# Proof that deduction is valid in all possible realities?

I was wondering, how can we know that deduction is a valid way to argue for something in all possible realities? How do we know that, in some alternative universe, something is not both ~P and P? How can we know it is valid in our reality? It certainly sounds impossible to us, but in quantum physics, it is accepted that electrons may spin up and down at the same time.

I am studying the philosophy of science and realism. I am curious if an antirealist would have grounds to reject deduction.

Is deduction true in all possible realities? If so, how do we know that?

• Where did you find that electrons may spin up and down at the same time? I think that's not correct. – Jo Wehler Aug 18 '15 at 19:54
• It was in my linear algebra textbook. I also discussed it with a couple professors concerning deduction, they told me to look into quantum logic. – zagadka314 Aug 18 '15 at 19:58
• @jobermark Schroedinger's original presentation of the cat-example is in §6 of "E. Schroedinger: Die gegenwärtige Situation in der Quantenmechanik. Die Naturwissenschaften, Heft 48 (1935)." It is well-known that Schroedinger was a sharp critique of the Kopenhagen interpretation, I recalled in my comment implicitly. - Nevertheless I do not see a difference between saying the spin is undefined and saying it is superimposed. Besides, the latter parlance seems a bit cloudy to me. But I know to which mathematical property of the state description in quantum mechanics it alludes. – Jo Wehler Aug 18 '15 at 20:52
• For Quantum Logic, see : Quantum Logic and Probability Theory. – Mauro ALLEGRANZA Aug 19 '15 at 7:56
• It is unquestionable that photons may be in a superposition of two states. But that does not mean that the photon has all properties which can be measured when it is in exactly one of the states. Other authors conclude from the self-interference result: A photon in a superposition state has no location at all. - In any case, many thanks for the interesting reference, – Jo Wehler Aug 19 '15 at 10:11

Deduction does not necessarily mean classical logic, but something more basic, which does not necessarily embody the rule of non-contradiction.

We use deduction rules in Universal Algebra or Category Theory for instance, which have little to do with classical logic, but help us to keep formal operations in order. The result does not necessarily establish truth or falsehood, but helps us decide when two items in a formal system are equivalent under a certain set of transformations.

You can look at these as logics with whole mathematical models like groups or rings or algorithms or other categories as truth values. But you don't have to. An application that is perhaps a bit broad is that of abstract semantics of computer type systems.

In that sense, some variant of deduction has to apply if you have grammar and sequences.

An anti-realist might easily imagine we live in a reality where grammar or time are not reliable touchpoints, don't behave consistently, or lack real meaning. This makes deduction still possible, but not trustworthy.

In fact many theories of meaning like those of Desaussure and Lacan insist all grammar is only approximate and not fully shareable between people, or is relatively constructed.

And some philosophical approaches to physics question whether time is real or only an aspect of perception -- most famously, Kant, and one version of the theories of Boltzmann.

@virmaior asked for an example, and the examples are fun but really boring to describe. I will try to describe one from 'fully abstract semantics', a Chomskyian linguistics model that gets adopted by computing. I hope it somewhat appeals to human beings.

One place deduction rules apply to something other than logical truth are in looking at abstract grammatical transformations (especially in computer languages, but also in human ones). The rules take us from one sentence to another, just as one might expect, and preserve meaning. Since this is not quite a logic, but only a deductive system, meaning has a broader sense than 'true, false or otherwise', and includes things we might really consider meaning.

It can be true that meaning is preserved but rendered very hard to access. So you can make a subcategory of 'reasonably efficient' such transformations. (Where reasonably efficient has to not penalize the steps in the grammar itself, so Big-O or something similar.)

It can be the fact that a transformation preserves meaning in way that sometimes is and sometimes isn't efficient, depending on the route through the transformation system. So important cases are characterized by the local equivalent of P & ~P. You can efficiently parse the statement as long as you don't take a certain detour, which will make you balk at it.

