Slides / Bullets

• General first-order languages
  • Sometimes we’ll ask you to translate English sentences into a first-order language that you design yourself.
  • This sort of task can be done in many ways.
    • Consider how we might translate ‘John prefers logic to mathematics’. The most natural way to do it is to use a ternary predicate and three individual constants: ‘Prefers(john, logic, mathematics)’.
    • But you could do it with a binary predicate and two constants: ‘PrefersToMathematics(john, logic)’, or ‘JohnPrefers(logic, mathematics). Or even a unary predicate: ‘JohnPrefersToMathematics(logic)’
• • The three-place predicate is obviously more flexible. In general, when you’re doing this kind of exercise, you want to aim for naturalness and flexibility.
• Function symbols
  • Some FOL dialects include an additional sort of vocabulary, function symbols.
  • Function symbols can be used to make complex terms, which function grammatically just like individual constants.
    • EG: favouriteactor(cian), favouriteactor(father(cian)), favouriteactor(favourtieactor(cian))
  • A function symbol has an arity just like a predicate, but we’ll write function symbols in lower case so there’s never any confusion.
• • (In prefix notation:) A complex term is the result of writing an n-ary function symbol followed by n terms (in parentheses, separated by commas), which may themselves be simple, i.e. individual constants, or complex.
  • An atomic sentence is the result of writing an n-ary predicate letter followed by n terms (in parentheses, separated by commas).
    • Happy(father(joe)); OlderThan(father(joe), joe).
    • This is nonsense: Happy(Happy(joe)).
• • Just as we require all individual constants to denote exactly one thing, so we require all complex terms to denote exactly one thing.
    • So we can’t have a function symbol ‘sonOf’, unless everyone in the domain we’re talking about is a person with exactly one son.
  • Just as the identity predicate ‘=’ is traditionally written in ‘infix’ notation, so certain function symbols are traditionally written in infix notation.
• The language of arithmetic
  • Binary predicates: =, <
  • Individual constants: 0, 1
  • Binary function symbols: +, ×
• Proofs
  • We’ll introduce three methods for showing that a certain claim is a logical consequence of certain premises: informal proof, formal proof and truth-tables.
  • In a proof, we start with the given premises, and step by step we establish intermediate conclusions that obviously follow from things we’ve already said, until eventually we reach the desired conclusion.
    • Actually that’s only the most basic sort of proof: later on we’ll introduce more complicated methods of proof that rely on subproofs.
• • In an informal proof, you’re allowed to make any step provided it’s obvious to your audience how it follows from what you’ve already said.
    • Unless your audience is a logic teacher, in which case the standard for ‘obviousness’ is higher.
    • Informal proofs in logic should be completely rigorous. You’ll have to develop a special writing style: a useful skill.
  • In a formal proof, the allowable steps are codified into a fixed set of mechanical rules.
• Informal proofs using atomic sentences
  • Given what we know about the meaning of the predicates in the blocks language, there are plenty of obviously valid arguments, e.g.:
    • ‘LeftOf(a, b)’ entails ‘RightOf(b, a)’
    • ‘LeftOf(a, b)’ and ‘LeftOf(b, c)’ entail ‘LeftOf(a, c)’
    • ‘LeftOf(a, b)’ and ‘SameCol(b, c)’ entail ‘LeftOf(a, c)’
  • Too many to list or codify in a formal system of proof.
• • The identity predicate ‘=’ is of special interest: it’s one of the bits of vocabulary that logic has traditionally been especially concerned with.
    • The most important method of proof involving identity goes by the names identity elimination, substitution, the indiscernability of identicals and Leibniz’s Law.
      • Roughly: if we have established from our given premises that a=b, we can infer that whatever is true of a is true of b.
      • Given a premise that involves a certain name, say a, and a premise of the form a = b, we can infer the result of substituting b for a in the first premise. Familiar from algebra.
• • We also have the rule of identity introduction, aka the reflexivity of identity: this lets us infer a sentence of the form ‘a=a’ from whatever premises we please, or from none at all.
    • Logically true sentences are sentences that must be true. They follow from every other sentence. We can also say that they follow from the null set of premises.
    • The assumption that all names have referents is playing a crucial role here. Is ‘Santa is identical to Santa’ a true sentence in English?
• • Other useful principles:
    • the symmetry of identity (from ‘a = b’ we can conclude ‘b = a’)
    • the transitivity of identity (from ‘a = b’ and ‘b = c’, conclude ‘a = c’)
  • These can in fact be derived from the first two principles. (B&E derive symmetry on p. 50)
• Problems for next week:
  • 1.9 (30%)
  • 1.11 (25%)
  • 2.6 (15%)
  • 2.8 - 2.13 (30%)