Conditionals

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Conditionals
Language, Proof and Logic
Chapter 7
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Material conditional symbol ?
7.1
P Q P?Q T T
T T F F F
T T F F
T
Truth table 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.
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Biconditional symbol ?
7.2
P Q P?Q T T
T T F F F
T F F F
T
Truth table 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.
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Conversational implicature
7.3
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, Ill check to
see if there is any ice cream left.
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Truth-functional completeness
7.4.a
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.
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Truth-functional completeness
7.4.b
Express ?(P,Q,R), defined by
P Q R ?(P,Q,R) T
T T T T
T F T T F
T F T F F
F F T T
T F T F
F F F T
T F F F F

• Using only ?, express
• ?P
• P?Q
• P?Q