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em·pir·i·cal / emˈpirikəl/ (also em·pir·ic) • adj. based on, concerned with, or verifiable by observation or experience rather than theory or pure logic: they provided considerable empirical evidence to support their argument.

From SEP

Relevant logicians point out that what is wrong with some of the paradoxes (and fallacies) is that the antecedents and consequents (or premises and conclusions) are on completely different topics. The notion of a topic, however, would seem not to be something that a logician should be interested in — it has to do with the content, not the form, of a sentence or inference. But there is a formal principle that relevant logicians apply to force theorems and inferences to “stay on topic”. This is the variable sharing principle. The variable sharing principle says that no formula of the form A → B can be proven in a relevance logic if A and B do not have at least one propositional variable (sometimes called a proposition letter) in common and that no inference can be shown valid if the premises and conclusion do not share at least one propositional variable.

To me this seems to suggest a conditional A → B is meaningless if there isn't a propositional letter shared by the antecedent and consequent.

Empirical Basis for the Meaning of the Connectives

Consider a child just learning to speak. Take a bowl, hold it up and say "this thing is the bowl.". Then take some apples and some bananas and put them all in the bowl. Then say, "if the thing is an apple then the thing is in the bowl and if the thing is a banana then the thing is in the bowl." Then take out all the bananas, and say "if the thing is an apple then the thing is in the bowl, and if the thing is a banana then the thing is not in the bowl." At this moment the child will understand the meanings of the words NOT, AND.

Now put a few bananas in the bowl, and say "if the thing is in the bowl then the thing is an apple or the thing is a banana." The child won't know if the meaning of OR is inclusive or exclusive. But now, without changing a thing, say "the thing is not an apple or the thing is in the bowl". He can then understand that OR is inclusive, since some bananas are in the bowl and also they are not apples.

At this point the child can figure out on their own that (if the thing is an apple then the thing is in the bowl) is simultaneously true to (the thing is not an apple or the thing is in the bowl). And they differ only in the presence of the words IF, OR, NOT. Thus the child can formulate the equation If A then B = not A or B. The point is, the antecedent and consequent have no variable sharing, but still the meaning of implication shines through.

In a related thread I was informed that the equation If A then B = not A or B does not hold in relevance logic.

So my question is, "which is better an empirical definition of the logical connectives, or a purely formal logical definition of the logical connectives, such as that used by relevance logicians?"

Note: To communicate the meaning of "if and only if" Take all the bananas out of the bowl, and say "the thing is in the bowl if and only if the thing is an apple."

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    Better for what and according to whom?
    – Conifold
    Commented May 19 at 11:37
  • @conifold, better for describing a changing reality, and according to everyone. All reasoning agents have equal access to the connectives. In the case of a child, they merely have to assign a word in a language they have no control over its agreed upon meaning. A temporal interpretation of the logical connectives is suitable for describing a changing reality, and no other.
    – lee pappas
    Commented May 19 at 12:14
  • "According to everyone" not even the sky is blue, and "best" descriptions change along with "changing reality". Think harder and spell out something more cogent.
    – Conifold
    Commented May 19 at 12:27
  • Maybe we should just assign meanings that people have no control over and be done? Would save no end of trouble. That's how engineering works. Social Engineering? Hmmmm...
    – Scott Rowe
    Commented May 19 at 14:47
  • Better for describing Reality? This is really Platonism VS Materialism. A materialist would say that the empirical approach would better, while a Platonist could say that the Pure Logic Approach is better. Commented May 19 at 23:03

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You are oversimplifying. You cannot learn the meaning of logical connectives just from a couple of simple examples. So your question is a false dichotomy. Depending on your preferred understanding of the epistemology and metaphysics of logic, you might regard logics as having a purely rational and a priori basis, or you might regard them as ultimately empirical. But one particular system of logic isn't more or less empirical than others.

Our understanding of connectives, and of logic generally, develops over many years of experience, combined with teaching and correction. It is quite a complex activity and it has been studied extensively by cognitive psychologists. In fact, there are many psychologists who have devoted their entire careers to understanding how people reason and use logic. There isn't any general consensus on the subject, probably because people think differently.

