## Is logic undoubtable? What can we know for certain?

• 774
Well there is a bit of A is A but also B but not at the same time, I don't have the preparation to explain it better, I'm sorry, I studied it but my explanation are not exhaustive.

The law of identity just states that A is A. The law of non-contradiction states that A is not and cannot be Not-A (the Not-A could be anything from B to Z as long as it's not A). The law of excluded middle states that A is A and cannot be Not-A, or anything in between. As far as I'm concerned there is no sense of conflation.
• 23
A Quantum is a particle or a wave?
• 774
A Quantum is a particle or a wave?

Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be partly described in terms not only of particles, but also of waves. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the behavior of quantum-scale objects.
- https://en.wikipedia.org/wiki/Wave–particle_duality

Therefore, from A is A, we get quantum entity=quantum entity. The conditioning does not alter the identity. Also, the only problem in that query is the means of defining a quantum entity. They (quantum entities) are and have always been what they are. That we can't distinctly determine their nature does not mean they're illogical, only that our methods thus far have not been successful.
• 23
You bring up a good point, go tell them, they changed the non-contradiction law for this and picked up another one, which is a bit more flexible.

But still, my point was, apart from trusting logic, which still is a question to me, what can we know really?
Can we reduce our knowledge without answering "because yes" to something and stopping there?
We don't really know much, we believe
• 774

Everything I know starts with "I" and then "I AM". I may not know everything about myself but the only certainty I have is that "I" am, regardless of whether the nature of my existence is real or illusion. My knowledge is just a relationship between my experience of "I" and that which is not "I". For me, logic is just the path of least chaos or the tool of best fit, so I trust it implicitly.
• 733
Everything I know starts with "I" and then "I AM"

If I had said something like that, which I have before because I think it’s true, I would have used a few dozen more words, which wouldn’t have made it any better.
• 2.6k
if you touch ice you do not get burned (I realize these are terrible examples but they'll do)
But you can never use logic on logic, to prove it right nor wrong so can we trust it? And how do we know?
Actually, if it's cold enough, you do. And indeed you do use logic on logic: it's called (the) proof. And you both trust and know, because it works. In its esoteric forms - formally - by compliance with its own criteria, and more generally, by a fitting with how the world works.

we use logic to understand relations of events, the road's wet so....
The key here is use. The state of the world isn't in itself either logic or necessarily logical (it may well be logical, but it is not clear to me that it is necessarily so). Logic, then, is in use in the world a tool, and as a tool in use is subject to possible misuse and abuse.

was only about trusting logic as an absolute tool a priori, without explanation, as I believe that we cannot express thoughts without one absolute at least.
Key here is what you mean, exactly, by "explanation," and by "absolute."

Example: a carpenter's saw just is a saw. How would "explanation" and "absolute," as you use them above, apply to your thinking about the saw - or perhaps you do not think about the saw at all?
• 763
My question for you is: can we be certain that the laws of logic are valid? Or is logic to be taken as an absolute a priori?

This needs to be disambiguated otherwise it's not coherent to my ears. Validity is a property of a logical *system*, not to the axioms of that system. Validity, roughly speaking, refers to the set of possible argument forms that can be made given some set of logical rules, known as the logical consequence relationship. Now, perhaps by 'valid' here what you mean is "true". But that's not the domain of logic at all, logic stands independent of truth. We know there are lots of logical systems: Classical logic, intuitionistic logic, Paraconsistent logic, many-valued logic, etc etc. Asking if any of them are "true" is something you need to thing about making sense of first before asking that question.

What would it mean for a logic itself to be true? To me that's either an incoherent suggestion or else it has to mean something about the structure of a universe mapping on to the abstract relationships sketched out by some particular logic. And the latter of those is necessarily relative to a specific world anyway. There's really nothing a priori here. But what should be clear is that in any case logic and (physical) reality are not about the same thing. It's a case of the abstract vs the material.

Can we, so to say, ”trust” the laws of logic? Are they absolute or rather just to be taken as if they were?

Trust them how? The logical consequence relationship is an abstract object and that by definition can't change, whether you understand logic semantically or syntactically. That seems a pretty good thing to trust, as it's not like it can randomly change or something. "Absolute" is a term I would avoid here. There's way too many things that can be taken to mean and most of them wouldn't be true.

And my second question for you is: can absolute relativism be logically acceptable?
Taking the laws of logic as true, is it possible to consider everything relative without contradiction? I mean, if I say that ”everything is relative”, then the fact that ”everything is relative” is not relative anymore, it is absolute, and if I say that even that is relative, so that ”even that everything is relative is to be considered as relative” I’m still considering the relativism of the relativism of everything as absolute, thus contradicting myself.

