Slides / Bullets
 Formal proofs and the quantifiers
 Straightforward rules
 Here x can be any variable, c can be any individual constant [or variablefree term] in the language, S(x) stands for any formula whose only free variable is x, and S(c) stands for the result of replacing all occurrences of x in S(x) with c.
 Generous use: you can eliminate more than one universal quantifier at once.

 Here again, x may be any variable; c may be any individual constant (or variablefree term); S(c) may be any sentence containing zero or more occurrences of c; S(x) is the result of replacing some or all of these occurrences of c with occurrences of x.
 Generous use: you can introduce more than one existential quantifier at once.
 Methods of proof
 Generous interpretation: eliminate more than one at once, using more than one boxed constant

 ∃elim corresponds to the informal method of existential generalisation. This lets us, having established something of the form ∃xS(x), introduce a new ‘dummy’ name c and assert S(c).
 Example: suppose we are given the premises ∃x(Cube(x) ∧ Small(x)) and ∀x(Cube(x) → (Small(x) → Larger(a,x))); we want to prove that ∃xLarger(a,x). We can argue as follows: by the first premise, there is a small cube. Call it Tiny. By the second premise, if Tiny is small, a is larger than it; but since Tiny is small, it follows that a is larger than Tiny. Hence a is larger than something.

 Corresponds to the informal method of general conditional proof. Suppose that, just from the assumption that P(c), where c is some new dummy name, you can derive (given other things you’ve established) that Q(c). Then you’re entled to conclude that all P’s are Q’s.
 Examples are very common in mathematics.

 As a special case, we allow this rule to be used with an empty assumption, thus:
 As with the other rules, has a generous use, on which you can introduce more than one universal quantifier, using more than one boxed constant.
 For next week
 Read: through chapter 13
 Do: 13.2, 13.3, 13.4, 13.7, 13.11, 13.12, 13.13, 13.16; (8% each); 12.2, 12.3 (10% each); write informal proofs based on your formal proofs of 13.2, 13.7, 13.13 and 13.16 (4% each).