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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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2 Answers 2

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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
    Nov 3, 2021 at 2:34
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    @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
    Nov 3, 2021 at 5:52
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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
    Nov 3, 2021 at 10:38
  • @kaya3 Thank you both for your help. I would like to clarify again all that has been said. Statements with unexpressed quantifiers are ambiguous: “S are P” can mean either “All S are P” or “Some S are P”; “S are not P” can mean either “All S are not P” or “Some S are not P”. “S are P” and “S are not P” are contradictory through and only through one of two interpretations: either 1) “S are P” means “All S are P”, and “S are not P” means “Some S are not P” (or “Not all S are P”) or 2) “S are P” means “Some S are P”, and “S are not P” means “All S are not P” (or “No S are P”). Right?
    – Falcon
    Nov 6, 2021 at 15:06
  • @kaya3 In both interpretations, the statements are each other’s wide-scope negations, in that if the one is Q (“It is the case that Q”), then the other is necessarily ~Q (“It is not the case that Q”). It all comes down to the fact that the conjunction of wide-scope negations and only the conjunction of wide-scope negations constitutes contradiction; the conjunction of narrow-scope negations can only ever guarantee contrariety. My error, thus, lay in my having held my two ambiguous examples as contradictory while interpreting them as narrow-scope negations. Is all of this correct?
    – Falcon
    Nov 6, 2021 at 15:06
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The english sentence snow is white doesn't translate to the FOL sentence \forall . x S(x) -> W(x). There are two reasons for this:

  1. The sentence snow is white is a generic, and generics don't express universal quantification.
  2. snow is a mass noun, and FOL predicates can't capture the semantics of mass nouns.

But (2) isn't really relevant to your question, so let's focus on (1). Generics are sentences like the following:

  1. dogs are mammals
  2. jewelry is expensive
  3. ducks lay eggs
  4. people are over three years old
  5. primary school teachers are female

They intuitively express generalizations of some kind, but linguists haven't had a lot of success uncovering the details of how their semantics works. However, one thing that's clear is that they do not express universal or existential quantification. They don't express universal quantification because e.g. (4) and (5) are true, but (4') and (5') are not:

4'. all jewelry is expensive

5'. all ducks lay eggs

(4') is false because some jewelry is cheap (I once bought a necklace for a dollar) and (5') is false because male ducks don't lay eggs. Notice that the same thing holds for your sentence: (8) is true, but (8') is not (just look at some dirty snow on the side of the road):

  1. snow is white

8'. all snow is white

This shows that generics don't express universal quantification. It follows that the fact that (8) and (8'') are contradictory, but (9) and (9') are not, has less to do with translating sentences of propositional logic into FOL and more to do with the nuances of generics:

8''. snow is not white

  1. \forall x . Snow(x) -> White(x)

9'. \forall x . Snow(x) -> ~White(x)

It's also worth mentioning that generics don't express existential quantification because e.g. (6) and (7) are both false, but (6') and (7') are both true:

6'. some people are over three years old

7'. some primary school teachers are female

If you want to learn more about generics, you can check out Ariel Cohen's chapter in this handbook or Sarah Jane Leslie's SEP entry.

EDIT: I'm answering @falcon's followup questions here:

Question 1: "Thank you for your answer! So, what you are saying is the following? “S are P” does not necessarily translate to “All S are P” or to “Some S are P”, neither does “S are not P” necessarily translate to “All S are not P” or to “Some S are not P”. This is because “S are P” and “S are not P” are generics, and generics express neither universal nor existential quantification."

Answer: Exactly.

Question 2: "Generics don’t express quantification because a generic might be true and its corresponding quantified-statement false, and vice versa--your two examples: “Ducks lay eggs” is true, but “ALL ducks lay eggs” is false; “Primary school teachers are female” is false, but “SOME primary school teachers are female” is true."

Answer: Yes, but with one caveat. Generics don't express universal or existential quantification, but many linguists think that they do express a distinct third kind of quantification called generic quantification.

Question 3: "And if all of the above is correct, would the following also hold, do you think? On account of differences in truth-value between generics and their corresponding quantified-statements, the inference from any generic to its corresponding quantified-statement is not tautologous. Because of this untautologousness, no generic is guaranteed any particular quantifier: generics tend thereby to be ambiguous in regards to quantification, and we must supply the missing quantifier through recourse to context in natural language."

Answer: No, I don't think that's right. An expression is ambiguous if there are multiple, distinct interpretations associated with it. In such a case, you should be able to paraphrase each interpretation and find scenarios in which only one of them is true. For example, the sentence all of the students failed an exam is ambiguous because it has exactly two distinct interpretations associate with it. And we can paraphrase them:

First interpretation: there is an exam that every student failed

Second interpretation: every student failed some exam or other

And finally, we can identify situations in which only the second interpretation is true (e.g. there are two students and two exams and the first student passed the first exam, but failed the second, while the second student failed the first exam, but passed the second).

However, this is not the case for generics. i.e. It's not the case that the sentence snow is white means all snow is white in some contexts, but some snow is white in other contexts. (just try thinking of such contexts!). Rather in every context, the sentence seems to have a single interpretation which simply is not equivalent to all snow is white or some snow is white.

With that said, here's something interesting that's not directly relevant to your question: the same surface syntactic structure can be used to express existentials and generics. For example,

1 a. racoons are mammals.

b. racoons are eating our trash

2 a. a cell phone is a portable means of communication

b. a cell phone is sitting on the table. do you know who it belongs to?

(1a) and (2a) are generics, but (1b) and (2b) are existentials. There are also empirical tests for telling them apart. Existentials are upward entailing, but generics are not. And the predicate in an existential must be a stage level predicate, but the predicate in a generic must be an individual level predicate (caveat: this isn't entirely true for indefinite singulars). But this isn't directly related to your question.

Question 4: "My error was in failing to account for this ambiguity, an ambiguity arisen from, as you put it, “the nuances of generics”."

Answer: I mostly answered this already, but no, I don' think that's right. I think the sentence snow is white is unambiguous. Your error was to interpret it as a universal when it's really (unambiguously) a generic.

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  • Thank you for your answer! So, what you are saying is the following? “S are P” does not necessarily translate to “All S are P” or to “Some S are P”, neither does “S are not P” necessarily translate to “All S are not P” or to “Some S are not P”. This is because “S are P” and “S are not P” are generics, and generics express neither universal nor existential quantification.
    – Falcon
    Nov 7, 2021 at 3:58
  • Generics don’t express quantification because a generic might be true and its corresponding quantified-statement false, and vice versa--your two examples: “Ducks lay eggs” is true, but “ALL ducks lay eggs” is false; “Primary school teachers are female” is false, but “SOME primary school teachers are female” is true.
    – Falcon
    Nov 7, 2021 at 3:59
  • And if all of the above is correct, would the following also hold, do you think? On account of differences in truth-value between generics and their corresponding quantified-statements, the inference from any generic to its corresponding quantified-statement is not tautologous. Because of this untautologousness, no generic is guaranteed any particular quantifier: generics tend thereby to be ambiguous in regards to quantification, and we must supply the missing quantifier through recourse to context in natural language.
    – Falcon
    Nov 7, 2021 at 3:59
  • My error was in failing to account for this ambiguity, an ambiguity arisen from, as you put it, “the nuances of generics”.
    – Falcon
    Nov 7, 2021 at 3:59
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    @Falcon I did my best to answer your questions in an edit
    – user236343
    Nov 7, 2021 at 20:17

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