If logic is constructed from axioms, and axioms are depended on observation which in term could be subjective, does this means that logic could be limited to our observation, and not really absolute and fundamental?

  • I'm intrigued by the claim "axioms are dependent on observation". Where are you getting this claim from?
    – virmaior
    Jan 20 '16 at 5:11
  • How do you arrive at an axiom?
    – ecorvo
    Jan 20 '16 at 5:12
  • 2
    In general, why must axioms be "arrived at" at all? Axioms are posits by definition and could be posited for any of a number of reasons. You've tagged your question philosophy of mathematics are asking specifically about axioms in math or do you have something broader in mind? (All of this needs to be resolved before an answer can really be given)
    – virmaior
    Jan 20 '16 at 5:15
  • 1
    Well I guess it boils down to how we arrive at axion. You say be definition in order to define it an observation must be made. So in essence how we make such observation can be questionable, can it?
    – ecorvo
    Jan 20 '16 at 5:19
  • "Logic" is a vague term. For example, when Spock in Star Trek used the term it rarely involved axioms and syllogisms but rather wisdom, which is subjective.
    – WGroleau
    Jan 21 '16 at 13:08

Many have argued that logic is empirical, or as you describe it, logic's "axioms are dependent on observation".

Quine, in his paper "Two Dogmas of Empiricism" questioned the analytic-synthetic distinction, and suggested that even analytic propositions were dependent on empirical evidence. Since the rules of logic were analytic propositions par excellence, they too, were ultimately dependent on empirical data, and were not absolute laws.

Birkhoff and Von Neuman proposed in the 1930s that the paradoxes of Quantum Mechanics can be explained if we abandoned classical logic and used some form of Quantum logic instead. Such a Quantum logic would change or abandon all together some of the rules of classical logic, and would be a perfect case of logical axioms arrived at by observation.

Hilary Putnam discussed this in depth in his paper "Is Logic Empirical?", later republished as "The Logic of Quantum Mechanics.". In it he argued that, just as empirical physical results - relativity - forced us to abandon Euclidean geometry, so it is possible that the results of quantum mechanics will force us to abandon classical logic.

Although Quantum logic is still an active field of study up to the present day, it is does not get much attention from most philosophers and had been abandoned completely by physicists. Those who do study the topic view it mainly as a mathematical tool for studying Quantum phenomena, not as some sort of fundamental logic to replace our current classical rules of logic.

The main problem that is faced by Quantum logic (or any such radical revision of logic, empirically justified or otherwise), is that we tend to think and communicate in classical logic. It would be very difficult, or in Kantian fashion, outright impossible for us, to perceive and discuss the world in anything other than classical logic - it seems to be hardwired into our brains. Although the logical atomist program failed as a metaphysical theory, it did show us just how ingrained classical logic is into our linguistic and mental structure. As Wittgenstein stated, the limits of language are the limits of the world: No one can place themselves outside of logic, and then pick among different logics to reason and argue with, even if those alternative logics are justified.

Those non-classical logics which have been successful (fuzzy logic, modal logic, intuitionistic logic) are those that extend classical logic, as opposed to replacing it, or at least respect classical truth tables in the limiting case.

As an after thought, one of my favorite Sci-fi short stories discusses the idea that while logic is indeed subjective, we learn classical logic at a very young age and once we grow into adults, we are incapable of unlearning it. If we were to somehow come across non-classical logics at a very young age, we would be capable of all sorts of superhuman feats. The story is of course, just sci-fi, but I do find the idea compelling.

  • 1
    If a logic is hard wired into us I do not think it is classical. Students' brains have to be broken over a knee to install the material conditional, and the law of explosion it begets is struggled with. People are also hesitant to apply excluded middle to undecidables, like future contingents. Dummett made an argument that the way logic is learned is best described via Gentzen's natural calculus, which is intuitionistic. projectbraintrust.com/cogburn/draustralasianpreprint.doc Classical logic is "wired" in the first year of college, or perhaps at school, but it's not nature's doing.
    – Conifold
    Jan 21 '16 at 20:58
  • "Brain logic" is something more like intuitionistic relevance logic rather than classical. But I don't think even that is truly hard wired, brain is known for its plasticity, the installation of classical logic is an illustration of that. Philosophers and mathematicians develop "working intuitions" for other logics to do their work and it diffuses (Searle claims that people disagree with him b/c they "unlearned" the "right" stuff). I think Kant overestimated the depth and scope of synthetic a priori on both geometry and logic.
    – Conifold
    Jan 21 '16 at 20:58
  • @Conifold yeah but intuitionistic logic "recovers" the same truth tables as classical logic, and is more of an extension than of a revision. The material conditional is more of a thing that laymen don't really think about before they are exposed to formal logic. Quantum logic on the other hand is truly weird: Even a trained logician can't really wrap their head around QL concepts like (p and x) or (p and y) != p and (x or y) . Jan 22 '16 at 2:03
  • IL is a subset of CL, IL theorems are CL theorems but not vice versa. It is non-compositional though, i.e. indescribable by truth tables, truth values of conditional and disjunction are not determined by truth values of the terms only, as in normal reasoning. QL is certainly way apart from both, but I suspect that a child raised around macroscopic quantum objects would internalize it, and a human society encountering them for generations would start teaching it at schools :) Once you delink from logic as Fregean bookkeeping of classical objects distributivity failing is not that hard to grasp.
    – Conifold
    Jan 24 '16 at 2:45
  • @Conifold "If a logic is hard wired into us I do not think it is classical. Students' brains have to be broken over a knee to install the material conditional, and the law of explosion it begets is struggled with". 1. You are making my point. The material conditional is the elephant alien in the room. - 2. No one's brain needs to be broken to accept Aristotelian logic. It is 100% natural. 3. The only "classical logic" there is is Aristotelian logic. Calling the material conditional "classical" is an insult to comment sense. Oct 31 '20 at 14:37

No, logic is not subjective.