Logical consequence is already defined relatively. An argument is relative to a set of logical rules specifying which Propositional transformations can be performed. Intuitionistic logic does not permit double negation elimination while classical logic permits it. The issue you're running into is a failure to state things correctly. I wouldn't necessarily say "Everything is relative", but in this context I would say "Every valid argument is relative to a set of rules specifying them as valid". That's a true statement and doesn't create any contradictions because I'm not saying every statement is relative or something off like that.

Yes you are right, I should have been more clear, I was referring to classical logic from Aristotle onwards, so I guess Syllogistic logic and friends, like I wrote to BrianW

One should note that Aristotle did not use classical logic, he created and used Syllogistic logic. Classical logic is a poor name because it was created in the 1870s by Frege and considers a different set of arguments to be valid than what Aristotle did, so they are different logics for sure.
• 23
All of your answers show a lot of interest, and I appreciate that, and also show me a lot of new things that honestly I didn't know about.
But I think you misunderstand my question, maybe my level of English is not as good as it should be for this, I only meant to bring about a question that may be wrong in its own roots:
Whatever logic you may want to refer to, whatever you prefer to use or even whatever you want to call it, how do you know it is the best tool to analyze relations between things?
I mean, putting aside names and even concepts, whenever you formulate thoughts you must use some kind of logical process, right? Any kind, Jack is a boy, therefore he is not a girl.
So, for those who say that endless logics work on endless different system, which one is to use when?
Because by saying that neither of us can be nor wrong nor right because everyone uses a logic that suits his system and this whole thing makes no sense at all.

For those who say that classical or Syllogistic logic is the one, how do you know?
Most of you agree that there is nothing absolutely true, ontologically I mean, like God for the Christians or The Idea of Good for Plato, yet logic works always and it is always right, thus absolute, how is that? You believe it is.
• 13
In regards to trusting logic (outside of mathematics), I find it doubtful that logic is completely trustworthy as it is ultimately an extension (or rather interpretation) of the senses which are traditionally considered fallible. So logic is no more trustworthy than the senses it is reliant upon to exist. Not to say that I think it is impossible to learn the truth so much as impossible to know it has been learned.

In regard to the second question, I see no problem with the logic of absolute relativity. Perhaps there are absolute truths, it is even likely... But how can knowledge of them be claimed by fallible logic?
• 9.9k
Yet, I sincerely don’t understand what your point is with the different and incompatible logics.

If they're incompatible then we're obviously not dealing with certainty for them.
• 232
I would maintain that at least the law of non-contradiction is indubitable in just this sense: it cannot intelligibly be doubted.

Try to imagine any situation that violates the law of non-contradiction. My sense is that I just can't do it. I can't even understand what A and NotA both obtaining is supposed to involve. Some people say that various physics results should be interpretted as involving such a situation, but I think even the people who defend that interpretation will admit that they have absolutely zero idea what it means. I think it is unintelligible, and won't be made any more intelligible by inventing pretty new logical symbols and defining their relations to other symbols.

Non-contradiction is, in that way, a necessary condition of intelligible thought. Of course you can invent abstract systems that violate it, by defining various symbols in various ways, but substitute symbols for actual concrete things and what you get is meaningless.

PA
• 767
I would maintain that at least the law of non-contradiction is indubitable in just this sense: it cannot intelligibly be doubted.

Yes, even the law of non-contradiction can be intelligibly doubted: see Dialetheism

The a priori status of logic has been under attack for quite some time (as you can see above, even the sacrosanct non-contradiction is not safe). This is true even of the so-called "laws of thought," which is what you must really mean when you talk about logic, because formal or mathematical logic is as diverse and open-ended as mathematics.
• 763
Try to imagine any situation that violates the law of non-contradiction. My sense is that I just can't do it. I can't even understand what A and NotA both obtaining is supposed to involve. Some people say that various physics results should be interpretted as involving such a situation, but I think even the people who defend that interpretation will admit that they have absolutely zero idea what it means. I think it is unintelligible, and won't be made any more intelligible by inventing pretty new logical symbols and defining their relations to other symbols.

Non-contradiction is, in that way, a necessary condition of intelligible thought. Of course you can invent abstract systems that violate it, by defining various symbols in various ways, but substitute symbols for actual concrete things and what you get is meaningless.

