General Propositions: Universal and Existential

In one line

Modern logic reads a universal proposition as a conditional about anything whatever, and an existential proposition as a conjunction about at least one thing.

In the wording a student can write in an examination: a universal proposition asserts that for anything whatever, if it belongs to the subject class then it has the predicate; an existential proposition asserts that there is at least one thing which belongs to the subject class and has the predicate. The first is a quantified conditional and the second a quantified conjunction.

The apparatus

A variable, written x, stands for anything whatever. It names nothing in particular.

A predicate letter, written Sx or Px, says that x has some property. Sx reads "x is a contract"; Px reads "x is an agreement".

The universal quantifier, written ∀x, reads "for anything whatever" or "for every x". Some books write it simply as (x), and MU's reading list uses that older form; both mean the same.

The existential quantifier, written ∃x, reads "there is at least one x such that".

That is the whole apparatus. Everything below is built from it.

Propositional functions, and the variable

Added after the past-paper check, because the examiner asks for this in these exact words and MU's own multiple-choice paper defines quantification by it.

A propositional function is an expression containing a variable, which is not itself a proposition, but which becomes one when the variable is dealt with.

"x is a contract" is not true and is not false, because nobody has said which x. It is a propositional function, written Sx. It becomes a proposition in either of two ways.

By substitution. Put a name in place of the variable: "this deed is a contract". That is now true or false.

By quantification. Bind the variable with a quantifier: "for anything whatever, if it is a contract then it is an agreement". That is now true or false as well, and it is a general proposition.

So quantification is an operation performed on a propositional function. MU's own examination paper of June 2022 puts it in exactly those terms: quantification consists in asserting a propositional function of all or some of the values of the variable. Asserting it of all the values gives a universal proposition; asserting it of some gives an existential one.

Individual variables and individual constants. The letter x is an individual variable: it stands for any individual whatever and names none. A letter such as a or b used to name a particular individual is an individual constant. So "Sa" is a proposition and "Sx" is a propositional function.

Free and bound variables. A variable is bound when it falls within the scope of a quantifier that governs it, and free when it does not. In Sx the variable is free, so the expression is a propositional function; in ∀x(Sx ⊃ Px) every occurrence of x is bound, so the expression is a proposition. An expression with at least one free variable is always a propositional function and never a proposition.