quantification and quantifiers

quantification and quantifiers - 12/4/2007 PROPOSITIONAL...

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12/4/2007 1 QUANTIFICATION Chapter 9 PROPOSITIONAL LOGIC What we have been doing over the last couple of weeks is called propositional logic . Propositional logic deals with the logic of simple atomic sentences with named objects. Cube(a) refers to a property of an object a . We know which object that is. We are now going to extend our understanding of logic to include more complex cases. WHAT IS WRONG WITH PROPOSITIONAL LOGIC? Suppose we want to say “All objects are cubes.” What we can claim with propositional logic. 1 atomic sentence: 1 property of 1 object: Cube(a) 1 relationship of a fixed number of objects: Between(a,b,c) Connectives Finite number of relationships (properties), each with a fixed number of objects Conjunction, disjunction, conditional symbol, … Cube(b) ((Tet(a) (Dodec (c)) → LeftOf(a,b)) PROPERTIES OF PROPOSITIONAL CALCULUS In a propositional calculus (a formal system): Objects must be named. “Finite” claims We have a limited number of objects and relations Can we use propositional logic to claim something like the following? All objects are small. No objects are tetrahedron. Some object is large. We can express sentences like “all objects are small” in a limited domain like Tarski‟s World simply by listing them all individually: Small(a) Small(b) Small(c) This becomes impractical and impossible if we expand the domain to the real world. How would we express a sentence like “all actors are rich” in propositional logic? You would have to list all actors individually, which is clearly impossible to do. If we don‟t have a limited domain, then we will need a new mechanism, one which allows us to say things like “all actors are rich” without having to list them all. This new mechanism is called predicate logic and it involves the notion of quantification . Predicate logic is also called first-order logic .
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12/4/2007 2 CREATING QUANTIFIED SENTENCES Redefining terms Defining well-formed formulas (wff) Redefining sentences ATOMIC SENTENCES ” ( THUS FAR) Formed by a single predicate followed by one or more terms: e is larger than b Larger(e,b) e is identical to a e = a A sentence expresses a claim that is either true or false . QUANTIFICATION To be able to work with ideas such as “all”, “most” or „none” we need to be able to talk about objects as a group (set), rather than as individual objects. We do that by introducing the notion of a variable , which is a reference to an object without the need of specifying what that object is. It is similar to the use of a variable in algebra.
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quantification and quantifiers - 12/4/2007 PROPOSITIONAL...

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