Slides / Bullets

• Formal proofs and the quantifiers
• Straightforward rules
  • Here x can be any variable, c can be any individual constant [or variable-free 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 variable-free 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).