition itself has moved forward. We cannot model what we have now. – Jan 14 '19 at 16:27
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And so we disagree. You cannot axiomatise naive set theory as it has been initially formalised. You haven't articulated any good reason to think we can't do better. – Speakpigeon Jan 14 '19 at 16:39
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@Speakpigeon Naive set theory is not currently formalized. So that statement makes absolutely no sense. We cannot do better because the abstract intuition of containment itself gives rise naturally to a contraction. If we removed part of it, it would no longer be the same thing. I have not said we cannot axiomatize something different. But that would not be doing better, it would be doing something else. If we don't, we have axiomatized a contradiction, and we cannot use it with the rest of classical logic. – Jan 14 '19 at 16:47
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@Speakpigeon So I have given a reason, a proof by contradiction is generally considered a reason. And you are just refusing to take me (and along with me a lot of the philosophy of math, because I didn't just make this up) seriously. If you don't have any approach other than simply ignoring what I say and shifting the burden of proof, this is not really a discussion, it is you ignoring me. – Jan 14 '19 at 16:50
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I provided only one reason, namely that several hundreds of millions of years of evolution is a better guarantor of our intuitive sense of logic than a couple of thousands of years of mathematics could be of current formal logic. If the human brain's processes can do it, there's no apparent reason that formal logic couldn't do it, too. Your proof seems to be we've tried and failed. Sure, for how long? I don't have a solution myself and I don't have the time to look into it. All my argument is in evolution. Make the best of it. And I'm prepared to admit it may be way harder than I imagine. – Speakpigeon Jan 14 '19 at 18:41
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@Speakpigeon If it seems that way, you are reading it wrong. The proof that the angles of a triangle are 180 degrees is not flawed and will not go away just because it does not apply to triangles on spheres. A contradiction in a concept is necessarily always going to be there until you get a different concept. At which point it is a different concept. It does not matter what happens to future triangles, or whether we ultimately stop caring about planes. What we have proved about triangles is not going away. – Jan 14 '19 at 19:09
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It isn't like we tried to have 120 degree triangles and failed, and we are going to evolve our way out of that. So, if you want to respond with exactly the same objection that makes no sense again, don't – Jan 14 '19 at 19:12
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To recap, I maintain my assertion that it is reasonable to think that we should be able to come up with a concept of set that is both an accurate reflection of our ordinary intuitive notion of set and immune to Russell's paradox. – Speakpigeon Jan 15 '19 at 10:29
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I take your point that we've tried and failed. As I understand your argument, though, it is not relevant to my assertion. You seem to believe that it is our ordinary intuitive notion of set which is, and has been shown to be, logically flawed. Me, I don't believe that, for the reason that natural selection beats a few mathematicians hands down. I also haven't seen in what you say any reason, good or bad, to change my view. – Speakpigeon Jan 15 '19 at 10:30
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As far as I can tell, you are assuming, without justification, that the formal expression of our intuitive notion of set, the one that mathematicians have been able to produced so far and have shown that it was logically flawed, is our intuitive notion of set. I don't believe that myself and I haven't seen in what you say any reason to change my view. So, I accept your argument is valid, but we start from different premises, premises none of us could prove true. – Speakpigeon Jan 15 '19 at 10:33
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My point is that it seems rather important that anyone here, and particularly maths students, if any, shouldn't be misled into believing that anyone actually knows there isn't a solution. All we have for now are people, competent mathematicians, and more generally intelligent and well-informed people, who have given the matter serious consideration, who merely believe there isn't a solution. My view is that if you think you're smart, you may have something to find if only you're prepared to look. – Speakpigeon Jan 15 '19 at 10:43
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@Speakpigeon At the point you accuse me of misleading people, you are just playing rhetorical games and not the least bit interested in philosophy. I am sorry to have wasted my time talking to someone who will not listen or take anything seriously that he has not thought of himself. – Jan 15 '19 at 11:59
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Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/88287/discussion-between-jobermark-and-speakpigeon). – Jan 15 '19 at 12:04
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Automated theorem provers, such as Otter, or Prover9 usually use a subset of first-order logic. There have existed open mathematical conjectures which first got solved by theorem provers, such as the Robbins problem. There are some mathematicians, such as Ken Kunnen, who use theorem provers extensively in their work also. So, I think the answer to your question is 'yes'.