I don't need to invent new symbols or anything to give a semantics to a logical system which, yes, intelligibly violates Non-contradiction. Let's note something first. Define what you mean by "intelligible" here in a non-question begging way, such that it's not just a substitute for "consistent". Otherwise all you're saying is that an inconsistency has to be inconsistent, which, well, yes.

Now, just take the standard, classical propositional logic and make the following modifications. Drop (or in some way weaken) the Disjunctive Syllogism inference that way the Principle of Explosion is no longer a valid argument. Then, replace the truth-functional semantics with truth relational semantics. What this means is that instead of a proposition relating to only one truth value they can relate to any number of them. Thus, a true contradiction (that is, a proposition which is true and has a true negation, a dialetheia) is simply some proposition P such that P relates to the value 'true' and P relates to the value 'false'. This is perfectly mathematically coherent and uses well understood math (hell, relations are used in everyday language as well).

Semantics don't stand independently of a logic but are created to understand, so it would be more than silly to say the above is meaningless. Forget physics, I've no idea if such Paraconsistent logic and dialetheism will ever be used there. Maybe to solve the Liar paradox, or the vagueness paradoxes, or for alternative set theories, or perhaps for some fundamental ontology or mereology if you want something metaphysical (particularly when discussing the concept of nothingness). There's possible uses, but whether or not these uses pan out has nothing to do with a circularly defined notion of intelligibility. There's no such thing as an indubitable logical axiom, or one which contravening entails unintelligibility. Perhaps if it entails trivialism then we can dismiss it but that's why Paraconsistent logic exists, so that potential violations of the LNC do not entail that every proposition is true.
• 3.3k
How is it that you provide reasons to the effect that logic is NOT trustworthy.

Haven't you shot yourself in the foot?

After all, providing reasons (in this case to prove logic is untrustworthy) means you already trust logic to do its job of finding the truth.

Perhaps that's the beauty of logic, right? It doesn't exempt even itself from its courts.
• 655
Forget physics, I've no idea if such Paraconsistent logic and dialetheism will ever be used there.

Note that @PossibleAaran said that the idea of A and not-A obtaining is unintelligible. This follows Aristotle's use of the LNC as a rule for thinking about the world.

Per physics, it's possible for an electron to be in a quantum superposition of spin up and spin down. But the term "superposition" has a clear mathematical meaning and there is no implication that the electron is in a contradictory state.

It is really the idea of contradictory states obtaining in the world that is unintelligible (so it seems to me).
• 763
It is really the idea of contradictory states obtaining in the world that is unintelligible (so it seems to me).

I don't see this. If one has a coherent but inconsistent logic with the appropriate semantics, and they have a theory about the world which best explains the data which requires reasoning by that logic, then it seems to me there would a case for intelligibly understanding inconsistent states of the world.

I'm not saying this is actually the case. As far as I can tell, since physics uses the standard math formalism it's going to necessarily make use of the underlying logical principles there so contradictions cannot be intelligibly added because it would result in trivialism. But that's a case of the logic and theory causing that, not whether or not the LNC necessarily applies to the world itself.
• 655
I don't see this. If one has a coherent but inconsistent logic with the appropriate semantics, and they have a theory about the world which best explains the data which requires reasoning by that logic, then it seems to me there would a case for intelligibly understanding inconsistent states of the world.

Suppose you have a logic that could represent a switch that is both on and off in the same sense and same respect. Can you visualize or simulate a scenario where such a switch would operate? That is the test of intelligibility. It seems to me that that exercise would require changing how the switch is represented such that its states were consistent.
• 763
I don't think that is the test if intelligibility, as there are many examples of consistent situations that I cannot possibly visualize but surely they are intelligible. Even if one thinks some contradictions may be true that doesn't entail that any contradiction may be so. After all, just because some propositions may be true does not commit one to thinking any proposition can be true. It just doesn't follow, even for the dialetheist. There has to be a reason to motivate accepting it, just as with anything else.

An analogue of what you're asking however has been suggested by Newton da Costa as a potential interpretation of superpositions as being potential contradictions (though this is difficult to understand for me and seems to require a deep dive in QM formalisms that I cannot do).

So let's take an easier approach. I take it for granted that people can (and most often do) have inconsistencies in their set of beliefs (one's internal maps of where things are located are often inconsistent with other such mental maps, for example). Take the hypothetical switch you mention and put it under the control of a reasonably advanced A.I. which tracks the behavior of a hypothetical person with an inconsistency in their beliefs. Presumably this switch would operate once such a scenario was observed when a person was mistakenly operating under these contrary beliefs. The switch then, could be represented by three states. 0 for false, 1 for true and 0.5 for both. If the A.I. determines the subject is showcasing their inconsistent beliefs 0.5 would be the value indicated when queried.

