LPL 9.1-9.4 lecture

LPL 9.1-9.4 lecture - Quantifiers So far, all of our uses...

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Quantifiers So far, all of our uses of FOL have been limited to the following sorts of sentences: Max is home and Claire is at the library. Home(max) Library(claire) a is not a cube or c is not a tetrahedron. Cube(a) Tet(c) If b is small then it is a dodecahedron. Small(b) Dodec(b) Notice that all these sentences concern named objects. The reason for this is that we are only able, at this point, to build our atomic sentences from a variety of predicates and individual constants. This limitation affects what we are able to say in FOL. In English and other natural languages, we build sentences by combining noun phrases and verb phrases. The simplest noun phrases are names, such as Max , Claire , a , b , c . To build more complex noun phrases, we need to use determiners like a , the , every , some , most , one , and no . Using determiners, we can construct noun phrases like: a cube the tetrahedron every student in this class some painters most politicians one apple no actors These complex noun phrases are called quantified expressions and sentences that contain them are called quantified sentences . Only two forms of quantification have been built explicitly into FOL. This language uses the quantifier symbols " and $ to express the English determiners every and some . Though these two quantifier symbols only correspond directly to two determiners, many other forms of quantification can be expressed using combinations of these quantifier symbols and the truth-functional connectives we have already encountered. We will be able to say things like: Every cube is large There are three blind mice. No students are tall. However, some quantified expression will remain outside the scope of FOL, such as: Most students enjoy logic. Many cubes are large.
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Variables and atomic wffs FOL contains: predicates =, Cube, Small, Between, FrontOf, etc. individual constants a, b, c, d, e, f, …, s truth-functional connectives , , , , quantifiers " , $ individual variables x, y, z, t, u, v, w Variables are used with predicates just as individual constants are, but the resulting statements will be different in each case. As we already know, when predicates are combined with individual constants we get atomic sentences. However, when predicates are combined with variables we get atomic well-formed formulas, or atomic wffs.
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This note was uploaded on 06/06/2011 for the course PHIL 110 taught by Professor ? during the Fall '06 term at South Carolina.

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LPL 9.1-9.4 lecture - Quantifiers So far, all of our uses...

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