In all mathematical theories all experts obtain consensus about the validity of the theorems of an axiomatized theory. But the theories differ and sometimes the Pros/Cons of a theory are debated. E.g., there are two-valued logics and many-valued logic and fuzzy logic and etc. It's not the question about the validitiy of the theorems of the theory. At most it is a question which axioms should be taken as beginning.

It is an insight of the last 200 years that axioms cannot be derived from any foregoing insights, neither intuitive insights nor from results of scientific discoveries. Which axioms to choose may depend on observation. But axioms do not derive from observation like theorems derive from axioms.

Instead it is often a question of usefulness which axioms should be chosen to develop a mathematical theory which fits as the foundation of a scientific theory. E.g., there was a discussion whether classical 2-valued logic is suitable for the interpretation of measurements on the quantum level.

Logic is not fundamental in the sense that there is excatly one calculus of logic. But it is fundamental in the sense that every rational argumentation, in particular any scientific theory, presupposes a certain calculus of logic.

  • The first sentence and the last paragraph seems contradicting to me.
    – Student
    Jan 13 at 14:17
  • Please explain which statement exactly seems contradicting and why. Thank you.
    – Jo Wehler
    Jan 13 at 17:23
  • "But it is fundamental in the sense that every rational argumentation [..] presupposes a certain calculus or logic." Among all rational argumentation I have seen, this is indeed true. However, it does not explain why all should be the case. To me, this leaves it possible that logic is still subjective. -- We just cannot think of any rational argumentation that does not require logic.
    – Student
    Jan 13 at 20:58
  • Logic is a tool to justify rational argumentation and making conclusions. Rational argumentation uses expressions like “because of “, “if … then”, “therefore” etc. Hence it is part of the definition of “rational argumentation” that rational argumentation uses a logic calculus.
    – Jo Wehler
    Jan 13 at 21:58
  • So logic is objective once we agreed on your definition of rational argumentation. But why should we agree on that? (Don't get me wrong, I do want to agree, only do I lack of an objective reason.)
    – Student
    Jan 13 at 22:08

While the answers above pretty much cover it, the question goes so deep into philosophy that it can be seen from many angles, especially in respect to the meaning of subjectivity itself.While I hesitate to say this, and may get brutally corrected, I would argue that Kant can be interpreted as saying that: Subjectivity itself is a logical system.

As has been pointed out, we can have different logical systems. We can drop Euclid's fifth postulate and generate a different, perfectly coherent logical system. The various systems cannot, so it seems, be reduced to one another. So it might seem that the "subjective" aspect is the choice of axioms. The "subject" can stand, so to speak, inside or outside any given system, choosing axioms.

But things are not quite so simple. What separates all these systems, making them irreducible to one another? The various choices of axioms and applications,... hence subjectivity? Yet again, what makes them all "logical" systems? Something beyond subjectivity? Something common to all possible subjectivities?

This is where Kant's transcendental approach might shine some light. We can think of any individual subject freely "choosing" axioms. Yet such "subjective" intervention is quite strictly limited by acceptance of its "logic" to other subjects, or it is simply a coherent madness... a paranoia.

In truth, we do not have such a thing as "individual subjects" or purely singular homo sapiens. We have "subjectivity," as a kind of evolving, punctuated continuum or discontinuous identity. Which is not unlike Logos evolving and differentiating along axiomatic boundaries or niches into various "logical systems."

So we may choose axioms "subjectively" and operate "inside" or "outside" different logical systems. Meanwhile, these Logoi grow and evolve. They outgrow their own axioms and their own "final conclusions" or "self-proofs." If they were to become "closed" systems they become purely tautological and die. So the systems themselves begin to sound not entirely analytic, but subjective or perhaps "synthetic a priori."

Now, the Kantian subject. Any given subject, such as they are, may choose axioms. But can they "subjectively" choose their way out of all logical systems? Were they to do so, they would disintegrate or close themselves up in a paranoia. They would, in effect, lose their very subjectivity. So, in some sense, "subjectivity" is what all these logical systems have in common... and what subjectivity itself presupposes. And this would be that categorical structure of reason, freedom, and morals proposed by Kant.

The difference is that we cannot identify or reason about some noumenal "axioms" of this metasystem. We are forever "inside" its relational structure. This then is the open system that generates axioms or, we might say, from which axioms are derived. So the answer is: Yes, logical systems are subjective, but subjectivity and sensibility are in turn constrained to a logical structure.


I would argue that logic and mathematics really are subjective, but only at a species level (or to the degree they are simply wrong.)

The way that axioms are dependent upon observation is not the same way that scientific principles or other facts are dependent. Axioms are not so much "discovered", or "worked out", but (as the name means in Greek) "found worthy", by being easily called forth in another person's mind, and appealing to them at a deep, intuitive level. It does not matter whether they occur in external reality because they exist in internal reality.

The purpose of disciplines like this is to isolate what understanding is common across all different ranges of experience. They do so by appealing to intuitive response, and the emotion of 'clarity'. They refine intuition for communicability, but they rely upon the intuition itself as backing. Since the only intuition we can interrogate is our own, we can only determine what is shared among those with whom we can communicate.

What is found to differ in a meaningful way between people is assiduously and ruthlessly removed from these subjects. The parts of language and processing that are more linked to the environment are purposively pushed out of logic into grammar, linguistics and philology, and ultimately to psychology, and the corresponding elements of potential imaginary models are pushed out of mathematics into the other sciences and engineering disciplines.

So logic strives to be non-subjective, but there is no way to test the relative subjectivity except between humans, and we can therefore never be sure.

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