Slides / Bullets

• Quantifiers
• Quantified sentences in English
  • Basic sentences are made by combing a noun phrase (NP) with a verb phrase (VP).
  • Names are noun phrases; but there are others: all cats, some dogs, the teacher, some black dogs who chase cats, most Americans, something, everything....
    • Words like ‘all’, ‘some’, ‘the’, ‘most’ are called determiners.
• • The syntax of FOL works very differently. The work done by determiners (and various other forms of expression) in English is all done by just two symbols, ∀ and ∃ , which correspond roughly to ‘Everything’ and ‘Something’, together with variables like ‘x’, ‘y’, ‘z’, which correspond roughly to pronouns in English.
• • Some examples:
    • ∀xMeaningless(x) means ‘For every object x, x is meaningless’, or more colloquially, ‘Everything is meaningless’.
    • ∃xOmnipotent(x) means ‘For some object x, x is omnipotent’ — ‘Something is omnipotent’ — ‘There is an omnipotent thing’.
    • ∃x(Dog(x)∧Omnipotent(x)) means ‘For some object x, x is a dog and x omnipotent’ — ‘Something is both a dog and omnipotent’ — ‘Some dog is omnipotent’
    • ∀x(Man(x)→Mortal(x)) means ‘For any object x, if x is a man, then x is mortal’ — ‘For any object x, either x isn’t a man or x is mortal’ — ‘All men are mortal’.
• Syntax
  • So far, we’ve been looking at sentences that are built up from atomic sentences.
    • But ∃x(Dog(x)∧Omnipotent(x)) is not built up from atomic sentences. ‘Dog(x)’ is not a sentence at all: the symbol x is a variable, not a name. ‘Dog(x)’ is not the sort of thing that can be true or false.
    • Expressions like ‘Dog(x)’ and ‘(Dog(x)∧Omnipotent(x)) are called well-formed formulas or wffs. All sentences are wffs, but not all wffs are sentences.
• • Now that we’ve introduced the quantifiers, we’re in a position to give a precise account of the syntax of FOL. Let’s first deal with the kind of language that doesn’t contain any function symbols.
    • A variable is one of the letters t, u, v, w, x, y, z, with or without a numerical subscript.
    • A term is a variable or an individual constant.
    • An atomic wff consists of an n-ary predicate together with a list of n terms, separated by commas and surrounded by parentheses.
• • We define the notion of wff as follows:
    • If P is a wff, ¬P is a wff
    • If P1... Pn are all wffs, (P1∧...∧Pn) and (P1∨...∨Pn) are both wffs.
    • If P and Q are wffs, (P→Q) and (P↔Q) are both wffs.
    • If P is a wff and v is a variable, then ∀vP and ∃vP are both wffs, and all occurrences of v inside P are said to be bound.
    • Nothing else is a wff.
  • A sentence is a wff that contains no free (unbound) variables.
    • By convention, we can leave off the outermost parentheses.
• • Which of the following wffs are sentences?
    • ∃xDog(x)
    • ∃xDog(x)∧Omnipotent(x)
    • ∃x(Dog(x)∧Omnipotent(x))
    • ∀x(Dog(x)→∃y(Flea(y)∧IsOn(y,x)))
    • ∀x(Dog(fido))
• Semantics for the quantifiers
  • In different versions of FOL, the quantifiers have different domains. For example, in the language of arithmetic, the domain of the quantifiers is the natural numbers, so ∀x(Even(x)∨Odd(x)) is true.
  • In the blocks language of Tarski’s World, the domain of the quantifiers comprises the blocks in the given world.
  • Playing the game.
• For next week
  • Read: chapter 9; optionally, chapters 10 and 11.
  • Do: exercises 8.31, 8.33, 8.34 and 8.37 (don’t forget to look back at the informal proofs you gave in last week’s homework); 8.26 - 8.28 (you may use Taut Con to justify an instance of Excluded Middle); 9.1, 9.2, 9.6. (10% per exercise.)