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 examG= Gerald passed the exam1. ~(B · G)2. GC. ~B

Example

1. ~(B · G) prem2. G prem/~B

Example

1. ~(B · G) prem2. G prem/~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) prem2. G prem/~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) prem2. G prem/~B3.

So let’s replace line 1 with an equivalent statement form.

Example

1. ~(B · G) prem2. G prem/~B3. ~B v ~G 1, DeM

So let’s replace line 1 with an equivalent statement form.

Example

1. ~(B · G) prem2. G prem/~B3. ~B v ~G 1, 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) prem2. G prem/~B3. ~B v ~G 1, DeM

And now we can straightforwardly conclude the proof.

Example

1. ~(B · G) prem2. G prem/~B3. ~B v ~G 1, DeM4. ~B 2,3 DSQED

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 EI 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 cheeseC :: C · C

Am I right or am I right? (rhetorically)R :: R v R