Predicate Logic

Predicate logic introduces the ability to break apart arguments into predicates. Predicates are verbs like actions or states of being. For example, we know there are similarities between the following two sentences:

• 3 is greater than 2
• 2 is greater than 1

In propositional logic, we would have had to assign each of these a different letter, such as P and Q. However, in predicate logic, we can make use of their similarities to create two formulae that look identical.

• GreaterThan(3, 2) or G(3, 2)
• GreaterThan(2, 1) or G(2, 1)

Most books tend to not give their predicates full names, but we will do so frequently in the text lecture to give context for what is being written.

Syntax of Predicate Logic

Every formula in predicate logic is built from the following parts:

• Predicates: P, Q, R, S, ...
  • These stand for our verbs in our formulae
• Terms: a, b, c, ..., x, y, z, ...
  • Constants: a, b, c, ...
    • These stand for proper names (e.g. 1, 2, 3, Jim, Bob, Mary) in our formulae
  • Variables: x, y, z, ...
    • These stand for variables in our formulae
• Connectives: →, ↔, ¬, ∧, ∨
  • These are our logical connectives in our formulae (if then, if and only if, not, and, or)
• Quantifiers: ∀, ∃
  • These are our quantifiers in our formulae (for all, for some)
• Parentheses: (, ), [, ]

We have the following rules for putting together these well-formed formula (wffs).

1. If P is a predicate and x, ..., z are terms, then P(x, ..., z) is a wff.
2. If P and Q are wffs, then
   1. (¬P) is a wff.
   2. (P → Q) is a wff.
   3. (P ↔ Q) is a wff.
   4. (P ∧ Q) is a wff.
   5. (P ∨ Q) is a wff.
3. If P is a wff, then
   1. ∀x[P] is a wff.
   2. ∃x[P] is a wff.
4. Nothing else is a wff.

Examples

We show by composition of parts that the following wffs are permissible in predicate logic through syntactic trees.

P(x,y) ∧ Q(y,z)



∀x[P(x) ∧ ∃y[Q(x,y)]]

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