Lecture 12: Contradictions, explosions & negation

Contradictions and Elimination

  • In proofs we don’t use just premises of an arg but also sentences we have proved (new lines)

  • & extra sentences we have assumed in subproofs for → I and V E

  • it can happen that we end up with sentences in a proof

    that are actually incompatible with each other

Not elimination rule

  • Involves elimination of a negation symbol

  • First line must be negated sentence

  • Proof strag: If you have negation as a premise,

    try to derive a contradiction to eliminate the negation

Explosion

  • Can’t use for every proof bc you need a contradiction first which you can’t assume

  • t any argument

    in which the premises are with jointly unsatisfiable (i. e., contradictory) is valid.

    (Think about how a truth table for such an argument would look like!)

    Formulated differently, we get the principle: ‘from a contradiction, anything follows’

    — or, in Latin, ex falso quodlibet

This suggests another proof rule: whenever we have ‘ ? ’, we can justify anything

X Rule

m ?

k P X m

• Here, the ‘X’ here stands for ‘eXplosion’, although we could have called this rule also

? -Elimination (but we don’t have a ? -Introduction either).

• This rule cites a single line, which can only be ‘ ? ’.

This rule finally allows us to complete our proof of the Disjunctive Syllogism:

1 A _ B Premise

2 ¬B Premise

3 A Assumption

4 A R 3

5 B Assumption

6 ? ¬E 2, 5

7 A X 6

8 A _E 1, 3–4, 5–7

  • Proof strategy (9): (Warning) You might be tempted to use the explosion rule frequently

to derive whatever you want, but that often leaves you with unwanted open assumption

Introducing ‘¬’

  • The last two rules we introduced were:

  • Official name Alternative name

  • ¬E ?I

  • X ?E

• To justify ‘ ¬P’, show that ‘ P’ (together with all other premises) is unsatisfiable.

  • However ‘unsatisfiable’ is a semantic term, so we cannot use it in our proof system. . .

  • But, a sentence that is clearly unsatisfiable is a contradiction, which is defined syntactically (as the conjunction of a sentence and its negation).

  • Thus, we get the purely syntactic rule:

• To justify ‘ ¬P’, show that ‘ P’ (together with all other premises) proves a contradiction (‘ ? ’)

The ¬ I rule cites a range of lines, because it refers to a subproof.

• The subproof that is cited must end with ‘ ? ’ in the same subproof (column)

as the initial assumption.

Thus, we are licensed to conclude that that initial assumption P is unsatisfiable,

hence the contrary ( ¬P) must be true.

• Notice a close relation between ¬P and (P ! ?) ,

which we could have obtained from applying ! I to the same subproof.

  • Proof strategy (10): If you have a negation sentence as the conclusion, it is likely that the last rule in the proof will be ¬ I, so that you can start by assuming the unnegated sentence

Indirect proof

  • The -I rule is closely related to another rule naely the indirect proof rule

  • Corrosponds to a very old method of proof traditionally called reductio or reductio ad absurdum or reductio ad impossible

  • Idea is to claim P

  • By first assuming its negation, ¬P,

  • And then showing that this leads to a contradiction

  • The IP might be unintuitive, but it is very powerful!

Proof strategy (11): Whatever the conclusion is, it could be proved using the IP Rule

  • So, always keep this option as a possibility when thinking about how to prove something.

• You can think about the IP Rule as a combination of the ¬ I Rule and the elimination of double negation: (¬ ¬ P ! P)

• This is a very popular proof technique in classical mathematics, but it is not valid in constructive or intuitionistic mathematics! (If you think this rule is fishy, you might have come good constructive intuition)