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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.

Someone said that natural language doesn't require or imply classical logic, but I thought this example might point to the contrary.

I was thinking about the law of identity. This states that "A is A" - i.e., anything is identical to itself. In grammar, this is reflected in the use of definite articles (the) and pronouns to refer to specific entities.

Example: "The cat is on the mat." Here, "the cat" refers specifically to a particular cat.

So how am I wrong? Could you explain?

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    “You aren’t you when you’re hungry. Eat a snickers and stop violating the Law of Identity.”
    – David H
    Sep 24 at 21:26
  • There is nothing in grammar, that forces "The cat" to reference a single cat. The fact that cat's obey the law of identity isn't due to any grammar rule. In fact, schrodigner's cat may not even obey the law of identity at all, are you refering to the dead cat? Or the alive cat?
    – Michael Carey
    Sep 24 at 23:19
  • Gramatically, the law of identity doesn't even hold, properly. Equality is a symmetric, transitive, and reflexive relation, and we can substitute equivalent terms into sentences in logic. So, the = used in, "The cat on the table", isnt equality in the logical sense. As, if the cats name is Mimi. "The Mimi on the table" isn't correct.
    – Michael Carey
    Sep 24 at 23:31
  • Natural languages with their production rules are true languages containing myriad logics malleable for different people's own epistem, even paradoxical ones like the liar sentences. Non-paradoxical classic logic is not natural but artificial, meaning if you subscribe to naturalism it may not fit your view in any generic situation, thus its expressive power is much limited to compensate for its rigor, soundness and completeness. Ergo natural language only implies classic logic as one of many logics possibly true but not necessarily so, such as its highly controversial material conditional...
    – Double Knot
    Sep 25 at 5:26
  • "Example: "The cat is on the mat." Here, "the cat" refers specifically to a particular cat." Does it? Or did you just write that sentence even without referring to an actual cat?
    – JiK
    Sep 25 at 10:04

2 Answers 2

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You are running together several questions.

  • Do the grammatical rules of natural languages imply some logic or other?
  • If they do, is this logic classical?
  • If they do, is the law of identity a component part of this logic?

  1. Some logicians, e.g. Quine, regard logic as being fundamentally linguistic in nature, and they see logical rules as grounded in grammar. Since we use language to reason, or at least to express our reasoning, it is natural that languages develop so as to support expressions of logic. It is definitely the case that logic interfaces with linguistics and sometimes understanding the logic of a sentence requires a precise grasp of the grammatical rules of its language.

  2. I don't see a good reason to suppose that the grammar of natural languages favours classical logic over others. Natural languages typically include words for 'and', 'or', 'not', 'if', 'all', 'some', etc, since these are all highly useful. But the grammar itself doesn't force one particular reading of their logic. Notoriously though, Quine held that classical logic is the uniquely acceptable logic and that to change the logic is to change the subject. This position is not so popular today.

  3. The law of identity: "Every thing is what it is and not another thing" (Butler), is a fundamental feature of how we think about the world. There are things, distinct from other things, and things have properties, and there are relations between things. This is so common it is not surprising to find it reflected in language. That said, your "the cat is on the mat" sentence is not a clear case. Many (not all) philosophers distinguish between the 'is' of predication and the 'is' of identity. Some go even further and distinguish the 'is' of existence and the 'is' of subsumption. So, we might distinguish between, "Mary is tall", "Hesperus is Phosphorus", "there is a maximum speed limit in this state" and "the whale is a mammal". For those who accept these distinctions, "the cat is on the mat" is not an identity claim but a predication.

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Arguably, all thought about objects requires that one assume the law of identity. In fact, an object is sometimes defined by the law of identity; that is, an object is defined as a thing A such that A=A. So in that case, to the extent that language represents thought, it represents something that involves the law of identity. However, grammar is just the set of rules for what makes a well-formed sentence or phrase. It has nothing directly to do with meaning, much less with the laws of thought, except in the Pickwickian sense that sentences are objects, and so to talk about sentences, you have to assume the law of identity.

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  • +1 Identity is absolutely baked into NL. Doesn't a conjugation of a verb inherently convey semantics by differentiating on aspects of meaning such as mood and tense? It is a grammatical fact that he did means he did before he does and will do, no?
    – J D
    Sep 24 at 21:39
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    I'd argue that the law of identity, is baked into the definition of equality, rather than the law of identity being baked into thought about objects.
    – Michael Carey
    Sep 24 at 23:24

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