Systems of logic, such as classical logic, or relevance logic, are attempts to capture some of the main features of what constitutes good reasoning and to express them in a formal way. Ideally, we would like a formal system of logic to be simple, expressively powerful, syntactically and semantically unambiguous, to have a proof system that is sound and complete with respect to our chosen semantics, and to be decidable so that proofs can be generated and checked mechanically.

These requirements are in tension with one another. The more expressive a system of logic, the more difficult it is to devise a complete proof system for it. Simple systems of logic are decidable, but more complex ones are not. First order classical logic hits a pretty good sweet spot: it is adequate for a lot of purposes within mathematics and science, and it has proof systems that are sound and complete with respect to a simple model of truth. But it is only semidecidable: there is no effective computable process that is guaranteed to determine for a given formula whether or not it is satisfiable. Also, despite being fairly expressive, there are many things first order classical logic cannot do, which is why there are lots of extensions and variations of it. For example, higher order logics, free logics, etc.

In fact, the limitations of classical logic run deeper than that. There are semantic features of natural language that just don't lend themselves to being represented in a simple bivalent fashion. Our grasp of the real world isn't just concerned with what is true and false. There are things that are necessarily true, contingently true, possibly true, things I believe are true, that I know are true, that I wish were true, things that ought to be true, things that I promise to make true, things that are false but might have been true, things that are true to a certain degree, etc. Expressing all of this within a single system of logic has never been done and may well be infeasible.

The material conditional plays an indispensable role within classical logic, but it is a long way from being a general account of what we mean by if/then. The conditional in relevance logic is an attempt to capture what it means for one proposition to imply another in a more intuitive sense. In classical logic (A → B) ∨ (B → A) is a tautology, but it would be strange to suppose that for any arbitrary pair of irrelevant propositions one implies the other.

Relevance logic is only one attempt at the task of understanding conditionals and it comes with its own problems. There is an enormous literature on the logic of conditionals: it runs to thousands of papers and dozens of books. Other proposals for representing them include strict conditionals, variably strict conditionals, probabilistic conditionals, belief revision processes, various non-classical logics, etc. Some theorists even consider that it is a mistake to think of conditionals as logical connectives at all.

You mention specifically the relationship between A → B and ¬A ∨ B. For the material conditional of classical logic these are logically equivalent. But in intuitionistic logic A → B does not entail ¬A ∨ B. This is because intuitionistic logic has the semantics of constructive provability, and the possession of a proof of B from A does not entail that we possess of a proof of ¬A or a proof of B. In probability logic, "not-A or B" does not entail "if A then B" because a high probability for P(¬A ∨ B) does not entail a high probability for P(B|A). One could easily extend the examples to other logics and other conditionals.

The lesson is that we have to be careful not to use the material conditional where it does not belong. Mathematicians are particularly prone to doing this. Often they are like the proverbial boy with a hammer who thinks everything is a nail. The realities of conditionals are much more subtle and complex.

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  • @bumblethank you for this response. It's going to take me some time to digest it, so I won't upvote it just yet. And you evaded the question, which was of the two choices which is better, and why. Since one is better it's not a false dichotomy. I don't know all possible approaches to defining the logical operators, but the question didn't require that knowledge.
    – lee pappas
    Commented May 19 at 12:57
  • @leepappas what if one is tiny fractional smidgen better? Or, too small to distinguish in most cases? Or, different but makes no difference in practice?
    – Scott Rowe
    Commented May 19 at 14:49
  • Different logics do different jobs, so it is difficult to say definitively that one is better than another. A rough analogy would be to say you can have several different maps of London: an Ordnance Survey map, an A-to-Z street map and a map of the Underground. Each is useful for a particular task, but no one is perfect. Formal logics are like maps of a territory: we can have several, but we need to understand which is the appropriate one for a given purpose.
    – Bumble
    Commented May 19 at 22:22

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