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e ce 627 intelligent web: ontology and beyond. lecture 13: propositional logic – part II. propositional logic Gentzen system. PROP_G design to be simple syntax and vocabulary the same as PROP_H it has , , as standard operators,
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ece 627intelligent web: ontology and beyond lecture 13: propositional logic – part II
propositional logicGentzen system PROP_G design to be simple syntax and vocabulary the same as PROP_H it has , , as standard operators, and a much larger set of inference rules for introducing and eliminating the operators
propositional logicGentzen system PROP_G a different name – a natural deduction system
propositional logicGentzen system a special symbol (or ) that defines a sequent sequent is interpreted as a statement: when all formulae on the left side of the are true then at least one of those on the right is true
propositional logicGentzen system G D if all formulae of a set G is true then one of formulae of a set D is true
propositional logicGentzen system G1, G2 D1, D2 either D1 or D2 can be derived from G1 and G2
propositional logicGentzen system a symbol is used for making statements about what hypotheses a chain of inference is based on, and for couching inference rules so that the steps in a chain of inference can actually be performed
propositional logicGentzen system a sequent rule is written as a collection of sequents above a horizontal line, and a single sequent below it (if you have a collection of sequents that matches what is above the line, you can replace them by the single sequent below)
propositional logicGentzen system there are two groups of inference rules • for introducing logical operators • for rearranging sequent
propositional logicGentzen system – structural rules reordering
propositional logicGentzen system – structural rules weakening
propositional logicGentzen system – structural rules contraction
propositional logicGentzen system proofs are constructed by working from sequents of the form A A, via the rules (just shown), to a sequent consisting of just the desired formula on the right
propositional logicGentzen system – … Modus Ponens
propositional logicGentzen system anything which can be proven in PROP_H can also be proven in PROP_G what leads to a theorem: anything which is valid is provable in PROP_G
propositional logicGentzen system there are also theorems for soundness, consistency, decidability – both systems are equivalent additional: cut elimination theorem any theorem which can be proved in PROP_G has a proof which does not contain a use of the CUT rule
propositional logictableau system designed to support proofs by contradiction idea: since every proposition is either true or false, if we show that something cannot be false then it must be true
propositional logictableau system PROP_B has the same syntax and vocabulary as PROP_G
propositional logictableau system proofs are constructed in terms of an object called a semantic tableau – this is an attempt to enumerate the ways the world could be, given the hypotheses of the proof, and to show that in all of them the negation of the desired conclusion must be false, so the conclusion itself must be true
propositional logictableau system a tableau is a tree of formulae, built up according to the following five rules:
propositional logictableau system (rule i) if A1, A2, … An are the premises of a proof, then A1 A2 … An is a tableau
propositional logictableau system (rule ii) if some branch contains a formula Ai which is of the form Bi Ci then the tree formed by adding Bi and Ci on the end is a tableau p p q r q r q r
propositional logictableau system (rule iii) if some branch contains a formula Ai which is of the form Bi Ci then the tree formed by adding Bi and Ci on the end is a tableau p p p q p q p q
propositional logictableau system (rule iv) if Ai is Bi Ci then the tree is extended by adding new branches Bi and Ci so that r r p q p q p q
propositional logictableau system (rule v) if Ai is Bi for some non-atomic Bi, then the tree is extended by adding Ci Di when Bi is Ci Di Ci Di when Bi is Ci Di Ci Di when Bi is Ci Di Ci when Bi is Ci
propositional logictableau system each branch represents a partial description of the world which is consistent with the original set of premises if any branch contains both A and A for some A then it is clearly not feasible description of the world – we say the branch is CLOSED
propositional logictableau system if all branch is CLOSED – then there is no feasible descriptions of the world which are consistent with the premises on which it is based so, proof – adding negation of the goal to the premises and showing that the tableau based on that collection is CLOSED (every branch is CLOSED)
propositional logictableau system – proof 1 to show that r follows from p, q, (p [qr])
