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 wellformed 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 nary 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.)