Chapter 05 Summary

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, Fα, α=β / Fβ.
  • 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.

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