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There is one seeming issue I happened upon that bothers me to no end.

Take a proposition like “Snow is white”. “Snow is white” and its negation “Snow is not white” are obviously contradictory. However, when they are expressed in the predicate logic, ∀x(Sx → Wx) and ∀x(Sx → ~Wx) respectively, they cease to be contradictory and become contrary: there is no way to get these two propositions to entail a contradiction; they can only ever entail Sx's entailment of a contradiction.

What seems to be happening here is that contradictory propositions in the propositional logic are contrary in the predicate logic such that, because contrary propositions are not contradictory, contradictory propositions in the propositional logic are not contradictory in the predicate logic. Yet, if both the propositional logic and the predicate logic are legitimate, do we not find ourselves having to accept the absurd conclusion that no contradictory propositions are contradictory?

How can this issue be resolved? More importantly, is this even an issue or am I simply terribly confused about something?

Thank you in advance for your time and help.

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Something that is a contradiction in the propositional logic remains a contradiction in predicate logic. The problem with your examples is that they are not particularly clear as to whether you are speaking of all snow or just some.

"Snow is white" and "snow is not white" are not contradictions in the propositional logic. For that, you would need, "snow is white" and "it is not the case that snow is white". You could symbolise those in the propositional logic as P and ¬P and then they would be a contradictory pair and would remain a contradictory pair in predicate logic.

In predicate logic, ∀x(Sx → Wx) might be read as "all snowy things are white" and ∀x(Sx → ¬Wx) as "all snowy things are not-white". These are not contradictory, since both would be false in the event that some snowy things are white and some are not. Strictly, they are not contrary either, since both are true if there are no snowy things. If you wish to include the commitment that some snowy things exist, you would need to write ∀x(Sx → Wx) ∧ ∃ySy and ∀x(Sx → ¬Wx) ∧ ∃ySy respectively.

If you wish to understand "snow is white" to mean all snow is white, then its contradictory is, "it is not the case that all snow is white", which in predicate logic is ¬∀x(Sx → Wx) or ∃x(Sx ∧ ¬Wx). These are contradictory to ∀x(Sx → Wx) so the contradiction remains.

Your confusion seems to arise from failing to distinguish between "it is not the case that all snow is white" and "all snow is not-white".

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  • Thank you for your wonderful answer. It cleared up a lot of my confusion. I still want to clarify a few things, if you don’t mind. You write "The problem with your examples is that they are not particularly clear as to whether you are speaking of all snow or just some”. Why is there ambiguity? Is it not true that only particular quantifiers require explicit expression, so that statements with unexpressed quantifiers can only ever be universal? I don’t see how a statement like “Snow is white” can be interpreted as “Some snow is white” unless one were to add “some” before “snow”.
    – Falcon
    yesterday
  • @Falcon Yes, if you say "snow is white" without stating a quantifier, then in natural language we normally would assume you meant "all snow is white", indeed. But in that case "snow is not white" is not the correct negation in propositional logic. If by "snow" you instead meant some specific snow (e.g. the snow on your driveway), then "the snow is white" would be correctly negated as "the snow is not white". So perhaps what Bumble means is that it's unclear whether you're talking about all snow or some specific snow.
    – kaya3
    yesterday
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    Yes, the ambiguity is with the negation. You can have a wide scope negation: "[NOT]-all snow is white" or a narrow scope negation: "all snow is [NOT]-white". One advantage of predicate logic is that the negation particle is explicit, and the syntax does not allow for ambiguity. In natural language you have to rely on context to determine what is meant.
    – Bumble
    yesterday

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