Schema
The general schema of categorical propositions is:
Quantifier (subject term) copula (predicate term)
are logical truths, but not all logical truths are tautologies.
Quantifiers have scope, namely, the first whole proposition, simple or compound, to their right. In this sense, they have the same scope as the negation sign. "Bx" is inside the scope of the quantifier in "(x)(Ax Bx)" but outside in "(x)Ax Bx".
Variables inside the scope of a quantifier are bound by that quantifier; otherwise they are free. More precisely, a variable is only bound by a quantifier on the same letter; hence "x" is bound in "(x)Mx" but not in "(y)Mx", even though it is inside the scope of the quantifier in both cases.
When a variable is within the scopes of two or more quantifiers, then it is bound by the most local (least global) quantifier on the same letter, if any. Hence, "x" is bound by "(x)" in "(y)" and "(x)(Ax·(x)Bx)".
A variable may occur more than once in an expression, free in some occurrences and bound in others, for example, "x" in "(x)Ax Bx". Hence it is imprecise to speak merely of free and bound variables. We must speak of free and bound occurrences of variables. In "(x)Ax Bx", the first occurrence of "x" is bound, because it is within the scope of the quantifier, but the second occurrence is free because it is outside that scope.
A variable may also occur freely with respect to one quantifier and bound with respect to another. For example, in "(x)Ax (x)Bx" the "x" in "Bx" is free with respect to the universal quantifier, bound with respect to the existential quantifier. So we must speak of free and bound occurrences of variables with respect to a given quantifier.
A quantifier that binds no variables is vacuous. For example, the universal quantifier is vacuous in "(x)Mz" and "(x)Ma" but not in "(x)Mx".
A general proposition is one with a quantifier; it can be existential or universal. A singular proposition lacks a quantifier and variables, and uses only constants, for example, "Ms". Singular and general propositions with no free variables are genuine propositions in the sense that they possess truth-values. By contrast, a propositional function has at least one free occurrence of a variable, for example "Hx". Therefore, propositional functions lack a truth-value; we can't tell whether the unfilled form " (blank) is human" is true or false until the blank (or free variable) is bound by a quantifier or replaced by a constant, that is, until the propositional function converted to a genuine proposition.
(Now that we know what a propositional function is, we can define quantifier scope more precisely: a quantifier's scope is the first whole proposition or propositional function to its right.)
One of the components of "(x)(Ax Bx)" is "Bx", which is a propositional function without truth-value. Hence we cannot determine the truth-value of the general proposition "(x)(Ax Bx)" using only the truth-values of the components. Hence, in predicate logic we give up truth-functionality. Hence, we give up methods for testing validity which depend on truth-functional propositions, such as truth tables.
There are two ways to convert a propositional function (like "Hx") into a proposition. First, the free variables may be bound by quantifiers; this is called generalization. Second, the free variables may be replaced by constants; this is called instantiation.
We will introduce four rules of inference for predicate logic. Universal generalization allows us to add the universal quantifier. Existential generalization allows us to add the existential quantifier. Universal instantiation allows us to remove the universal quantifier. Existential instantiation allows us to remove the existential quantifier. The two instantiation rules also allow us, after removing quantifiers, to replace form
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