In that sense the system both does and does not handle the situation. You can address these by tuning the rules, or you can use the cases in a given language to explain why people choose given grammatical transformations over others or why grammar changes in some ways and not others.

Finding such situations is important when the language is artificial and the equivalent of determining equivalent meaning is that a given variable's type matches a function signature. (The theory arose in verifying the type system in 'ml' was not impossible to compile.) The compiler should not spend an inordinate quantity of time merely matching types and not actually compiling any code. (A real fear in the early entries in the class of languages that eventually led to Haskell.)

It has also been used to prove whether a given optimization is helpful or not. (The optimized code computes the same thing, but should do so more efficiently and not less, but the compiler should not get lost in the weeds trying to figure that out.)

(These examples are from lectures on abstract semantics by John Gray at the University of Illinois in the late 80's and early 90's, which may or may not have ever reached publication.)

If you focus on the application to a real language, there are lots of more subtle ways of looking at meaning than logics allow.

You might include considerations like the later Wittgenstein's, where meaning is focussed on given motives and segmented into 'games' in a way that does not allow all sentences to combine and reduce to truth-values, but only to 'moves' in the game.

A lot of the power of arguments in the "Investigations" derives from determining when consistency does not apply and does not matter -- where P and not P is not relevant due to the sophisticated equivalent of a category error. And I would not even consider Wittgenstein and anti-realist.

More pragmatically you can inject the idea that comprehensibility is contingent, and that incomprehensible statements still have 'best accessible meanings', so reality is constructed by your statements whether you are understood or not.

Either of these remains a world where deduction matters, but does not constitute a logic, and where consistency is not necessarily important. Since both are reasonable, we might just live in such a world.

• This is an interesting answer, but it leaves me with several questions. (1) can you give an example of a system of thought that doesn't incorporate non-contradiction and is non-trivial? (and preferably not about QM since that's the question here). (2) Can you defend the claim Kant is not a realist about time? The "Refutation of Idealism" is on most interpreters views a realist account of time even if it says we always experience time as part of the manifold of sensibility. – virmaior Aug 19 '15 at 2:51
• @virmaior 2) No. Well only partly... I meant an anti-realist could start from Kant. And, yes, only partly, I think part of Kant's agenda in making time a human aspect is to allow for miracles and escape the opprobrium of his church. In that sense every orthodox Christian of previous eras has to be to some degree an anti-realist. 1) I will work on finding one that is not a fluffy paraconsistent topos no one can comprehend, but if I fail, those count. – user9166 Aug 19 '15 at 12:43
• Fair enough. Both of those responses make sense to me. – virmaior Aug 19 '15 at 12:54
• @virmaior: the easiest suggestion is an alternative description of infinitesimals as an infinitesimal rod; this is closer to Newtons methods of fluxions when thought geometrically; ie a tangent vector; one can see the contradiction here: it has zero extension - it is infinitesimal; but it also has extension - it is a rod; this, I think isn't actually how non-contradiction manifests itself when thought formally; but it is a very useful stand-in. – Mozibur Ullah Aug 22 '15 at 16:30
• @MoziburUllah That is a reasonable example, but it belongs on a question with a different definition of deduction. If all you want is 'there is a place where 'P and not P', yet the world does not fall apart, ask any lawyer. Law is totally overdetermined, and works principally by navigating contradictions, rather than resolving gaps. If the law of non-contradiction applied to Law, we would no longer need attornies. – user9166 Aug 23 '15 at 4:08

The usual logic which includes the principle of non-contradiction is our base of thinking about the real world and all possible worlds. It has been proved successful in explaining our world.

In addition, no logic calculus has been established which successfully rejects this principle.

Apparently we cannot know, whether the principle also holds in some alternative universe or “in all possible realities”. Because up to now we do not know anything about such universe. And the only way we can speculate is by using our logic calculus and by employing the principle of non-contradiction.

