Is Constructivism (philosophy of mathematics) against classical logic? I might be wrong, but mathematics' main branch of logic is based on classical logic, and I was wondering if Constructivism was against that main branch of logic, or it wasn't against that main branch because its views were not directly opposed to it in any shape or form. Also, if Constructivism allows us to get rid of the Liar's paradox, doesn't that mean that classical logic is flawed and there's a more flawless logic than classical logic?

  • What is deemed "classical logic" and modern logic are the same. Both are under the proper label "Mathematical logic". Aristotelian logic predates Mathematical logic. It is not like there are thousands of logics before Mathematical logic. Aristotelian logic had been modified several times before Mathematical logic was invented around 1845. You question is about Mathematical logic constructivism against Mathematical logic? Not sure I follow it how anyone can follow. Both are about math correct? Do you think the word philosophy included in the question makes this different? – Logikal Dec 22 '20 at 5:15
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    Constructivism as philosophy does reject classical logic in favor of intuitionistic logic, although constructive mathematics can be interpreted as part of classical mathematics (and vice versa). Classical and intuitionistic logic are generally bi-interpretable, i.e. one can build within each a model of the other, just like one can build models of non-Euclidean geometry within Euclidean spaces and vice versa. As for the Liar paradox, intuitionistic "resolutions" of it are no more satisfactory than classical ones, and both logics have their "flaws" and benefits in different areas. – Conifold Dec 22 '20 at 6:15

Mathematical constructivism typically involves the use of 'intuitionistic logic', which avoids LEM (Law of Excluded Middle) and which is central to classical logic, so in that specific sense, the two are incompatible. Not relying on LEM means that constructive mathematics can be regarded as one sort of 'generalisation' of classical mathematics, in the same way that not relying on Euclid's fifth postulate results in a generalisation of Euclidean geometry.

That said, one can do constructivist mathematics without being inherently opposed to LEM - it might be, for example, that one is seeking a constructive proof for computational reasons, rather than for any particular philosophical reasons.

i'm not sure what you mean by your reference to the Liar's paradox, as there have been a few different resolutions proposed: cf. the Stanford Encyclopedia of Philosophy entry on it: https://plato.stanford.edu/entries/liar-paradox/#SomeFamiSolu

Personally, i think it's not useful to conceptualise things in terms of 'more flawless' logics, as different logics can have different uses (cf. e.g. Wikipedia's list of applications of paraconsistent logics, https://en.wikipedia.org/wiki/Paraconsistent_logic#Applications), and are interesting in their own right regardless. Andrej Bauer has a post about the many worlds of mathematics: http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/

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