Right, spacetime a real concept, just like unicorn is. The fact that it's extremely useful separates it from the concept of a unicorn, which is not so useful. However, this just places it more like the concept of Santa Clause, or the perfect circle, a very useful concept. — Metaphysician Undercover

False, spacetime is real as in it's part of the model of physical reality as understood by both QM and Relativity. It's not merely extremely useful, the fact that it can be distorted by matter is a fundamental finding in modern physics. Ideologically-motivated rejection of this, a century later, is ridiculous.

No it doesn't it just gives us the means for modelling the effects of gravity. General relativity gives us no understanding of gravity itself, none at all. If it did, it could point us to the graviton. — Metaphysician Undercover

Whether or not a graviton exists isn't even understood. There are numerous problems in trying to even add the graviton to the Standard Model of physics (namely the current inability to renormalize certain results at certain levels of energy). Quantum field theory can explain gravity in terms of particle exchange but this doesn't work out at Planck scales so it's obviously not workable currently. GR's understanding of gravity is that it's not a force in the usual sense (just a feature of space in the presence of matter), so no gravitons are needed to explain what we observe at large scales. Is there more to learn? Sure, but that doesn't make the current models inaccurate, just incomplete. That's a non sequitur.

I've talked to many physicists, and your claims, that space-time is more than just a conceptual tool, is just not consistent with what these physicists tell me. You're just taking an extremely speculative metaphysical proposition, and claiming that physicists believe this proposition. Maybe some do. — Metaphysician Undercover

"I've talked to many" (likely just a few) is the textbook example of an anecdote. What I'm taking is the standard model and understanding of gravity and spacetime.

No, the structure was all F that E can be predicated of are E all F that G can be predicated of G, therefore some E is G. The predicates winged-horse and horse are not the same predicate. Hence it doesn't follow.

The issue of the arrow of time is well known in physics. The idea that it's not something that's ever been critically looked at or thought about by physicists is absurd. That sort of statement can only be made in ignorance. It's not a novel insight to say truths are conditioned by presuppositions. That's a trivial observation that holds everywhere, an actual point would be to show those presuppositions are false or questionable. I'll let you be the one to argue that physics has the wrong view about time and never critically examines it's assumptions.

Right, it's part of the model, not what is modeled, that was my point. It's theoretical like a perfect circle is theoretical. So we could take a model of a perfect circle, and map real things against it like the orbits of the planets, and see how they vary from the perfect circle. The circle is conceptual, the orbits are real — Metaphysician Undercover

Spacetime is modelled. Like what are you talking about? When I say it's part of the model I mean we have a set of propositions in a theory based on observable evidence which is closed under logical consequence. It's not just this background thing that is immaterial to the meat and potatoes, Spacetime is a real thing.

Actually, the model is deficient in its capacity to account for things like gravity and acceleration, so principles are added to allow for the model to be flexible. This gives the appearance that an aspect of the model, space-time is fluid, behaving. In reality the model just changes itself in an attempt to account for the things which it can't properly model. So if you happen to believe that space-time is a real entity, you'll believe that it changes according to those principles which have been added to allow for flexibility of the model. — Metaphysician Undercover

What? General relativity gives us an incredibly accurate understanding of gravity and acceleration. Spacetime is deformed by massive objects, the model isn't changing. That's a prediction of the model and one which is true. We have to account for the deformation of space by the planet Earth in order for our satellites to orbit properly. We literally observe this warping in distance pictures of galaxies because dark matter causes gravitational lensing, the distortion of Spacetime. Distortion is behavior, spacetime isn't somehow unaffected in the way you are insisting, no argument is given, nor evidence presented. Like come on, you're not giving anything serious to overturn the overwhelmingly minority position you hold as compared to physicists on the issue.

Right, the problem is the form in that the form doesn't guarantee that the conclusion is true.
[...]

But it's not impossible for the premises to be true and the conclusion false, because we can formulate a version of the argument where the conclusion is that some orange is bouncy. — Terrapin Station

But that's exactly the point. An invalid argument doesn't mean the conclusion is false, it means the form of the argument is such that the truth of the premises does not necessarily entail the truth of the conclusion. And as Darapti does not guarantee it's rightly deemed invalid.

