Introduction to Biconditionals and Disjunctions

Agenda

1. Quiz
2. Biconditional Proofs

Quiz

A.

1. ~P>R
2. (P>~Q)>~R    
3. P>~Q     /:. P<>~Q

B. 

1. Unless you BRUSH your teeth and SHOWER you shouldn't WRESTLE with me.
2. If you neither did the HOMEWORK nor came to OFFICE hours you're going to have a BAD time.
3. You should PANIC only if you don't know how to do MODUS ponens and modus TOLLENS.

Setting Up Proofs with Biconditional (Biconditional Strategy)
You will use this method anytime a biconditional is the bottom line unjustified line of your proof. Think of <> as & for two >s. In other words, I going to have to do two proofs: one proof for one direction and one for the other.
Suppose you have to solve for P<>Q. P<>Q is logically equivalent to P>Q & Q>P. This means in order to prove P<>Q I'm going to have to do two proofs: one for P>Q and one for Q>P then I can put them together using <>Int.

Step 1: Write P<>Q at the bottom of your proof.

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x. P<>Q

Step 2: I know that P<>Q is the same as P>Q and Q>P. Scan the given assumptions above to see if I'm given either of the conditionals. If I'm not given any then I'm going to set up two proofs: one for P>Q and one for Q>P. If one of the two conditionals is given to me in the assumptions then I only need to set up a proof for the conditional not given in the assumptions.

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e. Q>P
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m. P>Q
x. P<>Q        <>Int e, m

Step 3: Now that I've set up my proof I go about independently solving for both P>Q and Q>P beginning with CP as we've learned from previous lessons.

a.    Q    ACP
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d.    P     ____
e. Q>P  CP a, d
f.    P    ACP
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l.    Q  ____
m. P>Q  CP f, l
x. P<>Q  <>Int. e, m.


Proofs with <> + MP, MT, DN, &Int, &Elim, CP
A.

1. ~P>R
2. (P>~Q)>~R    
3. P>~Q     /:. P<>~Q

B.   

1. P&~Q  /:. P<>~Q

C. 

1. ~(C&~B)>~A    /:. (A&~B)<>A

D.

1. (S>R)&~R
2. (~S>Q)&P
3. P>R        /:.  (P&Q)<>(~S>R)

E. 

1. (C<>~D)&C
2. B<>~A      /:.  (A>~B)&(C&~D)

F. 

1. ~S>(~Q&R)
2. S>~(P>~Q)    /:.  ((P>~Q)&R)<>~S