Or are you asking the switch to be an inconsistent physical object? I'm not sure the representation of the logic is supposed to have all the same properties of the formal system. By way of example, standard computers do not instantiate the exact model of classical logic since classical predicate logic has a model that is infinite, where clearly no actual machine can be made to represent that.
• 232
thanks for the clarification Andrew. Yes that is exactly what I meant to say.

Yes smart people deny the law of non-contradiction, but even they do so only because it solves certain paradoxes and not because any of them can imagine any concretely obtaining contradiction. When it comes to thinking concretely about the world, and not about abstract formal systems, my sense is that there is just no choice but to think under the law of non-contradiction. It is indubitable in that sense, whatever we say about the liar paradox.

my reply to your post is the same as Andrew M's, so I'll move to your latest post if you don't mind.

Andrew asks us to imagine a switch which is both on and off at the same time, which I think is plainly inconceivable, as he notes. But your example is much more complex.

I take it that your aim is to describe a conceivable situation where a contradiction obtains. I'm not sure your example is really detailed enough. How does the switch work? The switch is hooked up to a person's brain and tracks their inconsistent beliefs. What exactly is the switch reporting? It "operates once a person is operating under contrary beliefs". Does that mean that the switch reports "true" when the person is operating under contrary beliefs? If so, why would the switch show 0.5? I don't get it. In any case, suppose that the switch does report 0.5. Where is the contradictory state of affairs? We have a person who has two different beliefs that contradict one another. Having the belief that A and the belief that -A is not a contradictory state of affairs, any more than having a blue pillow and a red pillow is. We also have a switch that is reporting "0.5", and that isn't contradictory either.

Regarding the charge that I used a question beginning notion of intelligibility, I didn't. Say that something is intelligible if and only if you can conceive how it would be.

PA
• 349
i am unable to visualise or demonstrate a semantic notion of logical inconsistency.

I can demonstrate what might be called psychological inconsistency, for example by holding a self-negating belief, such as "This sentence is false. Therefore it is true. Therefore it is false... etc", but this isn't any different from writing {-1, 1, -1, 1,...} as a consequence of iterating the equation x(t+1)=-x(t) starting from x(0) = -1.

This is hardly what one might call the semantics of logical inconsistency, which requires two incompatible statements to be held simultaneously. But this isn't imaginable by definition. At most, I can imagine a driver encountering two signposts for a town which point in opposite directions, and him being unable to make a decision. Or two people disagreeing as to which word applies in a situation. Or a computer program failing a software test.

So logical inconsistency at most refers to a syntactical convention of communication of which nothing else needs to be said. It has no significant implications.
• 763
I take it that your aim is to describe a conceivable situation where a contradiction obtains. I'm not sure your example is really detailed enough. How does the switch work? The switch is hooked up to a person's brain and tracks their inconsistent beliefs. What exactly is the switch reporting? It "operates once a person is operating under contrary beliefs". Does that mean that the switch reports "true" when the person is operating under contrary beliefs? If so, why would the switch show 0.5?

Sort of. As I said, I don't take it as controversial that people have inconsistent belief sets. What is the switch reporting? Well let's make it simple. Say the subject reports believing some business is located in certain location relative to their home and they draw a map of how to get there. They believe the locations are correct. Now they repeat this drawing of different maps to different locations and again voice their belief that they are correct. But say some of the maps are inconsistent with others because they place various locales in slightly wrong locations, such that the maps cannot all be take to be true. Whatever program is combing through these maps will reach this contradiction and when queried about some business being at a particular location will throw out the value 0.5 since the underlying logic is three valued (this is essentially the logic underpinning the database language SQL, although it's not really for contradictions). Its not true that the location is correct because one map says it isn't, but it's true that it's there since another map says it is. So to resolve this in a normal computer it's easier to throw out that value rather than try to continue the computation.

Now, the reason I said "sort of" is because this isn't necessarily a physical contradiction because this is about ones knowledge. But it's hard to say because if one is a physicalist I'm not sure how one talks about sets of beliefs in the mind. Is it contradictory because it's in the mind? I don't know. But the point is that switch would operate in this case, whether or not the contradiction is a bona fide physical one. The machine implementing the logic need not have contradictory properties .

Having the belief that A and the belief that -A is not a contradictory state of affairs, any more than having a blue pillow and a red pillow is. We also have a switch that is reporting "0.5", and that isn't contradictory either.

