We use the logical system that we know from observations (empirical data) holds true in the world we live in (please correct me if I am wrong). Hence the axioms of logic we choose are themselves dependent on our observations. Does this mean that logic is also limited to observations, and is neither the absolute or eternal truth nor fundamental?

I have currently learned that we have various types of logic. The two-valued logic teaches us for example:

1.The pot is red (A)
2.The pot is not red. (~A)

These are the only two cases possible in classical logic. But the logic used in the East before colonization was the multi-valued. In Buddhist Tradition Dīgha Nikāya provides an example. As the Buddha explains in the Brahmajāla Sutta, there are four alternatives:

(1) The world is finite, this is one case.
(2) The world is not finite, this is another case.
(3) The world is both finite and infinite, this is the third case.
(4) The world is neither finite nor infinite, this is the fourth case.
(5) There are no other cases.

The quantum logic has already shown that (p and a) or (p and b) is not equal to p and (a or b). The distributive law fails in quantum logic. Now if you say that we have to pick a suitable logical system for the particular area we are working in, then how can mathematics be the same everywhere, it will also be empirical then.

In quantum logic an electron can be both at Position A and B at the same time. Classical Logic does not permit it. When we prove something by contradiction we take advantage of the above condition. What I mean to say is we prove that if root of 2 is not rational then it can be irrational, or if root 2 is irrational it cannot be rational. But in quantum logic such proofs will fall flat.

Please See This Question : Is Logic Subjective

What I am not able to understand is: If logic can vary, how can Mathematics be universal?

Why Do Not We Allow Empirical Proof In Mathematics which gradually become more precise with each observation (The way it is in physics) if both the Logical System and the Axioms are themselves are dependent on our observation, they are based on our empirical observation?

EDIT : How the Distributive Law Fails ? (source)

p and (q or r) = (p and q) or (p and r), where the symbols p, q and r are propositional variables.

To illustrate why the distributive law fails, consider a particle moving on a line and let

p = "the particle has momentum in the interval [0, +1/6]"
q = "the particle is in the interval [−1, 1]"
r = "the particle is in the interval [1, 3]"

(using some system of units where the reduced Planck's constant is 1) then we might observe that: p and (q or r) = true in other words, that the particle's momentum is between 0 and +1/6, and its position is between −1 and +3. On the other hand, the propositions "p and q" and "p and r" are both false, since they assert tighter restrictions on simultaneous values of position and momentum than is allowed by the uncertainty principle (they each have uncertainty 1/3, which is less than the allowed minimum of 1/2). So, (p and q) or (p and r) = false. Thus, the distributive law fails.

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    it doesn't seem surprising if quantum theory can be expressed in alternative logics. what would be surprising to me is if it forced us to abandon the idea that all of reality obeys the rules of classical logic
    – user6917
    Commented Sep 10, 2016 at 23:21
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    @MATHEMETICIAN Aristotle already abandoned this idea, in his tomorrow's sea battle discussion he restricts the law of excluded middle, en.wikipedia.org/wiki/Problem_of_future_contingents Actually, the idea that all of reality obeys classical logic is fairly recent, it is an artifact of textbooks written after Russell's Principia. Natural reasoning significantly deviates from classical logic, e.g. on future contingents, conditionals, vague terms, etc., material conditional wasn't even invented until late 19th century.
    – Conifold
    Commented Sep 10, 2016 at 23:41
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    @Conifold Why do we still use classical logic ? I mean we should abandon it all together.
    – Suraj Jain
    Commented Sep 10, 2016 at 23:43
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    If you could , then it would i would really grateful to you. I really want someone to look at this paper and express his non-biased views. Paper is : ckraju.net/papers/Zeroism-and-calculus-without-limits.pdf
    – Suraj Jain
    Commented Sep 10, 2016 at 23:44
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    @MATHEMETICIAN Why do we always have to look at Plato , time changes and so does situations why are we still connecting maths to metaphysical domains dealing with infinite sets ?
    – Suraj Jain
    Commented Sep 11, 2016 at 0:03

