# Can any correct logical reasoning in natural language sentences be translated into a formal mathematical proof?

Since natural languages (e.g. English) are prone to ambiguities and misunderstandings due to their constant evolving nature and lack of rigorous formalization, and given an arbitrary philosopher X who wants to show the validity of his argument, would it be possible for that philosopher to translate his argument into a formal, non-ambiguous mathematical proof written in some kind of formal mathematical language in order to convincingly and indisputably show everyone else his argument's correctness?

• It looks like you're asking some great questions regarding language and proofs. Just be ready for even formal logical languages to have some quirks when you dig at them. There's some really interesting little frayed edges that show up when you really put their feet to the fire and try to figure out what they really are. – Cort Ammon May 13 '18 at 16:46
• When you say "formal mathematical proof", mathematicians do not normally write proofs in a purely formal style. Rather they use a combination of formalism and natural language. Do you mean "formal logical proof" as in philosophical logic (and certain areas of mathematical logic) where proofs are purely formal. – Nick May 13 '18 at 19:33

What you describe is the general structure of a mathematical paper. Words regarding the mathematical discovery are paired with precise mathematical notations to demonstrate correctness. However, a general version of this is tricky. If a philosopher says "Here's natural argument A and here's formal argument B. As the author of both, I say they are intended to be the same" that's one thing. However, if we try to say "here's a process that lets you turn the crank on A to produce B," we have to question whether the process works right.

Indeed, proofs by folk like Tarski showed very particular limits on processes like you describe. It's very hard for a language to prove it's own semantics, and most of the languages we are interested in when talking about proofs are simply not capable of it.

Of course, there is a way to do this. What defines a "correct logical reasoning in natural language?" If the definition is "there exists a corresponding formal logical proof," then the answer to your question becomes yes!

You might be interested in looking at Attempo Controlled English (ACE). ACE does the dual of what you seek. It is a way to write precise formal logic in a format which native speakers of English can read as though it were natural English. If you read it naturally, you get the right intuition. If you read if formally, you get the right formal meaning.

• Would it possible to define "free will" using ACE? For example, if a philosopher wants to translate his paper about free will into ACE, would he be able to do it? – xwb May 13 '18 at 2:30
• @xwb The answer to that depends entirely on the philosopher's definition of free will, and whether it admits being put into formal logic or not. More specifically, ACE will easily admit phrases like "Every person has free will," but whether it can describe what the philosopher wants to say about it depends greatly on what the philosopher wants to say. – Cort Ammon May 13 '18 at 5:07
• But if what someone says cannot be translated into formal logic, wouldn't that mean that such person is only playing with words and haven't actually said anything logical at all? – xwb May 13 '18 at 14:17
• @xwb How would you go about arguing that? Does the argument depend on defining logic to be something translatable into formal logic, or can the argument be made without such a rule? – Cort Ammon May 13 '18 at 16:05
• (also, getting ahead of myself a bit to save some time: do you consider arithmetic to be part of a logical argument, or is arithmetic something that is separate, and cannot be used in a purely logical proof?) – Cort Ammon May 13 '18 at 16:28

Hypothetically, yes. For any non-trivial argument though, the logic you’d employ would be much more complicated than the textbook logic you learn in a first course in formal logic.

Most important philosophical arguments will employ concepts like conceivability, parthood, necessity, causation, and so forth. To formalize your argument, you have to give a precise logical definition of how these concepts are to be used.

But that is only the first step. What one has “proven” by formalizing such an argument is syntactic, this argument is well formed, the conclusion follows from the premises. Whether the concepts one has formalized accurately capture reality is a semantic question—and that’s where the real philosophy happens. Put a different way: it is easy to invent concepts and show how they can be combined and what follows from those combinations. The secret sauce is figuring out whether those concepts describe reality.

• So basically you are saying that real Philosophy happens when mathematical reasoning is tested against reality. But isn't that the definition of Physics? – xwb May 13 '18 at 1:44
• Formalizing a theory is the same in philosophy and physics. You start off with the concepts, give them precise mathematical definitions, etc. The difference is that philosophy and physics have different subject matters. The subject matter of physics is matter, motion, energy, and so forth. The physicists can then go and perform observations and experiments to see if their theories work. Philosophy’s subject matter concerns features of reality which are not empirically observable, so you don’t get the kinds of evidence in philosophy as physics unfortunately. – user5172 May 13 '18 at 15:24

In my opinion, formalizing an the argument can be done — and it can even be done straightforwardly.

Recall, however, that an argument consists of

• A list of hypotheses
• A conclusion
• A proof leading from the hypotheses to the conclusion

In the (claimed) straightforward process of formalizing an argument to make it completely rigorous, what is going to happen is that all of the disputable parts are going to appear in the "list of hypotheses" section, rather than the "rigorous proof" section.

So, what's going to happen is that you will have an argument that is convincingly and indisputably valid, but you haven't really gained much — you now have the problem of convincing people that the list of hypotheses are satisfied by something someone cares about.

Mathematics is somewhat unusual in that there is a wealth of notions for which you can write down a list of hypotheses that will generally not be disputed, but yet are complete and precise enough to enable formal proof of interesting conclusions.

if it is possible to translate an ordinary language statement into mathematics then it must have been precise in the first place. This was the flaw in the idea of translating philosophical statements into symbolic logic and the reason Russell's attempt didn't prove useful. Where it's possible it isn't necessary. One cannot make a statement in English more clear by translating it into French.

See Stephen E. Toulmin’s The Uses of Argument. He claims that the standards for valid arguments are field specific and arguments in some fields need not be reduced to analytic arguments to have validity.

In particular he writes (page 235):

What has to be recognised first is that validity is an intra-field, not an inter-field notion. Arguments within any field can be judged by standards appropriate within that field, and some will fall short; but it must be expected that the standards will be field-dependent, and that the merits to be demanded of an argument in one field will be found to be absent (in the nature of things) from entirely meritorious arguments in another.

Should one get a position that validity of substantive arguments in all fields can “be translated into a formal mathematical proof”, that is, that all substantive arguments can be written as analytic arguments with entailment justifying validity then one should test that position against what Toulimin offers to counter it.