• What do you think of paraconsistent logic? – Moritz Aug 18 '15 at 21:14
• @Moritz Do you know a reference to a formal calculus of paraconsistent logic? What are the axioms, what are the theorems? – Jo Wehler Aug 18 '15 at 21:33
• You can see Paraconsistent Logic : paracons logic reject the "principle of explosion" (ex falso quodlibet) : {A , ¬A} ⊨ B but usually validates the Law of Non-Contradiciton : ⊨ ¬(A ∧ ¬A). – Mauro ALLEGRANZA Aug 19 '15 at 7:38
• @Mauro Allegranza Does this mean that paraconsistent logic accepts the principle of non-contradiction like standard logic but rejects the principle ex falso quodlibet? – Jo Wehler Aug 19 '15 at 8:48
• According to Stanford's entry : "many paraconsistent logics validate the Law of Non-Contradiciton (LNC) even though they invalidate ex contradictione quodlibet (ECQ)". – Mauro ALLEGRANZA Aug 19 '15 at 8:54

By deduction, I guess that you mean the modus ponens: knowing that A holds, and knowing that if A holds then so does B, one can deduce that B holds.

Modus ponens is what allows you to "link" different kind of properties to each others, acting like a bridge between what one knows or considers true (A, A implies B) and some unknown fact (B). It is therefore the very tool allowing people to prove things, so don't ever expect to be able to prove it. Proving some form of modus ponens would certainly imply that you used it at some places.

The answer to your title is therefore quite simple: you cannot prove it, and you shouldn't even expect to prove it. Fortunately, we are smart enough to understand that all reasoning is based on the modus ponens, and therefore we better believe that it holds if we ever want to conclude or deduce anything from what we know.

One suggestion might be via Lewis's Plural Worlds or Kripkes Possible Worlds; one could ask do modus ponens always hold in every world; I would suggest it does; but I'm no expert in his thought.

but in quantum physics, it is accepted that electrons may spin up and down at the same time.

P and ~P then would be of the form

`P = an electron spins up and down at the same time.`

`~P = an electron is not spinning up and down at the same time.`

Still only one can be true at once. For our reality at least deduction seems to hold. For other realities, who knows, there really is no way for us to tell. It seems like it is impossible for us, based in this reality, to imagine a round square, or a square circle, for example.

A reality that does not follow the rules of deductive logic might be impossible for us to visualize in any form. As for it existing, that is a whole different can of worms.

• But the P in question is (or at least seems to be) a contradiction by itself if we assume "up" and "down" to be contrary states, so yes, we have either P or ~P with that definition of P, but then by substitution we have either (U & ~U) or ~(U & ~U). Which means either a world where a contradiction is true (whatever that would mean) or a world where the law of non-contradiction works. – virmaior Aug 19 '15 at 2:02
• @virmaior I won't pretend to know how quantum mechanics work, but I know on an subatomic scale things behave in what seems an odd fashion. I have heard accounts of electrons spinning both ways or holding seemingly contradictory states simultaneously. However they are also capable of holding just one of the states (spinning just up, or spinning just down) but they cant hold just one of the states, and both of the states simultaneously, at least in so far as I have heard in quantum mechanics. – hellyale Aug 19 '15 at 2:17
• Here is a source : matterandinteractions.wordpress.com/2012/12/04/… " Thanks to major theoretical and experimental work over the last few decades, we know for certain that until Alice makes her measurement, her electron remains in a special quantum-mechanical state which is referred to as a “superposition of states” – that her electron is simultaneously in a state of spin up AND a state of spin down. This idea is very hard to accept. " – hellyale Aug 19 '15 at 2:21
• There's a few different interpretations of superposition, but that's not particularly relevant to the comment that I made, which is solely about the use of logic in your answer. Or rather, logically consistent interpretations don't assert that an electron's spin is up and is down but rather that its spin is a superposition of up and down. – virmaior Aug 19 '15 at 2:45
• But it can't be both a superposition of up and down, and not a super position of up and down... How did you get to be a moderator again? – hellyale Aug 19 '15 at 4:48