6 Answers 6


Not exactly. Of course, there is a broad reading of "empirical", which includes anything somehow extracted from experience, upon which the answer is trivially yes. But on this reading God is also empirical because some people experience communicating with him. On the more conventional meaning of "empirical", the opposite of empirical is not necessarily innate, absolute, eternal or fundamental. In particular, non-empirical can be fallible. A law is empirical if it can be "derived" from observations by induction/generalization, like Kepler's laws from observing the heavens. The laws of logic are not of course unrelated to experience, or "absolute", but they can not be so derived. See also Is geometry mathematical or empirical?

At the dawn of modern psychology, in the 19th century, the opposite point of view was advocated by many of its founders, and even by some philosophers, like Mill. It came to be known as psychologism about logic, and some of the arguments were elaborations of yours. It died out after Frege and Husserl showed that it leads to inconsistencies. First of all, the certainty of logical laws, while not absolute, far exceeds that of any psychological laws. So the former can't possibly justify the latter. Second, if logic is "subjective" it is a miracle how we manage to communicate successfully while using it. Here is more from Husserl's Prolegomena:

"Extreme empiricism, therefore, since it only basically puts full trust in singular judgements of experience... eo ipso abandons all hope of rationally justifying mediate knowledge. It will not acknowledge as immediate insights, and as given truths, the ultimate principles on which the justification of mediate knowledge depends; it thinks it can do better by deriving them from experience and induction... If, however, all proof rests on principles governing its procedure, and if its final justification involves an appeal to such principles, then we should either be involved in a circle or in an infinite regress..."

In other words, "deriving" logic from empirical experience involves employing the logic itself in the derivation. After the downfall of psychologism logical positivists offered an alternative proposal, that logic is adopted by convention. It is interesting that it fails for essentially the same reason, as Quine pointed out in Truth by Convention:"In a word, the difficulty is that if logic is to proceed mediately from conventions, logic is needed for inferring logic from the conventions".

So if logic is neither empirical nor conventional, what is it then? First, we need to distinguish, after Peirce and scholastics, two different logics: "logica docens" (doctoral logic), and "logica utens" (logic in possession), of instinctive reasoning. Like Kant's "a priori" Euclidean geometry, the latter may very well be biologically innate, or imprinted in early childhood, while the former, like axiomatic geometry, is chosen and followed systematically. But even the logica docens is neither empirical, nor subjective, nor conventional. The word is "theoretical" or "constitutive". Like theoretical entities (quarks) and constitutive laws (Newton's second) logic is not observed, measured, or inferred from experiments, it has to be adopted prior to experiments to make sense of what is observed, measured, or inferred.

But while it is a priori, it is not eternal or infallible. "Falsifying" logic empirically is not exactly meaningful, but we may still choose to abandon it if it is deemed counterproductive overall using holistic (i.e. extra-empirical) criteria of theory selection, and adopt an alternative. So far, this did not happen with quantum logic, after a burst of interest in 1970s it faded into a niche activity. But there is a recent proposal, which would adopt it as the logic of quantum gravity substrate, out of which the spacetime emerges, see Raptis' Presheaves, Sheaves and their Topoi in Quantum Gravity and Quantum Logic.

