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boolean connectives

boolean connectives - More complex claims of logic Boolean...

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12/4/2007 1 LING 178 Boolean Connectives More complex claims of logic So far, we were only able to do simple logical operations. In chapter 3 we will extend our scope by learning about boolean connectives which allow us to make more complex logical claims. How would we express the statement in Tarski’s world that Cube a is large ? What is wrong with Large(Cube(a))? We are trying to put a relation in the place of the argument and we saw last time that that doesn’t work. We are trying to say two things about Object a, namely that it’s a cube and that it’s large. Another way of saying this is that Object a has two relations that are both true of Object a. a is a Cube a is Large Connectives For cases such as this, boolean logic has a variety of so-called connectives that allow us to make complex claims. The connectives we are using for now are ¬ “not” “and” “or” Cube(a) Large(a) Properties of connectives These connectives have specific meanings in boolean logic. Even though we can describe them with English words, that doesn’t mean that they are exactly equivalent to English words. The boolean connectives only deal with truth values , while their English counterparts can also express other nuances of language. Example There is no difference between, for instance CameHome(max) Left(minnie) and Left(minnie) CameHome(max) Both have the same truth value. In English, there is a difference between Max came home and Minnie left and Minnie left and Max came home . English and carries a sense that things are done in order. If you have a sentence of the form S1 and S2, the sense is that S1 occurs before S2.
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12/4/2007 2 Truth These connectives manipulate the truth value of sentences. There are rules as to how the truth or falsehood of a relation is assessed For instance, If you believe Large(c) to be true, then ¬Large(c) is false We check truth values by constructing so-called truth tables . Negation
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