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How would one go about symbolizing the sentence "some cats are black?"

Would I use A: All cats are black and N: No cats are black, in the notation [~A & ~N] so as to say "It is not the case that all cats are black and it is not the case that no cats are black"?

Does that sentence sufficiently imply "some cats are black"?

  • As soon as "All" comes into the logic, quantifiers have come through the door. "All cats are black" is a conjunction of "This cat is black" applied to each cat. "Some cats are black" is just the disjunction of "This cat is black" applied to each cat. "All" and "Some" are both syntactic sugar. "All" just creapt in earlier because people were concerned about essences more than accidents in earlier millennia. – ben rudgers Jan 19 '15 at 14:36
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Yes, in propositional logic you could use ¬ A ∧ ¬ B where A = 'no cats are black' and B = 'all cats are black', but that's hardly an improvement from C = 'some cats are black', which is an atomic sentence (can't be split up) as well.

For things like this it's much easier to use first-order predicate logic and quantifiers: ∃c ∈ C [B(c)], where C is the set of all cats and B(c) ⇔ c is black. You can read this as 'there exists at least one element c in the set C of all cats for which B(c), that is, the cat being black, holds'.

Abstract: ∃v∈V [P(v)] ⇔ there exists an (at least one) v in V such that P(v).

Other quantifiers are ∃! (there exists exactly one ... such that ...) and ∀ (for all ... holds ...).

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  • So I would just use "some cats are black" as its own atomic sentence? – cass parong Jan 17 '15 at 22:32
  • @cassparong that's an obvious method, but doesn't really help much. The thing is that propositional logic doesn't really have a solution for stuff with quantities. For that we need first-order predicate logic, like I showed above. With propositional logic alone you can't get it nicer than ¬ A ∧ ¬ B where A = 'no cats are black' and B = 'all cats are black'. – user2953 Jan 17 '15 at 22:34
  • If you need more versatility than "no cats" "at least one cast" and "all cats" can provide, consider building your logic around sets of cats, and defining a line for how many cast are needed before "some cats" becomes true. This usually involves choosing a notation more expressive than First Order Logic, just as a matter of practicality. – Cort Ammon Jan 19 '15 at 0:45
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In Propositionals Logic 'Some cats are black', since it is atomic, will be formalised like this: 'P'.

In Predicate Logic 'Some cats are black' will be formalised like this: '∃x (Fx & Gx)', which should be read as 'There is at least one thing that is both a cat and black.

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Commonest types of formalization of the sentence 'Some cats are black'

The Aristotelian way

One way - historically, the first one - to formalize your sentence apart from those already given above (i.e. 1.through atomic or negated atomic propositions and 2. through quantifiers in predicate logic, which I will re-sketch below) is the one which is commonly used in the particular case of dealing with syllogistic logic in the Aristotelian tradition. It amounts to the following:

If 'C' stands for the term 'cat' and 'B' for the term 'black', then 'BiC' stands for 'some cats are black' or, equivalently, '"black" is predicated of some of "cat"' (this is Aristotle's preferred formulation, which explains the subject-predicate inversion in the formalization: but some commentators maintain the more natural order, whereby your sentence would be formalized as 'CiB' -- personally I prefer to stick more closely to Aristotle's formulations).

To explain why the 'i' between 'C' and 'B', a sketch will help of the commonly used abbreviations within Aristotelian syllogistic:

  • BaC (All cats are black): Universal affirmative (the lower case 'a' comes from the first vowel of the latin verb 'adfirmo'= to affirm)
  • BiC (Some cats are black): Particular affirmative (the lower case 'i' comes from the second vowel of the latin verb 'adfirmo'= to affirm)
  • BeC (No cats are black): Universal negative (the lower case 'e' comes from the first vowel of the latin verb 'nego'= to deny)
  • BoC (Some cats are not black): Particular negative (the lower case 'o' comes from the second vowel of the latin verb 'nego'= to deny)

This way of formalizing Aristotelian logic was invented during the Middle Ages and was common until the XIXth century.

Predicate logic

Today, if one is not dealing with Aristotelian logic, the most natural way to formalize this sort of sentences is through the quantificational machinery of predicate logic: as suggested by others, the resulting formalization would then be '∃x (Cx & Bx)', where in this case 'C...' and 'B...' are predicative letters, i.e. stand for predicates and not for terms (as in the Aristotelian logic), and these two predicates are respectively assigned the interpretations '...is a cat' and '...is black'.

Propositional logic

If instead you choose to formalize it as a "one-block proposition", then you'll have 'P' (atomic sentence letter), where 'P' directly stands for an atomic (i.e. unanalyzable) sentence, and the interpretation assigned to it will be 'Some cats are black'. By so doing, however, you'll lose in expressivity, e.g. you won't be able to define rules of inference from a sentence like 'All men are mortal' to a sentence like 'Socrates is mortal'.

Remaining within propositional calculus, I think you shouldn't use the formalization '~A & ~N', where 'A' stands for 'All cats are black' and 'N' for 'No cats are black'.

First reason: to get a logical equivalent of 'Some cats are black' you don't need to negate 'All cats are black' (unless you read 'some' as implying the negation of 'all', which is not the commonest view: when you say that some cats are black you are not necessarily excluding the possibility that all cats be black). Indeed, it is enough to negate 'No cats are black', namely to write '~N'.

Second reason: even in this latter case, I would say that what you'd be formalizing is not the English sentence 'Some cats are black', but the English sentence 'It is not the case that no cats are black'. But a formalization should be as closest as possible to the sentence it formalizes. Thus, formalizing 'Some cats are black' as '~N' would be justified only if you were forced to use a hyper-regimented language: for example, a language with only one atomic sentence letter, namely 'N' itself, and the interpretation 'No cats are black' forcefully assigned to it.

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  • This question was probably already adequately answered, but I voted here, because this was a really good summary – dwn Jan 29 '15 at 22:55

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