THE PREDICATE CALCULUS
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1 2 THE PREDICATE CALCULUS Slide Introduction 2.1 The Propositional Calculus 2.2 The Predicate Calculus 2.3 Using Inference Rules to Produce Predicate Calculus Expressions 2.4 Application: A Logic-Based Financial Advisor 2.5 Epilogue and References 2.6 Exercises
2 Slide 2.2
3 Slide 2.3
4 Slide 2.4
5 For propositional expressions P, Q and R: Slide 2.5
6 Figure 2.1: Truth table for the operator. Slide 2.6
7 Slide 2.7 Figure 2.2: Truth table demonstrating the equivalence of P Q and P Q.
8 Slide 2.8 The Predicate Calculus access the components of an individual assertion allow expression to contain variables Example single proposition P = it rained on Tuesday Using predicate weather weathertuesday,rain
9 Slide 2.9 The Predicate Calculus Continued general assertion about clauses of entities for all values of, where is a day of the week, weather,rain means that it rains every day
10 Slide 2.10
11 Slide 2.11
12 Slide 2.12 Examples Well-formed function expressions f,y fatherdavid pricebananas arity argument in fatherdavid, arity is 1 in plus2,3, arity is 2
13 Slide 2.13
14 Slide 2.14 Examples of atomic sentences A predicate relation is defined by its name and its arity likesgeorge,kate likesgeorge,sarah,tuesday friendsbill,george friendsfather_ofdavid,father_ofandrew helpsrichard,bill
15 Slide 2.15 atomic sentences are also called atomic expressions, atoms, or propositions combine atomic sentences using logical operators to form sentences in the predicate calculus operators are the same as connectives in propositional calculus
16 Slide 2.16 Quantifiers Universal quantifier is true for all values in the domain of the definition of Variable quantifier Y friends Y, peter is true if there is at least one object, indicated by Y that is a friend of peter
17 Slide 2.17
18 Slide 2.18 Examples let times and plus be functions symbols of arity 2 and let equal and foo be predicate symbols with arity 2 and 3 respectively plustwo,three is a function, not an atomic sentence equalplus2,3,five is an atomic sentence equalplus2,3,7 is an atomic sentence foo, two, plus2,3 equal plus2,3,5
19 verify_sentence algorithm Slide 2.19
20 Slide 2.20 Use of predicate calculus mothereve,abel mothereve,cain fatheradam,abel fatheradam,cain Y father, Y mother, Y parent, Y Y Z parent, Y parent, Z sibling Y, Z
21 Slide 2.21 Semantics for the Predicate Calculus It is important to determine well-formed expressions meaning in terms of objects, predicates and relations in the world To use the predicate calculus as a representation for problem solving, we need to describe objects and relations in the domain of interpretation with a set of well-formed expression The terms and predicates of these expressions denote objects and relations in the domain The database of predicate calculus expressions, each having truth value T, describes as state of the world
22 Slide 2.22
23 Quantification of Variables Slide 2.23 likesgeorge, substituting kate and susie for in likesgeorge, becomes likesgeorge,kate and likesgeorge,susie Variable can be replaced to Y without changing meaning, and so called as dummy Variables must be quantified in either universally or existentially A variable is considered free if it is not within the scope of either the universal or existential quantifiers An expression is closed if all of its variables are quantified A ground expression has no variables at all In Predicate Calculus, all variables must be quantified
24 Slide 2.24 Scope of quantification with parenthesis, is universally quantified in both p and r Relationship between negation and universal and existential quantifiers r Y q p Y q Y p q p Y q Y q Y p p p p p p Y q Y p q p
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