It is a valid argument form in Aristotelian logic because statements of the form All A is B must have one or more instances in order to be true (just as with Some A is B). — Andrew M

Sure but as I said empty terms show this to be improper. As your quotesaid, Aristotle stipulated that logic was to regard known existing things and thus empty terms were off the table by fiat, not by argument. But this is kind of ridiculous. It's just true that, for example, mathematicians both use formal logic and do not assume that every entity they quantify over is instantiated. Thus if we followed Aristotle we'd handicap mathematics in pretty ridiculous ways. Logic ought to work just just the same regardless of whether or not there are instances of the things referenced.

But what would make it valid or not would be for a precedent to be set by a judge ruling on the case. Prior to that it's a contradiction in the law. One law says they're married (they hold the certificate) the other says they aren't (they're a gay couple), and it's the law which establishes who is married or not.

But as I understand it, in PWS the existential and universal quantifiers are understood within each possible world, while the necessity and possibility quantifiers are understood across all possible worlds. — Banno

This is correct.

If an individual or group sometimes has a given property, sometimes not, then it is a possible property. — Banno

That's actually the definition of contingency. Possibility is just defined as truth in at least one world. Necessary truths, for example, are still possible truths because they fit that condition. Contingency means true in some worlds but false in others.

If an individual or group of individuals has the same predicate in all possible worlds, it necessarily has that predicate: Bachelors are unmarried in all possible worlds. — Banno

This has always been odd for me. It seems like one could have a married bachelor. What makes one married is to hold a certain legal status, yes? Well consider a state of affairs where there's a contradiction in the local laws. Law A says "Yada yada Those holding a marriage certificate are married" and Law J says "Etc etc Gay people cannot be married". Now some gay person managed to get married (certificate and all), and there is no judicial precedent in how judge which law overrules the other. On the usual assumption that law decides what is true in these cases (because that's how we know who is considered married), one would seem to have a married bachelor.

For example, if B is "orange" and C is "bouncy" (and As are bouncy orange balls) it doesn't make sense to say that some orange is bouncy. — Terrapin Station

What? The problem with that is the argument is just the form, not the truth value of the two premises. All bouncy orange balls are bouncy, all bouncy orange balls are bouncy, but that does not imply "some orange are bouncy", it doesn't follow. Of course the sentence doesn't make sense but the relevant logical issue is just the use of an invalid argument form.

In the winged horse example, we're not positing properties where it doesn't make conceptual sense to say that one property somehow is or has the other property. — Terrapin Station

I'm not sure I understand you here. How does it not make sense to say that a winged horse is a horse, that a winged horse has wings, or that some horse might have wings? I don't see the conceptual issue here, these seem like perfectly comprehensible properties some object might have even if they do not in fact have them

No no no, the premises ARE true. That's what you keep ignoring. All winged horses are horses. So too do all winged horses have wings. An argument's structure is valid or invalid bases on whether it *preserves* the truth of the premises to the conclusion. So with the Darapti argument (All As are Bs; All As are Cs; Therefore some Bs are Cs) we go from true premises to a false conclusion. That is the very definition of an invalid inference. It's not a valid argument in classical logic, we know it fails to preserve truth in particular models.

You're not making sense. Previously you said this:

It can suggest what's true of the actual world just in case such and such is true, but it can't tell you that such and such is true. — Terrapin Station

Which means that given the truth of the first two premises (All winged-horses are horses; All winged-horses have wings) and the falsity of the conclusion (Some horses have wings) that the inference is invalid. The logic cannot admit the inference because it does not preserve truth in all models.

Logic is about the relationships of the statements qua statements. It can't tell you what's true of the actual world. It can suggest what's true of the actual world just in case such and such is true, but it can't tell you that such and such is true. You have to look outside of logic for that. Logic is only about relational structure per se, and really only about how we think about that on an abstract level. — Terrapin Station

That's not right. I am looking outside the logic, so I am deriving "this is true in the actual world because such and such is true". Winged horses do not exist. Despite the truth of the two premises, the conclusion does not follow. Ergo there is at least one model where the inference does not preserve truth. So it cannot be a valid inference. Semantic logical consequence is not a new thing I'm making up.

