# Formal logic - a priori or a posteriori

I am aware of the classical classification of logical calculus as apriori. I have also read pretty much anything I could get my hands on regarding "logic", including "New Essays on the A Priori" edited by Paul Boghossian and Christopher Peacocke.

These readings often mentioned the idea of a deductive inference being carried out without the need of relying on any empirical knowledge. But I am still not convinced about the origin of these, so to speak, rules.

Let's take a basic "modus ponens" as an example. Whenever I know that "if p, then q" and "p", accepting "q" is so much imposing on me, I am not able to deny it. But what exactly does that imply?

Is it somehow within human nature to understand, accept and use deductive inference? Would a hypothetical human being that has NO empirical knowledge (let's say, a human being that has no senses whatsoever, only supplied with raw data directly to his/her brain) still be able to carry out logical inferences?

Or, is "logic" developed on the basis of experiments, exactly the same way any other natural science is? This view would accept as genesis of "modus ponens", the scenario, where i.e. some ancient greek observed that since whenever he threw an apple up, it fell down eventually ("if p, then q"); and because he'd just threw an apple up ("p"), it surely will fall down in a moment ("q"). Our greek repeated this experiment, say, 100 times, and so he'd formulated a law. It's an inductive reasoning.

I am curious about both points of view and maybe not so much about the "classics", as about personal understandings.

• Unfortunately "brain in a vat" theories like your middle paras aren't exactly testable. Commented Mar 7, 2021 at 14:34
• Related enough philosophy.stackexchange.com/questions/79413/… Commented Mar 7, 2021 at 14:43
• @Fizz I've read this discussion, which deals with modelling of the human reasoning, but my question is about its primary origin. Commented Mar 7, 2021 at 15:42
• Mkay, it seems to come down to a debate/discussion on Platonism vs psychologism (or something like that). Commented Mar 7, 2021 at 15:55
• @Fizz That's an interesting view, possibly grounding my a priori option, however I can't seem to sufficiently grasp the idea of logic being "grounded on human activities of [...]". Do these activities require any knowledge, fundamentally? And if so, what knowledge? I guess I need to read Wittgenstein's "Philosophical Investigations"... Commented Mar 7, 2021 at 17:40

This subject comes up quite a bit in different forms. Sometimes as the question, Is logical empirical? Or, What justification can be given for using logic? Or, Is logic subjective? There is a distinction between a particular system or calculus of logic, of which there are many, and speaking of 'logic' in a general sense. So, one should distinguish between asking, What is the justification for using this system of logic as opposed to that one? and asking, What justification is there for deductive reasoning generally?

You might like to look at Conifold's answers to these questions:

Is Logic Empirical?

Is there a deduction analog to the problem of induction?

and my answers to these questions:

What justifications have been given for using particular systems of logical calculus?

References for the justification of the use of Logic

• I appreciate these references very much, I'll dive into them and then, if some questions arise, maybe comment again here.. Commented Mar 7, 2021 at 20:18
• Your answers are very informative, thank you again for your time. And yes, it does pretty much exhaust the topic (at least for me). But what striked me the most, was the citation from Quine's '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". Commented Mar 8, 2021 at 16:37

What does "a priori" truly mean? To get at something so fundamental to thought, we must look at what is physically happening as you think. From a physical perspective, your brain's neurons have certain connection weights, which cause them to activate with varying intensity, and you make inferences and assertions as a result of these weights and activations. When you believe a proposition, or accept an inference rule, the inference rule or proposition must be stored as a pattern of neural connection weights. This connection-pattern causes you to behave in a certain way - if you have a connection-pattern corresponding to a proposition P, and then someone asks you, "Do you think P?", the neural connection weights for P may cause you to answer, "Yes, I do."

So this is our primitive notion of a believed proposition, inside the brain: it is a pattern of connection weights, linked to and causing a pattern of behavior.

Now, how did this pattern for P come to arise? Here we want to distinguish two cases. In the a posteriori case, the pattern for P arose as a result of inference in your brain. You at first believed Q and R ("had the brain patterns for Q and R"), and your attention was called to Q and R ("Q and R activated together with your brain's attention network"), and as a result, a precursor pattern to P arose (a neural activation pattern, part of short-term memory), and as a result of that, your brain acquired the connection-pattern P. Or we may say more concisely, from Q and R you inferred P.

The a priori case is simply any way you acquired P other than the above process. Perhaps you were born with the connection-pattern P. Perhaps you were not born with it, but your genes caused you to form it at a certain age. In these cases, you never deduced P from anything else - you simply knew it implicitly.

Now to apply this to your specific question. Is modus ponens a priori? This depends a bit on what you mean by "modus ponens." The brain does seem to come with some kind of inference baked-in. Even a rabbit has neural patterns that can be loosely described like, "There is a carrot over there. If there is a carrot over there, I can get the carrot by walking to it. Therefore, I can get the carrot by walking to it." The rabbit does not do this verbally - this is a loose description of a nonverbal neural process. But the rabbit (or any animal) does somehow represent the world, and draw deductions between different ideas about the world, in order to act effectively to obtain food, and these deductions could be described loosely in words as something like modus ponens.

So by that interpretation, the rabbit does a priori know a kind of nonverbal modus ponens, and so do humans.

Or we may interpret modus ponens a different way: as the specific symbolic skill, allowing us to take a material implication "A->B" (verbally) and an assertion "A" (verbally) and conclude "B" (verbally). Rabbits cannot do this. In fact, a lot of humans, even adults, don't understand the material implication either; see the Wason selection task. It must be learned, which makes it a posteriori.

• Regarding the rabbit example: basically, associative learning which is extremely widespread (if not the basis of most learned behaviors) is all about producing p->q types of rules in a neural network... and then applying them. Commented Mar 8, 2021 at 8:00
• Although more properly speaking this is Bayesian inference because rule derivation and application is probabilistic, i.e. not 100%. Here's one (open access) paper of the many one can find on this topic journals.plos.org/ploscompbiol/article?id=10.1371/… Commented Mar 8, 2021 at 8:09
• However, the Wason selection task is more complicated, as it involves both MT and MP. As I discussed in a linked question (that you asked), even non-human primates (never mind rabbits) are pretty bad at deriving MT-style rules, i.e. it seems to take a lot more neural network machinery/learning to produce MT rules than MP rules. It seems the extra hurdle for MT is that additional effort/coding is need for the fact that two conditions are mutually exclusive. Commented Mar 8, 2021 at 8:29
• @Fizz You are right that networks learn p->q type of rules. Some neurons are probabilistic, although this is not the same as Bayesian inference. Actually I would say that MT is just as natural as MP on a neural level. The chief way artificial neural networks are trained is with backpropagation, which has a similar structure to MT: the neuron produces a result, the result is partially wrong, therefore the neuron and/or the neuron's input is wrong (and changes in the direction of being more correct). Commented Mar 8, 2021 at 8:50
• @Fizz when the rabbit uses something like "modus ponens" though, this is not always so simple as supplying an activation A to a network whose weights represent A->B and getting the result B. Rather, A is a neural behavior pattern; neurons are activating in a certain pattern over time, which may involve temporally repeating spike trains. A->B and B are also neural behavior patterns. A->B specifically is a pattern with the property that when it appears in conjunction with A, the pattern for B increases in strength. Commented Mar 8, 2021 at 9:04