Applied Logic Lecture 15: First-Order Logic
CS 4860 Spring 2009 Tuesday, March 10, 2009
First-Order Logic is the calculus one usually has in mind when using the word “logic”. It is expres-
sive enough for all of mathematics, except for those concepts that rely on a notion of construction
or computation. However, dealing with more advanced concepts is often somewhat awkward and
researchers often design specialized logics for that reason.
Our account of first-order logic will be similar to the one of propositional logic. We will present
•The syntax, or the formal language of first-order logic, that is symbols, formulas, sub-formulas,
formation trees, substitution, etc.
•The semantics of first-order logic
•Proof systems for first-order logic, such as the axioms, rules, and proof strategies of the first-
order tableau method and refinement logic
•The meta-mathematics of first-order logic, which established the relation between the semantics
and a proof system
In many ways, the account of first-order logic is a straightforward extension of propositional logic.
One must, however, be aware that there are subtle differences.
15.1 Syntax
The syntax of first-order logic is essentially an extension of propositional logic by quantification
∀and ∃. Propositional variables are replaced by n-ary predicate symbols (P,Q,R) which may be
instantiated with either variables (x,y,z, ...) or parameters (a,b, ...). Here is a summary of the
most important concepts.
1. Atomic formulas are expressions of the form P c1..cnwhere Pis an n-ary predicate symbol and
the ciare variables or parameters.
Note that many accounts of first-order logic use terms built from variables and function symbols
instead of parameters. This makes the formal details a bit more complex.
2. Formulas are built from atomic formulas using logical connectives and quantifiers.
Every atomic formula is a formula.
If Aand Bare formulas and xis a variable then (A),∼A,A∧B,A∨B,A⊃B,(∀x)A, and
(∃x)Aare formulas.
3. Pure formulas are formulas without parameters.
4. The degree d(A)of a formula Ais the number of logical connectives and quantifiers in A.
5. The scope of a quantifier is the smallest formula that follows the quantifier.
In (∀x)P x ∨Qx the scope of (∀x)is just P x, while Qx is outside the scope of the quantifier. To
include Qx in the scope of (∀x)one has to add parentheses: (∀x)(P x ∨Qx).
Note that the conventions about the scope of quantifiers differ in the literature.
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