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I have a very basic question.

How would you answer to somebody that is asking you why in mathematics we use two-valued logic as the very ground of math reasoning instead of some multi-valued logic?

Is the reason purely practical, but – at the same time – based on conceptual/philosophical reasons that go way back in time?

Any feedback will be much appreciated!

EDIT: I feel the best way to think about this question is as what you would answer to a skeptical student that is always ready to drop the study of mathematics altogether, but that you would really like to convince on the beauty of it.

Kolmin
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    Because there is a "natural understanding" of the dichotomy True-False in human language and thought. – Mauro ALLEGRANZA Nov 21 '22 at 13:35
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    You can build math on 'fuzzy logic' or 'many valued logic' or 'probabilistic logic'. In the end the only thing that really matters is if other mathematicians follow suit. And most mathematicians are perfectly happy with two valued logic – Vercingetorix Nov 21 '22 at 13:44
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    Although when we look at Gödel's Incompleteness Theorems, and therefore metalogic in general, we sometimes need to think of things with at least the three options "yes", "no", and "unknown". – aschepler Nov 21 '22 at 14:16
  • @aschepler. Indeed. – Kolmin Nov 21 '22 at 14:19
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    I don't think that understanding of Gödel's theorem is right. Theorems can be proved true, proved false, both, or neither. The first incompleteness theorem shows only that in a system of sufficient expressiveness, there is at least one sentence in the "both" or "neither" category. And in any case it speaks about provability in an axiomatic system, not about truth. The theorem concludes that (if the system is consistent) there is a true but unprovable sentence. It can do this because it starts from the idea that the unprovable sentence must be either true or false. – MJD Nov 21 '22 at 14:25
  • Are there things that are inherently neither true nor false or are there things that are either true or false but we have no way of knowing which is the case? – John Douma Nov 21 '22 at 17:34
  • The incompleteness theorems are constructively valid, and as I recall, the reasoning about the unprovable sentence being 'true' arises from thinking about the informal meaning of the constructed sentence. E.G. one sentence 'means' "this sentence is not provable," and it actually isn't for the normal notion of proof. I don't think there is any need to assume the sentence "must be either true or false." – Dan Doel Nov 21 '22 at 22:56
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    There are countless applications of 2-valued logic. Not so of 3-valued logic. I don't know of any, but I am fairly certain that the few applications there are can also be implemented in 2-valued logic. – Dan Christensen Nov 22 '22 at 04:33
  • The SQL database query language has 3-valued logic. The interpretation of the third value, called "null", is somewhat confused. Nevertheless, it is ubiquitous. – MJD Nov 30 '22 at 06:36

2 Answers2

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Two-valued logic reflects our conceptualization of the world: we typically think of the world as a place where things are or are not the case. And that conceptualization often works, i.e. it allows us to make inferences and predictions that often come out true. So, it seems we are capturing something of significance about the world and how it works, and we can use it to great effect for many practical purposes. (indeed, the very fact that we conceptualize the world as such means that it we're getting something right, otherwise our brains would have rejected it a long time ago).

In this, logic is not any different from other branches of math or science: we come up with mathematical idealizations, and we see if they are applicable and useful to think and make predictions about the world about us. And even if they don't always work perfectly, as long as they work some of the time, and we get a pretty good sense as to when and where our idealizations apply or don't apply, then we'll use it.

Two-valued logic clearly works in this sense: it may not perfectly capture everything that's going on around us (think fuzziness, uncertainty, quantum weirdness, etc.), but it's pretty darn effective and useful in may real world situations. Its simplicity is of course another big plus.

Bram28
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  • Thanks! Thus, it is a bit like the rules for the sign of multiplications: they are neat (which is a very informal way to say that they satisfy a lot of nice algebraic properties) and – in the long run – they work. – Kolmin Nov 22 '22 at 08:27
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Mathematics can be viewed as unfolding of the kernel concepts of number and space through generalisations and abstractions over various operations and relations involving only them. The knowledge of mathematics grows with increasing sophistication, but remains always faithful to the primitive grasp of the kernel which links it to the empirical world.

Since mathematics always returns on itself, each time, enriched further, the question whether mathematical innovations are discoveries or inventions is quite wrong-headed; an answer would be, at best, "both", briefly to say. By this nature, mathematics is actually one of the leading motivations for the philosophical doctrines of innate ideas and rationalism, which recur in one or other form unabatedly in the contemporary philosophy as well.

So, mathematics realises its own judgement of truth. Hence, it is free to expel anything standing against its rationality as falsity or fallacy, while mathematical truths remain incorrigible, for there can be no empirical fact to falsify them. In other words, it could be said that empirical facts can only verify them. Consequently, the dichotomy of truth and the rest (i.e., falsity) is inherent in mathematics.

To explicate this point, an analogy can be drawn to our eyesight. Our visual perception is limited, but we can translate/transform the sight of things that are not in our visual limits into our limits, anyway. We can even give a visual counterpart to abstract objects. However, all of those are eventually our ways of seeing things, whatever their origins might be. In case that we become aware of something which is invisible to us, but we think it must be, even though we think it is beyond our ability, we aim at devising a method to carry its sight over somehow into the domain of our ability. Thus, there is no seeing for us other than our seeing to correct us. In this respect, the frequent characterisation of primitive mathematical truths as "self-evident" is rather misleading; we do not indeed find them out that they are true, we think within them, and there is no without. This aspect has led some philosophers to propound the view that mathematics actually describe an ontology (see, for example, Alain Badiou).

A clear example of the incorrigibility and the accumulative progress of mathematics associated with it is the history of Euclidean geometry. It has been understood that different geometries could be constructed, its presentation has been modified to fit in the current standards of rigour and precision; nonetheless, it has manifested its own development and remained a constituent of mathematical knowledge on a par with others. In this connection, Einstein's 1921 address titled "Geometry and Experience" would be a mind-broadening read.

How does this line of thought fare with intuitionism and constructivism in mathematics? We cannot go into the discussion of these topics, but it should be remarked that they are not about mathematical truth/falsity per se, but basically about the methods to attain and judge mathematical truth. In a similar vein, conjectures make an example for the pragmatical side of mathematical activity, rather than its essential theory.

Tankut Beygu
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