Should math logic reflect "real" logic

Question

In math we use logic. However, it seems mathematicians were free to define some of its rules. Say the OR. It is true, if either of arguments is true - or both.

Now we use math to prove some facts about real world. Take euclidean geometry. Proofs on the other hand rely on axioms and the mathematical logic and more precisely say on the OR operator the way it is defined, to prove even more facts about real world, or not?

So my question is, in order for this chain to work, the mathematical logic has to be correct also right: What I mean is that maybe mathematicians weren't so free to define the OR in that way, what if they had to use OR which is true only when one of the arguments is true and not both? We could do this theoretically but using this OR would we be able to prove further facts from basic axioms of euclidean geometry? So what dictates that the OR which is defined the way it is now, is the correct one?

Answer 1

Mathematician have both kinds of OR. The inclusive OR and the exclusive OR which is the one that you like better. Mathematical logic is OK and it works fine.

The more you learn about mathematics the more you appreciate the logic used in proving amazing theorems.

Keep learning and you will enjoy the mathematical logic as much as the geometry.

Answer 2

You’re referring to XOR. Yet,

A XOR B = (A OR B) \ (A AND B).

So if you really need the XOR operation, you can create it from the usual ones.

Answer 3

What you call "real logic" unfortunately involves ambiguities and depends on idiosyncrasies of the English language. For example, "or" can mean inclusive or (as in "whoever did xxx must be stupid or malicious" --- he might be both) or exclusive ("you can have ice cream or cake for dessert"). In contrast, Latin has different words ("vel" and "aut") for these. Ordinary language often leaves quantifiers implicit, leaving it to experience or common sense to guess which quantifier is meant. For example compare "Dogs must be carried on the escalator" with "Hard hats must be worn on the construction site", and consider what these mean if I arrive at the escalator with several dogs versus if I arrive at the construction site with several hard hats. And compare what they mean if I have no dog and no hard hat. The bottom line here is that "real logic" in English (or any other natural language) hardly qualifies for the name of "logic" (though it's certainly "real").

So how does mathematics, with its precise logical conventions, manage to produce information about the real world? Strictly speaking, there's no problem about the information; it's there in the precise mathematical statements. Where there's a problem is in expressing that information in English (or another natural language) so that most people (non-mathematicians) can use it. Here some work is needed. A literal translation (e.g., changing the logicians' symbol $\lor$ to the word "or") is likely to lose information (because that "or" is ambiguous), so one often needs a somewhat wordy translation (e.g., "X or Y or both") to express in English facts that have more efficient expressions in mathematics.

Natural languages developed to communicate information as needed in ordinary life, not necessarily with great precision but just well enough to work when supplemented with common sense, experience, social conventions, etc. Mathematics is intended to be much more precise than that, and, as a result, needs unambiguous concepts --- beginning with unambiguous logic.

Answer 4

While I endorse Andreas Blass's answer, a different perspective is the following which I think cuts more to the heart of the issue.

First, there are many quite distinct formal logics. That classical logic works some way doesn't mean every logic has to work that way. This means there's a plurality of possible options, and we can choose which seems to work best for our purposes. With regards to modeling informal reasoning, this is akin to having multiple physical theories (where physical theories model reality). To this end the answer to your question "what dictates that the OR which is defined the way it is now, is the correct one?" is "nothing" and different logics do make different choices. However, making a different choice is making a different logic, not changing what The One And Only Logic is.

To this end, different logics correspond to different ways of modeling informal reasoning. Just like we may use Newtonian Mechanics rather than General Relativity, say, logics that may not perfectly or completely capture every aspect of informal reasoning may nevertheless be good enough for our purposes and much more convenient than a more precise account.

In a different direction using the same distinction, nowadays we kind of make fun of Aristotle's physics. We tend to have the idea that he needed to go outside and do some real experiments. However, many of Aristotle's ideas do seem to fit observations better. Newton's ideas seem obviously wrong. In day-to-day experience, an object in motion does not remaining in motion. Of course, nowadays we know that applying Newton's ideas to day-to-day experience requires a much more in-depth modeling. Likewise, modeling a piece of informal reasoning in a formal logic may be (is) a much more involved and subtle exercise than a string replace that replaces "or" with "$\lor$" and so forth. Natural language is not entirely compositional or context-free.

Finally, informal reasoning in practice is filled with flaws. These mathematical models are also idealizations, so at times where they depart from informal reasoning, this may be to their benefit.

In a totally different vein, formal logics are mathematical structures and logicians and mathematicians have plenty of other reasons to study them regardless of how well they correspond to informal reasoning.