06 Slides--Quantifiers

06 Slides--Quantifiers - CS103 HO#6 Slides-Quantifiers...

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CS103 HO#6 Slides--Quantifiers April 2, 2010 1 CS103 Mathematical Foundations of Computing 4/2/10 TA Office Hours: Steve Today 3:15 – 5:15 Gates 200 Karl Sunday 5:00 – 7:00 Gates B12 (need ID to get in) The Language of First Order Logic Jim James a b We want to use logic to reason about some domain of interest. We will use names, called constants , to refer to objects in that world. An object is anything we can make claims about. -- Every constant must name an actual object -- No constant can name more than one object -- An object can have more than one name, or no name 2 3 The number of arguments is called the arity of the predicate. A predicate applied to its arguments is a proposition that has a truth value. The Language of First Order Logic In addition to being able to refer to objects, we need to say that -- objects have certain properties -- objects relate to one another in certain ways To do this, we have predicates in our language. Jim is tall Tall ( Jim ) a and b are circles Circle ( a ) Circle ( b ) Stanley is not moving ¬ Moving ( Stanley ) 3 is greater than 2 GreaterThan ( 3 , 2 ) The Language of First Order Logic We write the arguments of a predicate in parentheses after the name of the predicate, except for the special predicate for equality: a = b and mathematical relationships: 3 > 2 -1 < 0 The Language of First Order Logic Function Symbols Given an object, it is often convenient to be able to refer to another object that is uniquely connected to the first one in some way. For example, if we are talking about Jim, we might also like to talk about Jim's father, even though we don’t know his name. We do this using Function Symbols. For the example above, we would write Father ( Jim ) Function Symbol Argument The Language of First Order Logic The value of a function symbol applied to argument(s) is an object Father ( Jim )i s George SquareRoot ( 4 s 2 The value of a predicate applied to its argument(s) is true or false Tall ( Jim s True 2 > 3 is False Function values may be used where objects are expected, and predicate values may be used where propositions are expected.
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06 Slides--Quantifiers - CS103 HO#6 Slides-Quantifiers...

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