Are sets of beliefs not in the mind? The comparison to differently colored pillows isn't a legitimate comparison, they are not the same object. I take beliefs to be part.of the mind, and so if there's an inconsistency in ones beliefs (as there likely always is) then there's an inconsistency in the mind.

Regarding the charge that I used a question beginning notion of intelligibility, I didn't. Say that something is intelligible if and only if you can conceive how it would be.

That's not really explaining what you mean though. Is conceivability defined in terms of consistency? If so, it's question begging for the LNC. If conceivability is defined in terms of mental pictures, that's not going to work since lots of actual states of affairs cannot be pictured and mathematics has it's own notion of conceivability (basically deduction). Conceivability needs to be defined minimally in terms of logical deduction used to understand a concept (or something like that), and that's just as available to inconsistency-tolerant logics as consistent ones. Paraconsistent logics have their own model theories that have contradictions in the metatheory.
• 763
i am unable to visualise or demonstrate a semantic notion of logical inconsistencysime

Even if this is true it's not going to be a sufficient refutation of giving semantics to inconsistency. I cant visualize the expanse of a million miles, only a tiny scale of it. I can't visualize something infinite, much less point at it (perhaps space and time). But these are surely not refuted from possibility on that basis.

I can demonstrate what might be called psychological inconsistency, for example by holding a self-negating belief, such as "This sentence is false. Therefore it is true. Therefore it is false... etc", but this isn't any different from writing {-1, 1, -1, 1,...} as a consequence of iterating the equation x(t+1)=-x(t) starting from x(0) = -1.sime

That's not true. For one, the liar paradox is, well, a paradox. In other words, it is not the bald assertion of a contradiction, it's an argument from seemingly valid principles of reasoning which ends in contradiction, and the LP is just such an argument. It only requires 5 or so axioms and inference rules (capture, release, Excluded Middle, adjunction) to produce it. So the comparison to just a sequence of opposed values isn't the same.

That's not (just) psychological inconsistency if one accepts the argument, it's a logical inconsistency. If one wants appropriate semantics for a logic to maintain it, adopt a paraconsistent metatheory.

This is hardly what one might call the semantics of logical inconsistency, which requires two incompatible statements to be held simultaneously. But this isn't imaginable by definition.sime

By what definition? If one thinks the contradiction you mentioned (the LP) is veridical, then they hold it to be true and false simultaneously because there's a purported proof that it is. It's only unimaginable or incompatible "by definition" if your definition of imagination has the requirement of consistency in the definition you're using. But that's the very assumption questioning the law of Non-contradiction is challenging so it can't be used to defend the LNC on pain of circularity.
• 5.4k
can we be certain that the laws of logic are valid?

This isn't like being certain you have five fingers. It's more like being certain the bishop moves diagonally. Validity is defined by logic. How could they be invalid?
• 5.4k
can absolute relativism be logically acceptable?

Absolutely not. It contradicts itself in asserting its certainty.
• 13

I hear you lol. Logic is a dangerous and inaccurate gun, but I certainly didn't mean to imply that it ALWAYS misses it's mark. Lucky shots (probably) happen.

I would say I only mostly trust logic, but it certainly is limitted by the accuracy of one's knowledge. Something may check out logically with the knowledge one has and still be totally inaccurate in reality.

If logic were a pancea for human knowledge it seems like we would have more convincing answers for people's questions instead of many arguments that are not very convincing. At least it seems to point towards logic not being an innate talent... but I'm sure that doesn't shock anyone.
• 13
Thinking about it, I think the absolute part of relativism is sort of implied; as relative relativism is rather redundant and nonsensical, bordering on a double negative, (I relatively believe in relativism vs I absolutely believe in it). I don't (as I understand it) believe relativism implies there are no absolute truths unless we are talking about ethics (in which case morality is said to be situational). An existence with NO truth makes NO sense... Which I suppose is only a problem if you think some things should make sense.

Am I mistaken in this understanding?
• 629
as relative relativism is rather redundant and nonsensical

well maybe this is solvable by using different words for the two levels. Maybe someone could suggest a n empiricist who already did so.
• 2.6k
I would say I only mostly trust logic,
Which is to say that sometimes you do not trust logic. Presumably you're referring only to situations where logic, of some kind, would be appropriate. So what are the conditions to which logic might well apply, from which you nevertheless dismiss logic, because you do not trust it?
• 663
Doesn't "meaning" presuppose identity (the 1st law)?
How would our talk have much meaning without self-identity (of some sort or other), including the posts in this thread?
Seems mostly like the only justification to abandon identity would be if we found that in the world.
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