Doug Spoonwood
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As I understand now, mathematicians overwhelmingly stick to using their intuitive sense of logic to prove their theorems. Theorem provers don't seem to be used very much. And for what I read about them, they in effect apply some variation of the Gentzen method of proof, which seems to me to be strictly a generalisation of Aristotle. Further, Gentzen relies on the use of a set of rules of inference, which are themselves not proven but merely accepted as obviously true. – Speakpigeon Jan 14 '19 at 15:26
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@Speakpigeon From what I've seen many, and I would guess most, theorem provers use resolution. I don't think that's what you mean by a Gentzen method of proof. Additionally, though no rules of inference can get proven, rules of inference can often get checked for validity. The question as stated also doesn't concern what mathematicians do, but rather their personal confidence. If mathematicians were not confident that theorem provers proved things correctly, then, if asked at least, they will reject the results of those theorem provers or express doubts about those proofs. – Doug Spoonwood Jan 17 '19 at 18:18
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According to what I read recently on theorem provers, they will typically use a small set of fairly obvious logical truths used as rules of inference. Resolution itself seems to be a sort of generalisation of that method. As to proving the obvious logical truths used as rules of inference, as I see it, they are the empirical evidence that allow you to validate any method of proof, not the other way round as you seem to suggest here. – Speakpigeon Jan 17 '19 at 19:25
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My point was that confidence seems to rest almost entirely in sticking with the empirical evidence of the logical truths identified by the tradition, whether it's proof by mathematician or proof by theorem prover. The few exceptions I was able to find seem to be "wild explorations", for example assuming that A and not A implies not A. – Speakpigeon Jan 17 '19 at 19:26
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@Speakpigeon No rules of inference are not logical truths. A truth consist of an accurate statement. Rules of inference are not statements. Thus, to call rules of inferences truths makes for a category error. Rules of inference are also not empirical, in that they do not rely on sense data or perception in any way, though perhaps I misunderstand what you mean by 'empirical'. There is no way to observe the law of identity for example, since it applies to an indefinite, if not also potentially infinite, if not also actually infinite universe of statements. – Doug Spoonwood Jan 17 '19 at 19:46
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I'm quite sure it is true that p and q implies p and if it is true, then it is a truth, and since it's a logical proposition, then it's a logical truth. I agree that a truth is a statement true of some real thing, and so a logical truth is a statement true of some real thing, which we call an implication. So, no category error, unless you could prove it. – Speakpigeon Jan 18 '19 at 07:17
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Rules of inference are arbitrary and as such need not be empirical, but rules of inference which are also logical truths are empirical since logical truths themselves are empirical, and in the same sense that the Sun revolving around the Earth is an empirical proposition. – Speakpigeon Jan 18 '19 at 07:27
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The empirical evidence for logical truths comes through our sense of logic, which is a sense of perception in the properly understood sense of the word. Personally, I can observe the law of identity every morning at breakfast and every evening before going to bed. Hell, it seems we just have a very different semantics. I haven't found yet any difficulty with mine and I have got it since a long while. Unless you could prove implications such as p and q implies q don't exist or that we don't have a sense of logic. – Speakpigeon Jan 18 '19 at 07:33
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"p and q implies p" is not a rule of inference. There is no assumption, and there is no conclusion. It has the form of a statement. In symbols "p and q implies p" can get notated as CKpqp. Earlier you said "logical truths used as rules of inference". But that makes for a category error. Rules of inference are not arbitrary given a fixed semantics (such as truth-table semantics) for a system. Both the Sun and the Earth can get observed via sense perception. There is no way to observe a statement via sense perception, since there is no way to observe meaning via sense perception. – Doug Spoonwood Jan 19 '19 at 18:44
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Meaning and understanding happen at a different level than sense perception. Meaning requires a concept, not just a perception. And no you can't observe the law of identity, since it applies to more objects than you can perceive. Something like CKpqp (alternatively ((p^q)->p), again, has the form of a statement. There is NOTHING that can get inferred purely from that. If a statement having the form Kpq holds, then since CKpqp is a true propostion, p can get inferred validly. But, that doesn't make CKpqp a rule of inference. The rule of inference used was modus ponens. – Doug Spoonwood Jan 19 '19 at 18:51
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First, I didn't claim nor suggest that "p and q implies p" is a rule of inference. Second, rules of inference are arbitrary to the extent that what you call "semantics" are arbitrary, and they are arbitrary if not based on logical intuition. Truths table for conjunction, disjunction and negation have been produced to fit with our logical intuitions. The truth table for material implication is based on the arbitrary and unjustified assumption that logical implication is truth functional. You should shun the notion of category error unless you've made sure you could prove it. – Speakpigeon Jan 19 '19 at 19:08
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Don't call our sense of logic perception all you like, but I can observe my intuitions, of the logical sort and other sorts, and I'm obviously not alone in that. Further, our sense of logic works like a sense of perception and there's no reason not to call it perception. Same for memory for example. Meaning is observed in the same way as everything else. All our perception senses are very different from each other so difference is not a rational ground to dismiss our sense of logic. – Speakpigeon Jan 19 '19 at 19:15