But logic is used to analyze what actually exists and infer things about them. If I am taller than Terrapin, and Terrapin is taller than aletheist, then I am taller than aletheist. I am (for arguments sake) taller than Terrapin, therefore I am taller than aletheist. That's a valid inference.

But if what is inferred to exist or be true is done on a basis which yields a false conclusion from true premises, that means the argument form was not a valid one. That's the very definition of (semantic) logical consequence.

Yes. As I said, we know the argument form is invalid but Darapti was regarded by Aristotle and the medieval logicians as valid. You cannot validity make an inference from a domain of things to that domain having members without stating it as a separate premise.

I'm not really sure what you're pointing at. All the top answers say more or less what I do. It's always taken to be continuous and positing a discrete structure to these things would require adopting a lot of speculative ideas and even ditching assumptions like Lorentz Invariance which by all accounts appears to hold. Not saying it's indisputable but it's so heavily favoring a continuous view that it's way more reasonable to hold that view currently.

We know that no animal is magical in what context? — Terrapin Station

I'm not sure I even understand you here. Magical in the sense that it exists or acts in some manner inconsistent with the laws of the physical world. The content is really irrelevant though. The argument does not come out valid whether you do logical metatheory semantically or syntactically.

But this is really a problem. Answer me: how do you know that a winged horse doesn't exist? Unless you define horse as being something wingless, you can't know if there is a horse with wings. — Nicholas Ferreira

On this basis one can never ever at all give a truth value to any proposition that does not contradict itself. In which case you've lost the ability to use logic for anything useful. We have abundant evidence that not only would a winged horse be biologically silly, but that they don't exist. We've been everywhere such a creature could be on Earth and it's not here. Unless you redefine what a horse is (which would lose the argument) then this is known to be true unless you think knowledge is impossible, which would be another problem. It doesn't make sense to me to be skeptical to the degree that the normal mode of discourse is obscured.

Why woudn't both be universal? Russell says that "No Greek are men" is the same of "All Greek are not-man". For me, it's clear that both propositions "all greeks are man" and "no greek are man (all greeks are not-man)" are universal ones. For it to be a particular one, it would need to use existential quantification and, therefore, assume the subject existence, woudn't? — Nicholas Ferreira

"No Greek is man" is just to say that it is not the case that some Greek exists and is a man. That's an existential quantifier, not a universal one.

It does, though. It's the same as "All silver toasters are toasters. All silver toasters are silver. Therefore some toasters are silver." — Terrapin Station

No, it doesn't follow otherwise one could not posit a counterexample. The universal quantifier does not imply existence, this is a known fact about the theory of quantifiers. Just think about an obvious example.

All chimeras are animals. All chimeras are magical. Therefore some animals are magical.

But we know no animal is magical. If there is a known counterexample to an inference it cannot be a valid argument form. It doesn't preserve truth in all models.

But the conclusion does not follow from the premises. We know in the actual world that the conclusion is false so we have a counter example to the inference form to show us that. If the form we're valid there would be winged horses. I'm saying something about the actual world makes that inference invalid, but that it's a way we can know it to be invalid.

You're treating the premises in a purely logical manner, but assessing the conclusion with respect to whether it's contingently true in the actual world. — Terrapin Station

Ok? The point is the argument form is invalid because it can take one from definitely true premises to a definitely false conclusion. In modern logic it commits the existential fallacy, it's suppressed premise ("There exists at least one winged horse") is clearly false.

Honestly, how do you know that winged horses are non-existent? I'm noy saying that they exists or that I believe that they exist, but you can't affirm that categorically only based on "no winged horse has ever been seen". — Nicholas Ferreira

Based on this one either can't claim to know almost anything or else you have to change our understanding of biology and what creatures exist on Earth in order to credibly say we don't know them to not exist.

Well, actually he says it on Logic and Knowledge (p. 229), and I think it's kinda weird.

"If it happened that there were no Greeks, both the proposition that "All Greeks are man" and the proposition that "No Greeks are men" would be true. The proposition "No Greeks are man" is, of course, the proposition "All Greeks are not-man". Both propositions will be true simultaneously if it happens that there are no Greeks. All statements about all the members of a class that has no members are true, because the contradictory of any general statement does assert existence and is therefore false in this case. This notion, of course, of general propositions not involving existence is one which is not in the tradictional doctrine of the syllogism." — Nicholas Ferreira

Note the bit I bolded. "All Greeks are man" is a universal statement, while "No Greek are men" is a particular. One is about an abstract domain not involving existence while the other is explicitly about existence. Russell says this just before the bit you quoted:

I want to say emphatically that general propositions are to be interpreted as not involving existence. When I say, for instance, 'All Greeks are men', I do not want to suppose that that implies that there are Greeks.

