Remarks+on+Propositional+Logic++PL+ - Remarks on...

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Remarks on Propositional Logic English Formal Language Symbolization Simple Sentences : state single facts, We&ll use capital English letters to or contain a single whole thought. represent simple sentences (choose a l e t t e r t h a t r e m i n d s y o u o f w h a t i s b e i n g s y m b o l i z e d ) : E x a m p l e s : I t s r a i n i n g . R B l a c k h o l e s e x i s t . B L P S 2 9 i s f u n . F The only even prime number is 2. E John thinks that Mary knows that angels T have wings. All dogs go to heaven. A The last three sentences seem, on the face of it, complex. But they do contain single thoughts, so they count (for this class) as simple sentences. Note that unlike syllogistic logic (SL) ignore subject/predicate structure, so ±All dogs go to heaven² is symbolized as a single capital letter representing a simple sentence. Compound Sentences: In English there are many devices for building compound (or complex) sentences out of simpler ones. We say things like ±UCI is a great school, but there are too many students here². This evidently contains two distinct thoughts, and the sentence is made out of two simpler sentences connected together with the word but. Expressions in English that build compound sentences out of more basic constituents (parts) are called connectives . They come in many varieties, and the ones we study in propositional logic are called truth-conditional . The definition is best approached via an example. Consider phrases of the form ±_______ and _______.², where the blanks may be replaced by any declarative sentences of English (for example, ±California is in a budget crisis and Californians favor low taxes.²). For any declarative sentences A and B, if you know A and B are both true, you know ±A and B² is true; if one or both of A, B is false, you know ±A and B² is false, too. This description completely characterizes the functional behavior of the connective ±and², in particular, if you know the truth-values (i.e. either true or false) of A and B, then solely on that basis you can determine the unique truth-value of ±A and B². Notice that in these examples the specific content of A and B is irrelevant in calculating the truth-value of ±A and B². A and B can represent any proposition expressed by an English declarative sentence.
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Contrast this case with the following &connective±: &G. Bush believes that ________________&. This is also a connective, in the sense that replacing the blank with any declarative sentence of English produces a new sentence (either true or false). However, it is not truth-functional. One way to see this is to show that there is a true sentence S 1 such that 1 ± is true, and another true sentence S 2 such that &G. Bush believes that S 2 ± is false . This would mean that we can not infer Bush²s beliefs simply by knowing whether the beliefs he espouses are true. For example, let S 1 be &2 + 2 = 4± and S 2 some true proposition of sufficiently difficult mathematics like &there exists first-
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Remarks+on+Propositional+Logic++PL+ - Remarks on...

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