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Meaning doesn't require any concept, only association between hearing the word in an appropriate empirical context, if not, our species would be incapable of thinking of the meaning of a word. Meaning of statements require that the meaning of words you know come to you as intuitions, which they do. – Speakpigeon Jan 19 '19 at 19:21
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You can observe the law of identity in the same way as you can observe the law of gravitation, the 2nd law of thermodynamics, etc., i.e. from empirical evidence. All these laws are understood as potentially applying to an infinity of objects. Good, so, you've just denied that Newton's law of gravitation and the 2nd law of thermodynamics could possibly be observed. – Speakpigeon Jan 19 '19 at 19:26
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@Speakpigeon You might want to look up 'sense' in say Wikipedia sometime. There is no stimulus external to the human body for logic or memory. So, there is no 'sense of logic' to speak of in the same way that we speak of sense data. Given that physical laws apply to a potentially infinity of objects (which I'm not sure physicists would agree to), they cannot get observed, yes. I don't see any problem there. Observing something leads to knowledge. Physical laws make predictions about future events. But there is no knowledge of future events. So, why would physical laws be observational? – Doug Spoonwood Jan 21 '19 at 04:32
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As to sense, there are obviously inputs coming from other senses, like for example for memory, and also for a range of impressions that help us go through our day and take inputs from other senses. What is perceived in these cases, including our sense of logic, is, like in the case of pain and memory for example, not something outside the body. Again, all our senses are very different from each other and yet we are able to see how they belong together. I can't see what would make our sense of logic not belonging there. – Speakpigeon Jan 21 '19 at 07:53
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As to logical and physical laws, you still have to explain the difference. And logical laws are necessary to logical laws, we can only understand and use through application of the modus ponens. Physical laws are obviously observational. I really don't think I need to argue that. – Speakpigeon Jan 21 '19 at 07:58
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@Speakpigeon "As to sense, there are obviously inputs coming from other senses, like for example for memory, and also for a range of impressions that help us go through our day and take inputs from other senses." I think I already denied that. So, no, that's not obvious. Physical laws need tested via careful experiments involving observational phenomena. Experiments can support or reject any such laws at any point in time. It doesn't work that way with logical laws. – Doug Spoonwood Jan 23 '19 at 05:04
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I think it works that way. We've had 2,400 years of formal logic, mostly based on the evidence available, our intuitions. And I can look at the formal expression of logical truths identified by people since Aristotle and have the intuition they are true in the same way I can look at the tree in my garden. We all do it and mathematicians have reported doing it and even discussed at length how they use their intuitions. Science itself relies on maths and maths on those very same logical intuitions mathematicians have. You think mathematics is somehow less trustworthy than science? – Speakpigeon Jan 23 '19 at 12:46
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@Speakpigeon Laughs. I don't think you understand the history of formal logic. Aristotle's formal logic has an entirely different conceptual basis than Frege's logic. Logic got reworked from it's foundations by Frege and his followers. That propositional logic precedes other forms of logic didn't gain acceptance until Frege. Intuitions? Medieval people would tell us that our intuitions about there exist a precedence of logical systems are all wrong. The law of the excluded middle gets rejected by intuitionistic logic (and many other logical laws). – Doug Spoonwood Jan 23 '19 at 23:37
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@DougSponwood Stop patronising and try better English. Nothing you say is both factual and relevant. No one in the 2,400 since Aristotle found his syllogisms possibly not valid, not e.g. Copernicus, Galileo, Newton nor Leibnitz. Compare this with the notion of validity as defined by modern classical mathematical logic, definition consistent with the truth table of the material implication. Here, a well-known fact, most people with no training in modern mathematical logic find this definition contrary to their intuition for most implications involving a false antecedent or a true consequent. – Speakpigeon Jan 26 '19 at 19:29
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@Speakpigeon Jan Lukasiewicz did detail an error that Aristotle made in his book "Aristotle's Syllogistic From the Standpoint of Modern Formal Logic". So, Aristotle's reasoning has gotten found not valid on at least one point. Additionally, the Stoics had a different approach to logic than Aristotle, and basically disagreed with Aristotle one some points long before Frege. – Doug Spoonwood Jan 26 '19 at 20:17
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"Łukasiewicz's controversial views sparked a controversy over how to interpret the syllogistic. While the principles won an early adherent in Patzig (1968), subsequent criticisms by Corcoran (1972, 1974) and Smiley (1974) established clearly that syllogisms are not propositions but inferences, and that Aristotle had no need of a prior logic of propositions. That view is now universal among scholars of Aristotle's logic. In retrospect, it appears that Łukasiewicz was keen to wish onto Aristotle his own Fregean view of logic as a system of theorems based on a propositional logic." – Speakpigeon Jan 26 '19 at 22:58
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Łukasiewicz wrongly interpreted Aristotle's syllogistic in the terms of Russell's definition of the material implication, which implies the validity of the implication (¬(A ≡ B) ∧ (X ≡ A) ∧ (X ≡ B)) → (X ≡ B), which makes no sense and is not valid according to the prevailing interpretation of validity in Aristotle's syllogistic. Łukasiewicz just got it wrong. I don't know of any rationalist and empiricist ever agreeing with the Stoics prior to Frege and modern mathematical "classical logic". Modern mathematical "classical logic" doesn't make sense and so isn't logic at all. It is mathematics. – Speakpigeon Jan 27 '19 at 14:16