This is more an issue of syllogistic logic not modern logic. Syllogistic is supposed to be used for things known to exist and so it's fairly limited in a few ways including inferences involving empty terms. That's why the argument form is deemed invalid in classical logic despite Aristotle's logic deeming it valid.

Spacetime is literally part of the relevant models in physics. Spacetime has it's own behavior which is correctly predicted by current models, namely how it is deformed by massive objects. And I didn't separate space and time. Space is warped as is time, that's really the entire point of Relativity.

But the one-on-one correspondence procedure yields nonsense like Galileo's paradox — Devans99

Begging the question. You're saying it's nonsense because it results in infinities having the same size when you think it shouldn't, no argument given on your part. That's the very thing under debate, you cannot point to it as if it supported your point at all.

And the continuum does not have a cardinality... Cantor should never have made such numbers up. It's down to a deficiency in the core of set theory; the polymorphic definition of set supports two different object types: finite sets and descriptions of set. The first have a cardinality, the 2nd do not. They are different kinds of objects with different properties and need to be treated differently. Cantor tried to shoe-horn both objects into a common facade and ended up making up magic numbers for cardinality - definitely not the right approach. — Devans99

This is ridiculous and incoherent. The semantic method of defining a set is easily proven to be coherent. I definine a set P as the set of all red objects in my room. As it happens, that set has three members, so its cardinality is 3. This is just as coherent as defining a set P explicitly with the members {My phone case, my pen, an apple}. In fact, defining sets semantically is far more efficient and is used by people every day all the time. You give no argument that semantically defined sets don't have a cardinality and by any reasonable means they do have cardinality.

There is nothing "shoehorned" here, you object to it because it's patently obvious that semantically defining sets let's one define infinite sets as easily as one defines finite sets, e.g. the set N of natural numbers is defined as having every whole number 0 and greater as a member. No one is confused by what members populate that set, and the cardinality is infinite per Dedekind and Cantor.

Can you give an example of something illogical from nature/reality — Devans99

The point was logic is about abstract objects, it has nothing to do with nature. I didn't say nature was illogical, it's non-logical. Some models work better than others at explaining it, but that's the best one can do.

Yes but you cannot actually infinitely divide a line - it would take forever. So thats a potential infinity rather than an actual infinity you can describe at best geometrically. It's impossible to describe actual infinity geometrically, mathematically or otherwise so/as it does not exist. — Devans99

This is your fundamental confusion. No one is saying you can actually do a calculation an infinite number of times in a finite period because that's a process that is defined in terms of an activity done in small periods of time. But we know mathematically that the cardinality of the continuum is such that it can be put into a one-on-one correspondence with a proper subset of itself. That's the definition of infinity, hence a line (which is defined in terms of real numbers, i.e. the continuum) is infinite.

This categorization has absolutely nothing to do with some mechanical process. Your argument would be like saying the natural numbers are finite because I cannot count to a number called infinity. This is just a misunderstanding on your part on what these words mean.

Come on. Spacetime is not conceptual, not under any model in physics. You'd have to be seriously in denial to think models saying space is curved and correctly predict gravitational lensing and predicts that simulateneity is relative to reference frames is also saying that thing is not part of the world

But nature is logical so maths can explain it because it is logical also. Actual infinity is not a logical concept so does not fits in maths or nature. — Devans99

Nature isn't 'logical', logic is just useful in understanding nature when used appropriately. Mathematics is used to give us models of nature but no model ever really captures things perfectly because nature is just too complicated. And an actual infinity is perfectly consistent no matter how many times you just claim it not to be, whether or not it can be physically instantiated.

If we were to try that with a real line, we'd see discrete atoms. — Devans99

A real line is not composed of atoms or anything else so this is nonsense. A line is an abstract object, you have to investigate it's properties mathematically. And in basically any geometry you like a line is not finitely divisible.

