the logic of boolean connectives

the logic of boolean connectives - 9/30/2007 The Logic of...

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9/30/2007 1 The Logic of Boolean Connectives Truth functional The Boolean connectives are said to be truth functional , because they change the value of the complex sentence based on the truth value of the components. e.g. The value of P Q is T only if P is True and Q is true. The change of truth value is predictable, because it follows from the truth value of the connectives. We talked about that in Chapter 3. Truth table for Conjunctions are true only if BOTH conjuncts are true. P Q P Q T T T T F F F T F F F F Truth value of complex sentences In this section we will see how we determine the truth value of sentences with more than one connective. We will see how to construct complex truth tables to determine the truth value of larger sentences. We will use the program Boole for constructing truth tables.
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9/30/2007 2 Introductory ideas There are three cases that we must examine in some detail: Logical truth Logical possibility Logical necessity Logical truth There are sentences that are always true. That is, no matter what the circumstances or what your world looks like, the sentence is true. We have seen an example in Fitch: the rule Intro allows us to introduce a new object: a = a Precisely because this sentence is always true, we were allowed to introduce it. Logical necessity Sentences that are always true have the property of logical necessity . A logically necessary sentence is true in every possible world. a = a Logical possibility A sentence is logically possible if that sentence can be true in a given situation. It can also be false: it depends on the world you are talking about. Cube(a) is logically possible in T arski’s world. You can construct a world in which it is true, but also one in which it is false.
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9/30/2007 3 Example… ¬( Tet (b) ∨ Cube(b) ∨ Dodec(b)) This sentence is logically possible. It is possible to
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the logic of boolean connectives - 9/30/2007 The Logic of...

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