**Chapter 5: Full First-Order Logic**

**Summary**

- Full first-order predicate logic builds on monadic predicate logic by introducing relational or polyadic predicates.
- Relational, or polyadic, predicates are followed by more than one singular term. Dyadic predicates are followed by two singular terms. Triadic predicates are followed by three singular terms.
**F**is the formal object language of full first-order predicate logic.- The syntax of
**F**specifies its vocabulary and rules for making formulas. - The vocabulary of
**F**is identical to that of**M**; it includes lower-case letters, uppercase letters, the five propositional operators (~, •, ˅, ⊃, ≡), two quantifier symbols (∃, ∀), and punctuation marks (), [], {}. **F**has five formation rules: (1) a predicate followed by any number of singular terms is a wff; (2) for any variable β, if α is a wff that does not contain either ‘(∃β)’ or ‘(∀β)’, then ‘(∃β)α’ and ‘(∀β)α’ are wffs; (3) if α is a wff, so is ~α; (4) if α and β are wffs, then so are (α • β), (α ˅ β), (α ⊃ β), and (α ≡ β); and (5) these are the only ways to make wffs.- A quantifier’s scope is wider the more subformulas it contains; it is narrower the fewer subformulas it contains.
- The semantics of
**F**specifies the rules for interpreting the symbols and formulas of the language. - An interpretation of a theory of
**F**proceeds in four steps: (1) specify a set to serve as a domain of interpretation; (2) assign a member of the domain to each constant; (3) assign some set of objects in the domain to each monadic predicate, and assign sets of ordered*n*-tuples to each relational predicate; and (4) use the customary truth tables for the interpretation of the propositional operators. - An
*n*-tuple is a set with structure used to describe an n-place relation. It is a general term for pairs, triples, quadruples, and so on. - The semantic definitions of validity and logical truth are the same in
**F**as in**M**. - Our system of inference for constructing derivations in
**F**is the same as that for constructing derivations in**M**, with one addendum: never UG on a variable when there’s a constant present, and the variable was free when the constant was introduced. - Our system of inference for constructing derivations in
**F**can be extended to include identity relations. - The identity relation is a dyadic relation between two names of a single object. The notation ‘a=b’ is short for ‘Iab’, taking ‘Ixy’ as the identity relation.
- Negations of identity claims, such as ~Ixy, can be expressed in two ways: ~x=y or x≠y.
- A definite description picks out an object by using a descriptive phrase beginning with ‘the’.
- Our system of inference for constructing derivations in
**F**, extended to include identity relations, uses the rules of**F**plus three additional rules governing identity: IDr (reflexivity), IDs (symmetry), and IDi (indiscernibility of identicals). - IDr, or reflexivity, is an axiom schema that says that any singular term stands for an object that is identical to itself. For any singular terms α and β, α=α.
- IDs, or symmetry, is a rule of equivalence that says that identity is commutative: if one thing is identical to another, then the second is also identical to the first. For any singular terms α and β, α=β :: β=α.
- IDi, or indiscernibility of identicals, is a rule of inference that says that if you have α=β, then you may rewrite any formula containing α with another formula that has β in the place of α throughout. For any singular terms α and β,
*F*α, α=β /*F*β. - The indiscernibility of identicals should not be confused with the identity of indiscernibles, the contested principle that states that no two things share all properties.
- Full first-order predicate logic with functors builds on the logic of
**F**by introducing functors. - A functor is a symbol used to represent a function.
**FF**is the formal object language of full first-order predicate logic with functors.- The syntax of
**FF**specifies its vocabulary and rules for making formulas. - The vocabulary of
**FF**includes lower-case letters, uppercase letters, the five propositional operators (~, •, ˅, ⊃, ≡), two quantifier symbols (∃, ∀), and punctuation marks (), [], {}. - Uppercase letters A . . . Z are used as predicates.
- Lower-case letters a, b, c, d, e, i, j, k . . . u are used as constants; f, g, and h are used as functors; and v, w, x, y, z are used as variables.
**FF**has five formation rules: (1) an n-place predicate followed by n singular terms (constants, variables, or functor terms) is a wff; (2) for any variable β, if α is a wff that does not contain either ‘(∃β)’ or ‘(∀β)’, then ‘(∃β)α’ and ‘(∀β)α’ are wffs; (3) if α is a wff, so is ~α; (4) if α and β are wffs, then so are (α • β), (α ˅ β), (α ⊃ β), and (α ≡ β); and (5) these are the only ways to make wffs.- A semantic interpretation of a theory of
**FF**proceeds in five steps: (1) specify a set to serve as a domain of interpretation; (2) assign a member of the domain to each constant; (3) assign a function with arguments and ranges in the domain to each function symbol; (4) assign some set of objects in the domain to each one-place predicate, and assign sets of ordered*n*-tuples to each relational predicate; and (5) use the customary truth tables for the interpretation of the propositional operators. - Our system of inference for constructing derivations in
**FF**is the same as that for constructing derivations in**F**, with three additional restrictions regarding when you can and cannot introduce new functional structure into a formula: (1) you may increase functional structure when using universal rules (UI or UG), but you may not decrease it; (2) you may decrease functional structure when using existential rules (EI or EG), but you may not increase it; and (3) if you change the functional structure of a wff, change it uniformly throughout. - Functional structure reflects the complexity of a functor term or of the
*n*-tuple of singular terms in a functor term.