If we start with the common sense notion that there must be more points/intervals in a large line compared to a small line then a continua immediately violates this with ∞ = ∞. Continua are illogical, reality is logical, hence continua don't exist in reality. — Devans99

Lines are not composed of points. In a real-valued n-dimensional space, points are defined by distance from the original. But crucially, *points don't have a width* so an infinite sum of points would never give you a line. Points can be used to mark the beginning and end of a line but they cannot define them.

Further, all you're really saying is "If we assume my position that everything is discrete, then opposing views are incoherent", which, well, who cares? You're position is less credible because you're not telling anyone why they should accept your assumptions and your criticism of continuum and infinities has yet to start with accurstely representing infinity.

Whats logical about ∞ + 1 = ∞ (implies 1 = 0)? In fact infinity is invariant under all arithmetic operations; what's logical about something that when you change it, it does not change? — Devans99

That does not imply 1=0. Finite additions to an infinite number *by definition* cannot change the sum. It's the definition of infinity that it does not change by finite modifications. And saying infinity is invariant under all arithmetic operations is patently false. Take the smallest infinite number Aleph-null. Now take the power set of Aleph-null. The Cardinality has increased, it's size is now that of the continuum, Aleph-One.

Um, what? 'All winged horses are horses' isn't trivially true because the term doesn't refer to anything. It's trivially true because the initial referent makes explicit reference to belonging to the category of the second referent (namely, a winged horse is clearly a kind of horse).

And I don't see how Russell would consider "All winged horses are wingless" to be trivially true. His description theory of names doesn't say sentence with empty terms are by default true, especially contradictory ones. They are deemed false in his theory because they must posit the existence of some thing (winged horses) but we know the thing to not exist. Nothing satisfies the condition "winged horse" so the translation of the previous argument into classical logic would have a suppressed premise, namely:

There exists at least one winged horse.

Which gets the value false, leading to a false conclusion. Have I misunderstood you or perhaps Russell's theory?

This has to mean that we have metaphorically categorize our peers in such a way that trancends matter, otherwise we wouldn’t be able to even be a ”consistent” person for more than 10 years post-birth. — Pelle

How does this follow? As long as one looks similar enough to how they did previously it doesn't seem strange that we'd still regard them as being the same person. We easily recognize the causal chain that makes them the same person as before and visually they aren't too different, so we say they're the same person. Cell replacement seems to be completely immaterial here.

What do you mean by "empty terms"? Are you refering to arguments with undefined variables? — Nicholas Ferreira

Terms without referents. No pegasi exist, so pegasus is an empty term.

But why is the conclusion false? I mean, I know that horses doesn't have wings, but it's inductive, empirical constatation, isn't? It's not logically impossible that a winged horse exists, unless you define horse as something that doesn't have wings. But, if this the case, then both premises are nonsense, because you would be saying something like "all winged things that doesn't have wings have wings". I don't know if i understood... — Nicholas Ferreira

It's false because we know it's true that winged horses are non-existent. Logical impossibility is irrelevant, this isn't a discussion about possibility or any other modality. This is about existence or non-existence. The first two premises are objectively true and the conclusion surely false. For logic the content or possibility of the premise is immaterial. What matters is that the truth of the premises does not entail the truth of the conclusion. If a counter example exists we know the argument form is not valid.

Because empty terms show this argument form to fail and thus Aristotle was wrong to deem it a valid argument, hence Classical Logic was right to distance itself from Aristotle's logic. Following from Russell, take this argument:

All winged horses are horses,
All winged horses have wings,
Therefore some horses have wings.

Clearly the first two premises are true but the conclusion is clearly false, we know there are no horses with wings. So this ought not be regarded as a valid argument in the logical systems developed after Aristotle.

Whose logic? Suppose my logic tells me differently than yours, thus leaving us in a situation where we both claim logic but do not agree? — Carmaris19

Then you're using a different logic and will have to determine which logic is to be applied. But outside very deep disagreements in technical math and philosophy the difference in what arguments are considered valid aren't going to have an impact in everyday life. Classical logicians and constructivist logicians are going to agree on basically everything outside mathematical logic discussions, for example.

