Philosophy 200 substitution rules

Substitution Sometimes, when you translate a statement from English to SL, you translate it in a form that is less useful to a proof than some other, equivalent form.

Example “Biff and Gerald didn’t both pass the exam, and Gerald passed, so Biff must not have passed the exam.” B = Biff passed the exam G= Gerald passed the exam 1.~(B · G) 2.G C. ~B

Example 1.~(B · G)prem 2.Gprem/~B

Example 1.~(B · G)prem 2.Gprem/~B What to do? We have no rules to deal with negations, and that’s all we have to work with.

Example 1.~(B · G)prem 2.Gprem/~B This is where a different translation would have helped us. But you don’t really know ahead of time which translation will work in your proof. (unless you use a system with a good enough rule set to handle any given translation…)

Example 1.~(B · G)prem 2.Gprem/~B 3. So let’s replace line 1 with an equivalent statement form.

Example 1.~(B · G)prem 2.Gprem/~B 3.~B v ~G1, DeM So let’s replace line 1 with an equivalent statement form.

Example 1.~(B · G)prem 2.Gprem/~B 3.~B v ~G1, DeM This is an example of DeMogan’s law which proves that the statement forms on 1 and 3 are equivalent. (also that ~(P v Q) is equivalent with ~P · ~Q)

Example 1.~(B · G)prem 2.Gprem/~B 3.~B v ~G1, DeM And now we can straightforwardly conclude the proof.

Example 1.~(B · G)prem 2.Gprem/~B 3.~B v ~G1, DeM 4.~B2,3 DS QED

Important Tip Continue to think of the replacement rules as alternate ways that a statement could have been translated from English into SL.

De Morgan’s Laws Pip and Quincy don’t both speak Klingon. ~(P · Q) :: ~P v ~Q Neither Pip nor Quincy speaks Klingon. ~(P v Q) :: ~P · ~Q

Association The number in this box could be a five, a six, or an eight. F v (S v E) :: (F v S) v E I wear pants, hats, and shoes P · (H · S) :: (P · H) · S

Double Negation Bob doesn’t not smoke ~~B :: B

Material Implication If you pay me the protection money, then I’ll beat you up. P  B :: ~P v B

Exportation If the smurf has plumped up and if it has light grill marks, then it’s cooked perfectly. P  (G  C) :: (P · G)  C

Commutation Jack and Jill went up the hill. (M · F) :: (F · M)

Distribution You can get the burger and have mustard or mayo on it. (B · Y) v (B · D) :: B · (Y v D) You can make a deal or you can stand trial and go to jail. (A v B) · (A v C) :: A v (B · C)

Transposition If you don’t study, you won’t pass. ~S  ~P :: P  S

Biconditional Equivalence You are a mother if and only if you are a female parent. (M  F) :: (M  F) · (F  M) :: (M · F) v (~M · ~F)

Tautology There’s cheese and then there’s cheese C :: C · C Am I right or am I right? (rhetorically) R :: R v R