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 ’Äòall’Äô, ’Äòsome’Äô, ’Äòthe’Äô, ’Äòmost’Äô 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, ’àÄ and ’àÉ , which correspond roughly to ’ÄòEverything’Äô and ’ÄòSomething’Äô, together with variables like ’Äòx’Äô, ’Äòy’Äô, ’Äòz’Äô, which correspond roughly to pronouns in English.
• • Some examples:
    • ’àÄxMeaningless(x) means ’ÄòFor every object x, x is meaningless’Äô, or more colloquially, ’ÄòEverything is meaningless’Äô.
    • ’àÉxOmnipotent(x) means ’ÄòFor some object x, x is omnipotent’Äô ’Äî ’ÄòSomething is omnipotent’Äô ’Äî ’ÄòThere is an omnipotent thing’Äô.
    • ’àÉx(Dog(x)’àßOmnipotent(x)) means ’ÄòFor some object x, x is a dog and x omnipotent’Äô ’Äî ’ÄòSomething is both a dog and omnipotent’Äô ’Äî ’ÄòSome dog is omnipotent’Äô
    • ’àÄx(Man(x)’ÜíMortal(x)) means ’ÄòFor any object x, if x is a man, then x is mortal’Äô ’Äî ’ÄòFor any object x, either x isn’Äôt a man or x is mortal’Äô ’Äî ’ÄòAll men are mortal’Äô.
• Syntax
  • So far, we’Äôve been looking at sentences that are built up from atomic sentences.
    • But ’àÉx(Dog(x)’àßOmnipotent(x)) is not built up from atomic sentences. ’ÄòDog(x)’Äô is not a sentence at all: the symbol x is a variable, not a name. ’ÄòDog(x)’Äô is not the sort of thing that can be true or false.
    • Expressions like ’ÄòDog(x)’Äô and ’Äò(Dog(x)’àßOmnipotent(x)) are called well-formed formulas or wffs. All sentences are wffs, but not all wffs are sentences.
• • Now that we’Äôve introduced the quantifiers, we’Äôre in a position to give a precise account of the syntax of FOL. Let’Äôs first deal with the kind of language that doesn’Äôt 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’àß...’àßPn) and (P1’à®...’à®Pn) are both wffs.
    • If P and Q are wffs, (P’ÜíQ) and (P’ÜîQ) are both wffs.
    • If P is a wff and v is a variable, then ’àÄvP and ’àÉvP 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?
    • ’àÉxDog(x)
    • ’àÉxDog(x)’àßOmnipotent(x)
    • ’àÉx(Dog(x)’àßOmnipotent(x))
    • ’àÄx(Dog(x)’Üí’àÉy(Flea(y)’àßIsOn(y,x)))
    • ’àÄx(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 ’àÄx(Even(x)’à®Odd(x)) is true.
  • In the blocks language of Tarski’Äôs 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’Äôt forget to look back at the informal proofs you gave in last week’Äôs 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.)