# Should math logic reflect “real” logic

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?

• Mathematics describes a purely theoretical world that is bound by axioms. To the degree that the "real world" follows rules or axioms, there will be a mathematics that describes it. – Doug M May 15 at 20:59

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.

• I see your point so you mean in math there can be something proved with OR, but if you'd translate that in English you'd make that precise that you assume either X, Y or both are true. – user674432 May 16 at 16:55
• or when you prove math theorems from some basic math axioms and you use OR, it is also clear you assume X, Y or both are true. So as soon as you make precise what kind of OR you are using, there should be no problems, I think I am more grasping it now, it could be I hadn't formulated the question in my head clearly enough too. – user674432 May 16 at 16:58
• I always hear an inclusive "or" when asked if I want ice cream or cake.... – Barry Cipra May 18 at 6:34

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.

• Thanks for response, but you didn't understand the question. Math Proves fact about real world right? It uses logic in between, say using the OR which I mentioned. But mathematicians were free to define this OR in the way that it is true if one of args is true or both. Question was this OR is used to prove facts about real world, so the way this OR is defined must be true right? How did they know? What if they defined it in a way it is true if only one arg is true? Would we still prove theorems? – user674432 May 15 at 20:55
• As I mentioned, both kinds of OR are defined and their truth tables are agreed upon. There is not dispute. – Mohammad Riazi-Kermani May 15 at 21:04
• Good I see they are defined. But you use one of them in proving facts from real world don't you? And in that case the one you use must also be "correct" one right? By correct I mean match the real notion of OR...... – user674432 May 15 at 21:07
• In real life, if I say " you are smart OR you are rich" it does not mean that you can not be both. On the other hand If I say you are either rich or smart, then I mean you can not be both. – Mohammad Riazi-Kermani May 15 at 21:17
• so which one is the correct OR? as far as natural world is concerned and proving theorems about it? seems I am finding hard to formulate precisely what I mean .... – user674432 May 15 at 21:26

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.

• I am afraid you too didn't understand the question, maybe look at my comment to other answer – user674432 May 15 at 21:03
• @gdan: this answer shows you that both types of ORs are equivalent, in the sense that you can get one from the other. So whatever system of logic that is subsequently generated by either one, both generate the same system. This means that using XORs would not generate “further facts.” – Alex R. May 15 at 22:06

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.