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Does natural language like English make more assumption about logic than mathematics? In mathematics, there doesn't seem to be any assumption made about which logic system is true, and therefore it is logic neutral, but it seems like natural language makes certain logical assumptions that make it difficult to use it, for instance, English in all infinite number of logic systems like Paraconsistent logic. However, I am not sure if there's any discussion made about it and whether I am imagining things and that both languages are actually logic neutral and can be used no matter what logic system we decide to adopt, imagine we're in a multiverse and each universe uses its own unique logic system.

I asked the question, because I couldn't find examples to show it was true.

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    "In mathematics, there doesn't seem to be any assumption made about which logic system is true,"-- Mathematics already had a large body of work and methods to be embedded within set theory and logic. It even already had a notion of sets before set theory and it's accompanying classical logic. Classical logic best fit with already held mathematical discoveries, so I'd contest this statement. However, you are right that it's often said people's everyday reasoning is classical, and there's non-classical mathematics. Philosophers like G. Priest or JC Beall contest natural language is classical
    – J Kusin
    Sep 16, 2023 at 17:13
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    Standard mathematics is not logic neutral, it explicitly assumes classical logic (FOL). Natural language is a hodge-podge of informal dialects in constant flux with very loose standardization, so it does not "assume" anything beyond vague guidelines of dictionaries and grammar that are routinely ignored. This is why English can be used to express paraconsistent and all other logics, and why it is impossible to fully formalize.
    – Conifold
    Sep 17, 2023 at 5:21
  • Natural language statements can be logically evaluated either literally or figuratively. So the assumptions may be different depending on the context. Sep 17, 2023 at 15:41
  • For example the phrase "Business is business" is a literal tautology but not a figurative tautology. Sep 17, 2023 at 15:44
  • This may help illustrate: philosophy.stackexchange.com/q/102888/67687 Sep 17, 2023 at 15:57

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Languages, natural like English or French, or subject to specification like the mathematical language or formal logic itself, do not make any assumption, and this for the obvious reason that assumptions are made by logical beings which have a language, not by the languages themselves.

Mathematics is not a language but a collection of non-necessarily mutually consistent theories, each theory being based on its own assumptions, only some of them axioms, and couched in broadly the same language of mathematics.

Mathematical work requires mathematicians to make assumptions, but not necessarily the same as each other. Mathematics is more of a commitment to a general method of work than a unified system, method which essentially involves the use of one symbolic scheme and some inordinate rigour in the definition of mathematical concepts. Beyond that, and adherence to logical reasoning, each mathematician is on their own.

Like the mathematical language, a natural language in based on the speakers' assumption that the words used refer. That is, that any word used whatever means something and not nothing. Not even the word "nothing" means nothing. All speakers assume that all words mean something to all speakers of the language, which is perhaps just wishful thinking. All languages are also based on the assumption that the syntax is shared among different speakers of the same language.

However, such assumptions are essentially technical, somewhat like a code of politeness between civilised people whereby you allow the prisoner to make a last wish before being hanged. The form is assumed, but the content is left to your unfettered imagination, and do humans make the best of that!

Languages are also supported by the speakers' unvarying assumption that the audience shares some minimal common perception of reality. Talking would have no utility if not. It would be a waste of time and energy.

So, broadly, languages require common-sense assumptions.

Assumptions about logic as such? No. People used natural languages, and some mathematical language, long before the notion of logic emerged from the travails of the Greek Antiquity. Humans are innately logical. That is, they have some logical capacity. However, this is not an assumption, it is just a fact of human nature. Without it, there would be no language, natural or otherwise, but the same could be said of our linguistic capacity, or even about our capacity to perceive the world.

Formal logic is something else. Like all sciences, formal logic requires logicians to make assumptions about the nature of their object, but this is not language-dependent.

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