  • @Suraj What I tried to explain is that "we use the logical system that we know in observation (empirical data) holds true in the world we live in" is not the case. It is not that it is false, it might even be "morally right", but it can not be made sense of in the way you are trying to. It's not that we do not "allow" empirical observations concerning logic, but rather that it is unintelligible what such allowing amounts to. We use logic to interpret physical observations. We would need it to interpret "observations about itself" too, but then it's prior to the empirical.
    – Conifold
    Commented Sep 22, 2016 at 20:43
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    Why Do We Choose One Logical System Over Other ?
    – Suraj Jain
    Commented Sep 23, 2016 at 1:49
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    So how can we take math as universe language . it is similar to other sciences and get regularly updated
    – Suraj Jain
    Commented Sep 26, 2016 at 7:54
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    @SurajJain What was refuted is the universal applicability of Euclidean geometry as a whole to the physical world, not the axiom of parallels in isolation. In fact, even that was not "refuted" exactly. We are not obligated to identify the light rays with straight lines, they can be interpreted as deviating from straightness due to physical forces (and such interpretations were offered). As Poincare remarked, only the pair geometry + physics is empirically testable, not each piece in isolation. Decisions between alternative interpretations are made on non-empirical grounds.
    – Conifold
    Commented Sep 27, 2016 at 17:26
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    @SurajJain Not everything comes from empirical. To even have empirical input some non-empirical priors have to be adopted. They are not "self-evident truths", etc., and their adoption implies no metaphysical commitments. It is not just logic and mathematics, even empirical theories have constitutive parts that are not "derived" from observations. It is well-known, for example, that Newton's second law has to be assumed to make the force measurements. Empirical proof is possible only within theories, whose constitutive parts are generated non-empirically but adjusted through long practice.
    – Conifold
    Commented Sep 30, 2016 at 16:50

The tetralemma can be viewed and handled as two contradictory pairs orthogonal to each other and then it can be treated in the dialectic in the usual way, the 'laws of thought' applied and so on.

The vital issue, the issue which is so simple that many philosophers forget all about it, is that the rules of classical logic can be applied only for true contradictory pairs A/not-A defined as a pair for which one member is true and the other false. So, in your case of 'The pot is red or not-red' one would have to be careful since it may the case that it is not either. For instance, the pot may not exist.

We see this sort of problem when Heraclitus states 'We exist and exist not'. Plato sees this as a contradiction but Hercalitus is saying that both halves of this statement are false on their own, so there is no dialectical contradiction. The misuse of Aristotle's rules for the dialectic is rife in academic philosophy and everyday life and it causes havoc, and it all stems from forgetting that for a dialectical contradiction one member of the pair must be true and the other false. Examine any metaphysical dilemma and you'll notice that it cannot be shown to be a dilemma because both of the horns may be false. This is the metaphysical point Samuel Butler makes in Erewhon, that the 'illogical mean is better than the absurdity of the extremes', but he forgets that the mean is not 'illogical' if the extremes are both false.

I'd say logic is empirical in a way. For instance, it is an empirical fact that all positive metaphysical positions are logically absurd. This may be verified by any logician. So is it an empirical fact, a subjective fact, a logical fact or all three? I suspect this is just a matter of convention.

EDIT: I came back to add that it seems like a mistake to imagine that we derive logic from observing the world. To do this we would have to use inductive logic. Logic is therefore prior to observation. If we were born deaf, dumb and blind we'd still naturally follow the LNC and LEM. It just stands to reason...


There are quote a few questions in your question, so I'm just going to amplify on one.

First, some-one, I forget whom, quipped that mathematicians take something, turn it into their own language and then it's something completely different. This, though a quip, has a kernel of truth to it.

Mathematical logic is different from logic per SE; merely by formalising it one is forced to make choices, and then later one can argue about these choices; for example, should one formalise the principle of 'explosion'? That one inconsistency renders all sentences inconsistent?

Logic has, historically several sources, and one of this is language, where we state propositions; here if I state an inconsistency, we are hardly going to say well, everything that I have said and everything that I will say will be inconsistent. Instead, we suppose I made an error - either deliberately, or unknowingly; what we see here is that logic is understanding how to reason correctly.

Another sense, as pointed out by Heidegger associates logic with ontology; this was at first by Plato, and then much later and in a different form by Hegel. It's this echo of ontology in logic that probably leant its support for the principle of explosion in formal logic.

To return to the main point, once logic is formalised we can look for resemblences in a merely formal manner - and this is a mode of thinking that wasn't open before; further, we can drop or add formal laws as we see fit, and this again a new possibility.

