philosophy 200 substitution rules. substitution sometimes, when you translate a statement from...
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Philosophy 200
substitution rules
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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.
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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
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Example
1. ~(B · G) prem2. G prem/~B
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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.
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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…)
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Example
1. ~(B · G) prem2. G prem/~B3.
So let’s replace line 1 with an equivalent statement form.
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Example
1. ~(B · G) prem2. G prem/~B3. ~B v ~G 1, DeM
So let’s replace line 1 with an equivalent statement form.
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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)
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Example
1. ~(B · G) prem2. G prem/~B3. ~B v ~G 1, DeM
And now we can straightforwardly conclude the proof.
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Example
1. ~(B · G) prem2. G prem/~B3. ~B v ~G 1, DeM4. ~B 2,3 DSQED
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Important Tip
• Continue to think of the replacement rules as alternate ways that a statement could have been translated from English into SL.
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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
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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
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Double Negation
Bob doesn’t not smoke~~B :: B
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Material Implication
If you pay me the protection money, then I’ll beat you up.
P B :: ~P v B
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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
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Commutation
Jack and Jill went up the hill.(M · F) :: (F · M)
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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)
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Transposition
If you don’t study, you won’t pass.~S ~P :: P S
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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)
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Tautology
There’s cheese and then there’s cheeseC :: C · C
Am I right or am I right? (rhetorically)R :: R v R