1 Overview

In propositional logic, there is no concise method of encoding similar statements, such as: Amy loves her mother; Carolyn loves her mother; Michele loves her mother. These statements must be encoded asdistinctpropositions. Moreover, inpropositionallogic thereis not even an awkward way of encoding statements of the form: everyone loves his/her mother; someone loves his/her mother. This lecture, which is concerned with predicate calculus ¹ —first-order logic without quantifiers or variables—addresses the first of these concerns. The next lecture on first-order logic (with quantifiers and variables) addresses the second.

The ontological basis of propositional logic is the proposition. The ontological bases of predicate calculus are objects and their attributes. Some examples of objects are people, animals, plants, andbaseballgames. Therearetwotypes of attributes ofparticular relevancetopredicate calculus: (i)functions, such as amy, whichis a constant (i.e., a0-placefunction), and mother

of(amy), whichis a1-placefunction; and(ii) relations, such as female(amy), which is a property (i.e., 1-place relation), and loves(mother of(amy),amy), which is a 2-place relation. Note that propositions are 0-place relations.

2 Syntax

Predicate calculus generalizes propositional logic with terms, which represent objects, and predicates, which represent propositions, properties, and relations. The predicate calculus alphabet includes a set of constant symbols {a, b, c, . . .}(i.e.,0-ary function symbols), a set of n-aryfunction

¹ InCS141,wereferto first-orderlogicwith onlyground terms(novariablesorquantifiers) aspredicatecalculus. This is not necessarily standard nomenclature. First-order logic and predicate calculus are usually synonymous.

symbols {f, g, h, . . .}for n> 0, a set of n-arypredicate symbols {P, Q, R, . . .}for n ≥ 0, two select 0-ary predicates {⊤, ⊥}, and a set of logical connectives {¬, ∨, ∧, →}. Given this alphabet, the syntactic rules of the predicate calculus govern the construction of terms and formulas.

The following inductive definition describes the terms of predicate calculus:

• constant symbols a, b, c, . . . are terms
• if t1 ,...tn are terms and f is an n-ary function symbol

– f(t1 ,...,tn)is a term

The following inductive definition describes the formulas of predicate calculus:

• if t1 ,...,tn are terms and P is an n-ary predicate symbol

– P (t1 ,...,tn)is a(n atomic) formula

•⊤ and ⊥ are(atomic) formulas

• if φ and ψ areformulas,thenthefollowing are(complex) formulas:

–

¬φ is a formula

–

φ ∨ψ is a formula

–

φ ∧ψ is a formula

–

φ → ψ is a formula

As in the case of propositional logic, complex formulas involve one or more of the connectives; atomic formulas do not.

Example: Given the alphabet A = {0, 1, succ, add,<}, where 0 and 1 are constant symbols, function symbol succ has arity 1, function symbol add has arity 2, and predicate symbol < has arity 2, the following are sample terms of predicate calculus:

0, succ(0), succ(succ(0)),..., add(0, 0), add(0, 1), add(1, 0), add(1, 1),...

and the following are sample formulas:

< (0, 0),< (0, 1),< (1, 0),< (1, 1),< (succ(0), 0),< (0, succ(0)),...

Semantics

The semantics of predicate calculus give meaning to its formulas. This meaning is derived from aninterpretation of terms andpredicates,the syntactic units usedinthe construction offormulas. Terms are interpreted as objects and predicates as relations. An interpretation I is a tuple �D, M� where the domain D is a(non-empty) set of objects and the map M is defined s.t.

M

• for all constant symbols a, ais an object in the set D
• for all n-ary function symbols f, f^M : Dⁿ → D (i.e., f^M is an n-ary function on D)
• for all n-ary predicate symbols P , P ^M : Dⁿ →{T,F} (i.e., P ^M is an n-ary relation on D)

Example: Continuingour example,given alphabet Adefinedabove,in the standardinterpretation I = �D, M� with domain D = N, the constant symbols 0 and 1 would map to themselves, the function symbol succ would map to the function n �→ n +1, the function symbol add would map to the addition function, and the predicate symbol < would map to the less-than relation (i.e., {(0, 1), (0, 2),..., (1, 2), (1, 3),..., (2, 3), (2, 4),...}).

The terms of predicate calculus are interpreted under I as follows:

M

• for all constant symbols a, I[a]= a
• for all terms t1 ,...,tn and n-ary function symbols f

– I[f(t1 ,...,tn)]= f^M (I[t1 ],..., I[tn])

As in propositional logic, I|= φ is defined inductively. Only in the base case does the definition for predicate calculus differ from that of propositional logic:

I|= P (t1 ,...,tn)iff(I[t1 ],..., I[tn])∈ P ^M

The notions of satisfiability, validity, and unsatisfiability extendimmediately topredicate calculus.

Example: Continuing our example yet again, given alphabet A defined above, in the standard interpretation withdomainN, the term succ(0)isinterpretedas succ^M (0^M ), whichis1. Similarly, the term add(0, succ(0))is interpreted as add^M (0^M , succ^M (0^M )), which is also 1.