This is how the old quantum logic - by Birkhoff and Von Neumann was discovered; the new quantum logic isolates certain features of QM and thinks them through categorical logic; interpreted, they are a logic of processes - and here phenomena as no cloning, teleportation, or entanglement become more perspicuous.

The point is that formal logic, merely by its formality, requires interpretation for it to make any sense; and this might be a tenuous connection to the classical concepts of truth and falsity - which might help us make a renewed acquaintence with these somewhat jaded and out-moded concepts - or just as possible, be more offhand with them.

  • Why Should we relate mathematics to metaphysics ? I guess mathematics have practical value in the real world and that is what most of us needs .I heard somewhere that Mathematician Do mathematics for mathematics itself . How right is that ? I mean first we have to choose a logical system , and here due to historical influence classical logic is formalised with some changes here and there and now mathematics is being taken to a metaphysics domain dealing with infinities , why shouldn't be mathematics be restricted to real world applications ? Why should not we consider other logical system?
    – Suraj Jain
    Commented Sep 11, 2016 at 0:00
  • According to Heidegger, this was due to Plato attempting to discerning constancy in flux of the everyday; later, he turned away - or began to question that too as a method. Commented Sep 11, 2016 at 0:12
  • So why should we even teach that now, it is just like we are so busy solving a problem , generations of us are trying to do so , we are teaching students in university about that , they are finding it hard to understand and nonetheless we are still teaching them? But the fact is problem is imaginary , it did not have to exist at first place
    – Suraj Jain
    Commented Sep 11, 2016 at 0:18
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    It's an old problem, not a new one; Parmenides complained he had to wade through a huge heap of nonsense to find something of genuine truth and lastingness; this is a truth that is true in every age and epoch; maybe even more so in age of abundance of so called knowledge that is the internet age: a giant heap of sophistry, mostly; a theatre of spectacle and echoes and mostly echoing each other... Commented Sep 11, 2016 at 0:24
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    The foundation of mathematics is metaphysics as, for instance, Hermann Weyl makes clear in his book on the continuum. Meanwhile the tools of metaphysics are logico-mathematical. So the two are impossible to pull apart. .
    – user20253
    Commented Aug 5, 2017 at 18:37

The real numbers are a purely theoretical construct. While they were motivated by "real world" concerns, they exist completely independently of it. There is no* empiricism in their study.

However, they are a tool that can be applied to describe features of physical theories, and how to do so could be said to be empirical.

Similarly, lattices of logical propositions are purely theoretical constructs, but they can be applied to describe features of physical theories.

That's all that's going on here. It's not very different from doing geometry via vector calculus rather than by manipulating coordinates.

*: Not strictly true; empirical methods are a part of the mathematician's toolbox. (e.g. gathering numerical evidence to formulate a conjecture)

  • What I mean is , we decide rules for the game of mathematics to model the real world . I mean we first perceive things in real worlds , and then we according to them make axioms and logical systems. Axiom like only one line pass through a point can be parallel to another line are all we see through our eyes. We make logical system generalizing it so that we can guess the unknown with what we have known so far. What I really wanted to know is why do we then reject empirical proofs in mathematics? When the foundation of any science , no conditions apply , any science is empiricism.
    – Suraj Jain
    Commented Oct 10, 2016 at 8:21

if logic can vary, how can mathematics be universal ?

"Logic" varying does not mean that it is unreliable. What we have seen in the question is that there are different forms of logic which correspond to different mathematical structures. Contradictions only arise when one attempts to identify one with the other. Mathematics is thus universal in describing these structures unambiguously and finding their links.

