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Language, Proof and
Logic
Conditionals
Chapter 7
7.1
Material conditional symbol: 
Truth table:
P
Q
PQ
T
T
F
F
T
F
T
F
T
F
T
T
Game: PQ is treated as PQ
The following English constructions are all translated as PQ:
If P then Q;
P only if Q;
Provided P, Q.
The following are translated as PQ:
Unless P, Q;
Q unless P.
Q is a logical consequence of P1,…,Pn iff (P1…Pn)Q is a
logical truth.
7.2
Biconditional symbol: 
Truth table:
P
Q
PQ
T
T
F
F
T
F
T
F
T
F
F
T
Game: P  Q is treated as (PQ)(QP)
The following English construction is translated as PQ:
P if and only if (iff) Q.
P and Q are logically equivalent (PQ) iff PQ is a logical truth.
7.3
Conversational implicature
If the assertion of a sentence carries with it a suggestion that could be
cancelled (without contradiction) by further elaboration by the speaker,
then the suggestion is (just) a conversational implicature, not part of the
content of the original claim.
The waiter says: “You can have either soup or salad”.
The suggested exclusiveness is cancelled by the further elaboration:
“And you can have both if you wish”.
A father tells his son: “You can have dessert only if you eat all your
beans”.
The suggested promise of dessert is cancelled by the further elaboration:
“If you eat the beans, I’ll check to see if there is any ice cream left”.
7.4.a
Truth-functional completeness
Do we need to introduce any more truth-functional connectives?
Such as, say,
Neither P nor Q
P unless Q
If P then Q, otherwise R
… etc.
Not really!
Theorem: The collection {,,} is truth-functionally complete, in the
sense that these connectives allow us to express any truth function.
In fact, so is just {}, where PQ means “neither P nor Q”.
7.4.b
Truth-functional completeness
Express (P,Q,R), defined by:
Using only , express:
• P
• PQ
• PQ
P
Q
R
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
(P,Q,R)
T
T
F
F
T
F
T
F