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Introduction to Logic
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Tools for Thought
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Even though a set of sentences does not determine a unique world, it is often the case that some sentences are true in every world that satisfies the given sentences. A sentence of this sort is said to be a logical conclusion from the given sentences. Said the other way around, a set of sentences logically entails a conclusion if and only if every world that satisfies the sentences also satisfies the conclusion.
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What conclusions are logically entailed by these sentences? Quite a few, as it turns out. For example, it must be the case that Bess likes Cody. Also, Bess does not like Dana. There are also some general conclusions that must be true. For example, in this world with just four girls, we can conclude that everybody likes somebody. Also, everyone is liked by somebody.
Bess likes Cody. |
Bess does not like Dana. |
Everybody likes somebody. |
Everybody is liked by somebody. |
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One way to check whether a set of sentences logically entails a conclusion is to examine the set of all worlds in which the given sentences are true. For example, in our case, we notice that, in every world that satisfies our sentences, Bess likes Cody, so the statement that Bess likes Cody is a logical conclusion from our set of sentences.
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