  • Why should that which depends on "empirical things" not be universal? Commented Jan 8, 2017 at 16:11
  • It would not be fundamental then , i mean then it could not be consider more correct than empirical proof , like in physics we experiment , then saw something , develop theory , and then predict , if fails then develop new theory, i mean constant changing . Then math would be like physics .
    – Suraj Jain
    Commented Jan 8, 2017 at 16:15
  • Theories in physics are said to "fail" when their application (in a specific manner) to the world around us does not correspond to our experiences. It doesn't mean that those theories are "flawed" in any way, unless they were logically inconsistent. Why would something that depends on "empirical things" not be fundamental? Axioms are fundamental and yet their application may depend on "empirical things". Commented Jan 8, 2017 at 16:19
  • what do you mean by logic , and where does from axioms come ?
    – Suraj Jain
    Commented Jan 8, 2017 at 16:19
  • By logic I refer to the rules of whichever form of logic one decides to use, and axioms do not come from anything more basic; by definition they are axiomatic. Commented Jan 8, 2017 at 16:21

Excuse me, but first I should correct this statement:

In Buddhist Tradition Dīgha Nikāya provides an example. As the Buddha explains in the Brahmajāla Sutta, there are four alternatives: (1) The world is finite, this is one case. (2) The world is not finite, this is another case. (3) The world is both finite and infinite, this is the third case. (4) The world is neither finite nor infinite, this is the fourth case. (5) There are no other cases.

Buddha in Brahmajāla Sutta didn't say there there is four alternatives about World's finiteness, he says that some recluses and brahmins have have such and such opinions. Difference is, that Buddha describes other's opinions, and not stating his opinion. In this tetralemma position is nothing particularly Buddhist, this is just Indian logic at the time (see Catuskoti).

Why it's fourfold and not twofold (as is usual in A and ~A). Because, it is logic of natural language and not formal logic. In formal logic we have rule of identity to be also represented in language, so we can state that what is written as A is always meant as A (and vice versa), so there is only ~A possible as alternative (in formal writings). In natural language, we don't have such important convention, so it is possible to write that thing is A (in some sense), and in same time not A (in other sense), where sense of such writing should be derived implicitly. Non-formal writing would look contradictory if we miss context (i.e. if we misinterpret it). As a consequence, there is four alternatives instead of two to cover all possible 'syntax'.

For Example Let us take the quantum logic , in quantum logic An electron can be both at Position A and B at the same time. Classical Logic does not permit so.

Why it does not permit so? Just assume electron can have both positions and voila, it's permitted. Logic does not say that if electron have position A it can not have position B (it does not know anything about electrons). It's you saying that. If we put not valid propositions into logical inference it will produce not valid conclusion. It is not guilt of logic, because logic is method for avoiding mistakes in inference. Logic warns you to take correct propositions before inferring, so, if you put object which can have two positions and then assume it could have only one, this is your fault, not of classical logic.

If you can not construct mental object for which A=A is true, you can not put it into logical inference, because you breaking identity rule. Logic can not infer anything you would wish, it can only infer some valid things if you obey it's rules.

Logic also does not vary. Why there is multiple Logics then? Because this is multiple formalizations of logic and not (variance) of logic itself.

And finally, if A is bigger than B and B is bigger than C, is A bigger than C? When you know this is it empirical to you?

  • In India mathematics was done without formal proof and done rigorously , why do we now have to deal with proof ?
    – Suraj Jain
    Commented Sep 18, 2016 at 3:25
  • @SurajJain "And Please See This: ...Quantum_logic" At what statement of the article do you want to point at? I read that the main difference to classical logic is that the distribuive law does not hold. Does thisa contradict to the post of catpnosis? I can't see this.
    – miracle173
    Commented Sep 22, 2016 at 7:22
  • @catpnosis What an excellent summary! Your comment on the Buddha is important and what you say about logic has given me a new insight into why Aristotle's law of identity is necessary. Your point about electrons can be put more formally. For a dialectic contradiction one of A and not-A must be true and the other false. If the electron can be at two locations at once then clearly the two locations cannot be represented in logic as A/not-A. Aristotle's logic is foolproof since his instructions for their use ensures that the world can never break the rules. And knowledge is NEVER empirical.
    – user20253
    Commented Aug 5, 2017 at 18:52

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