Central to Russell’s “On Denoting” view is the propositional function. The propositional function is an abstract entity that comes about from the extraction of the denoting phrase such that a gap is produced or some variable, x, that is “essentially and wholly undetermined” (“On Denoting”, 480). In the proposition “an Athenian studied with Plato”, the logical form for such a sentence is “there is an x, x is Athenian, x studied with Plato”. Russell’s analysis of the phrase “an Athenian studied with Plato” would extract the denoting phrase “an Athenian” so that there is a gap where the denoting phrase was, making the sentence “gap studied with Plato” or “x studied with Plato”. Moreover, “gap studied with Plato” does not express a proposition,
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FOL offers an unambiguous symbolization that expresses English sentences. Kaplan believes that Russell had FOL in mind, and expressed quantifier logic such that if you want to know what the sentence says, then you must translate it to FOL (Kaplan). Russell holds that since denoting phrases do not represent denoting concepts, they cannot have a meaning in isolation and that the meaning of denoting phrases only contributes to the meaning of the whole sentence. For Russell, denoting phrases stand for universal and existential quantifiers in logic. Since there is no way of symbolizing denoting phrases such as “everything”, “every student”, “something”, “some student”, then they cannot have a meaning in isolation. In FOL, the denoting phrase “every student” cannot be symbolized without a predicate. A denoting phrase with a common noun such as “Every student is hungry” can be symbolized ∀x (student(x) → hungry(x)), but “every student” as an autonomous denoting phrase has no equivalent FOL symbolization. Sentences must meet the condition of a well-formed formula (Wff) in which there is a predicate with the variables. Without a predicate, the denoting phrase alone cannot be symbolized in FOL. This reflects Russell’s famous “principle” "that denoting phrases never have any meaning in isolation, although they contribute to the meaning of every sentence in which they occur”. Restricted quantifiers, such as “every student”, must be eliminated to unrestricted quantifiers such as “everything” and the unrestricted quantifier should be made conditional, such that “every human is mortal” becomes “everything is such that if it is a human then it is mortal” or ∀x (human(x) → mortal(x)). For Russell, denoting phrases structurally and mechanistically contribute to the whole meaning of the sentences that they are found in. An
FOL offers an unambiguous symbolization that expresses English sentences. Kaplan believes that Russell had FOL in mind, and expressed quantifier logic such that if you want to know what the sentence says, then you must translate it to FOL (Kaplan). Russell holds that since denoting phrases do not represent denoting concepts, they cannot have a meaning in isolation and that the meaning of denoting phrases only contributes to the meaning of the whole sentence. For Russell, denoting phrases stand for universal and existential quantifiers in logic. Since there is no way of symbolizing denoting phrases such as “everything”, “every student”, “something”, “some student”, then they cannot have a meaning in isolation. In FOL, the denoting phrase “every student” cannot be symbolized without a predicate. A denoting phrase with a common noun such as “Every student is hungry” can be symbolized ∀x (student(x) → hungry(x)), but “every student” as an autonomous denoting phrase has no equivalent FOL symbolization. Sentences must meet the condition of a well-formed formula (Wff) in which there is a predicate with the variables. Without a predicate, the denoting phrase alone cannot be symbolized in FOL. This reflects Russell’s famous “principle” "that denoting phrases never have any meaning in isolation, although they contribute to the meaning of every sentence in which they occur”. Restricted quantifiers, such as “every student”, must be eliminated to unrestricted quantifiers such as “everything” and the unrestricted quantifier should be made conditional, such that “every human is mortal” becomes “everything is such that if it is a human then it is mortal” or ∀x (human(x) → mortal(x)). For Russell, denoting phrases structurally and mechanistically contribute to the whole meaning of the sentences that they are found in. An