If logic establishes validity shouldn't our logic align as say our senses of sight and touch often do when we agree on the color and firmness of a rock? It seems that disagreements on objective reality presupposes invalid logic on one end or the other unless truth and validity are meaningless. — Carmaris19

Um, a logic establishes the validity of an argument given a set of axioms and inference rules, it says nothing of our experience in the world being correct or not. If I see a color and say it looks more red than orange, and my brother says it looks more orange than red, we have surely not therefore made a fundamental disagreement about logic.

Yes, that's the scenario that is unintelligible. — Andrew M

Then in that case I don't think it can exist. As I said, i doubt inconsistent physical objects can exist (though I'm unclear how to regard the mind), but this seems distinct from the abstract objects I mentioned previously.

As far as I can see, there is no actual contradictory state of affairs in this example. There is the computer and it's program. There are various maps which are drawn differently, and there is the person who drew the maps. None of this is contradictory, is it? — PossibleAaran

The person's set of beliefs are, and beliefs are part of the mind. This would make minds the sort of objects that can have contradictory properties, no?

The content of my beliefs is contradictory, but there is still no actual state of affairs that is incoherent, is there? Let's try to make this clear. If you have found a case (instantiated in the real world) where the law of non-contradiction is false, then there must be some proposition you can state, about the world, which is both contradictory and true. What would that be? — PossibleAaran

Are your beliefs not part of the world? It would seem strange to regard one's set of beliefs as a fundamentally different type of collection than other sets. Whether dualist or otherwise.

What's wrong with the mental pictures definition? You say lots of states of affairs cannot be pictured. Could you give an example? I should note that the picturing need not be absolutely precise. I can't really mentally picture what the atoms which compose my laptop are like, but I can at least picture billiard balls interacting in certain ways, and perhaps picture billiard balls that have smaller parts that produce certain effects. I can picture that much, and I know that the atoms in my laptop are a bit like that. — PossibleAaran

I think you're trying to have it both ways here. You say it need not be precise but then what you're saying implies some unimaginable things can still exist despite not being properly conceived of. Conceiving of a useful alternate picture isn't really conceiving of the thing itself, just an analogue that suffices for some explanations but fails others. Examples could include any example of unobservables in scinetific theories, fields, geometric objects that are of infinite size (like a Euclidean plane), huge distances (can one really picture the expanse between Earth and the Sun???), etc.

As to the point about mathematics, I don't see why it is relevant. Let mathematicians define conceivability however they like for their purposes - I have no objection. But that they define it one way does not show that there is anything wrong with defining it another way for some other purpose than mathematics. — PossibleAaran

Well the point is there's no real way (even in principle) to conceive of nearly anything large or strange in mathematics. Infinite sets? Nope. Or just large numbers, even (say 10^10^10 amount of anything at all, totally indistinguishable pictorally from 10^10^11 of something else). Or weird algebraic objects like groups or rings. Conceivability in math is really about have a way to construct or prove things about these objects by means of formally established rules of proof. The mental picturing theory just can't work for anything outside of everyday finite counting and even then it hits a limit. To me conceivability needs to include this otherwise it's fundamentally incomplete a view, so the inconsistent objects do make it on if standard mathematics does.

i am unable to visualise or demonstrate a semantic notion of logical inconsistency — sime

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.

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? — PossibleAaran

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. — PossibleAaran

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. — PossibleAaran

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.

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.

...what? I don't think time or space are discrete, but even if it were this:

Assumption 2: Every physical process can be expressed mathematically.

Then it follows:

The logical framework that underpins a theory of everything must be based on natural numbers. This means, by the incompleteness theorem, that this system cannot be complete and consistent at the same time — Karl

Is incorrect. Logical theories are not based on numbers, number systems are reasoned about by logical systems. Further, scientific theories use real numbers (decimal numbers) just as much as of not enormously more than natural numbers. Calculus is all about the real numbers, for example. And even further than that, your mention of Gödel's Incompleteness Theorems is probably false. Even in physics, theories aren't fully mathematical and are more like quasi-empirical. Gödel's Incompleteness Theorems apply purely to formal systems capable of expressing arithmetic, a theory of everything is usually thought of as a theory sufficient to explain all of fundamental physics (say by a theory of quantum gravity to unify GR and QM).

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

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.

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. — PossibleAaran

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.

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? — Towers

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? — Towers

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. — Towers

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 — Towers

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.