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Aristotle is wise (subject = Aristotle, predicate = is wise) The sky is blue (subject = The sky, predicate = is blue) Russell likes math (subject = Russell, predicate = likes math) Predicates can be thought of as open sentences, i.e. sentences with variables. For example, x is wise. is a predicate 1.
By predicate, I think he means a "property" of the entity, for example, the predicate of being tall. This is the meaning that I'm aware of and which is the meaning we use in mathematical logic. Exactly; in mathematical logic "existence" is a quantifier acting on a predicate; we read: ∃xPx. as: "there is an object having property P".
Not all presentations of the language of first-order logic use 0-arity predicates and functions. The definition of a first-order language at the Open Logic Project does not use them. The set of non-logical symbols contain the following: (page 145-6) a) A denumerable set of n-place predicate symbols for each n > 0. b) A denumerable set of ...
A predicate is a function that takes some arguments and returns a True or False. [That x is even] is a predicate, it's sometimes true, sometimes false, depending on x. Even [x = x] is a predicate; to make it into a proposition we can plug a specific x in, say [1 = 1] or we can quantify it: [forall x: x = x], both of those are true propositions.
Every tautology of propositional logic, like P ∨ ¬P, can produce an unlimited supply of valid predicate logic formulae through uniform substitution, i.e. by replacing every occurrence of a propositional letter by an atom of predicate logic language. For example, from P ∨ ¬P we can produce the valid formulae : ∀xP(x) ∨ ¬∀xP(x)
The need for them is transparently linked to the needs of formalizing logic into predicate calculus with quantifiers, where predication and identity are expressed by different means, P(x) vs x=y. There are even more options in set theory, where one can talk of inclusion of classes defined by predicates, and use "is" for that.
The mismatch with natural language is well-known, and is one of the arguments for alternatives to the predicate calculus that allow plural expressions with vague and varying domains, see e.g. Ben-Yami, Logic and Natural Language, ch.6:"the way quantification functions in the calculus shows that its semantics is fundamentally different from that ...
If =2 is the predicate, then =2(2) is perfectly fine; it is, in fact, the curried form of the usual 2 = 2 (= 2 2 in polish notation). And yes, putting such a type-restriction will result in a system without fixed-point ("Y") combinators (because these rely on terms of form xx), which are generally used to represent recursive functions.
If you want to rewrite a prop argument in predicate logic you have to replace uniformly prop variable with atoms: if B is (Ex)Bx in 4, then 3 must be e.g. (~ (x)Dx > (Ex)Bx) – Mauro ALLEGRANZA. May 2, 2022 at 7:54. Alternatively, replace prop letters with predicates (open formulas) and consider every line of the resulting argument universally ...
In technical logic, a predicate is an entire statement. In your usage, with respect to grammar, the predicate is the verb and object (or other parts) which apply to the subject; the predicate is usually some relation about the subject. I think the latter concept of 'predicate' is what you are referring to.