of 38 lecture 13: propositional logic – part II

of 38 propositional logic Gentzen 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 ece 720, winter ‘122

of 38 propositional logic Gentzen system PROP_G a different name – a natural deduction system ece 720, winter ‘123

of 38 propositional logic Gentzen 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 ece 720, winter ‘124

of 38 propositional logic Gentzen system  if all formulae of a set  is true then one of formulae of a set  is true ece 720, winter ‘125

of 38 propositional logic Gentzen system  either  or  can be derived from  and  ece 720, winter ‘126

of 38 propositional logic Gentzen 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 ece 720, winter ‘127

of 38 propositional logic Gentzen 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) ece 720, winter ‘128

of 38 propositional logic Gentzen system there are two groups of inference rules  for introducing logical operators  for rearranging sequent ece 720, winter ‘129

of 38 propositional logic Gentzen system – introduction rules ece 720, winter ‘1210

of 38 propositional logic Gentzen system – introduction rules ece 720, winter ‘1211

of 38 propositional logic Gentzen system – introduction rules ece 720, winter ‘1212

of 38 propositional logic Gentzen system – introduction rules ece 720, winter ‘1213

of 38 propositional logic Gentzen system – structural rules ece 720, winter ‘1214 reordering

of 38 propositional logic Gentzen system – structural rules ece 720, winter ‘1215 weakening

of 38 propositional logic Gentzen system – structural rules ece 720, winter ‘1216 contraction

of 38 propositional logic Gentzen 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 ece 720, winter ‘1217

of 38 propositional logic Gentzen system – proof of PH1 ece 720, winter ‘1218 A (B A)

of 38 Gentzen system – proof of PH2 ece 720, winter ‘1219

of 38 propositional logic Gentzen system – proof of PH3 ece 720, winter ‘1220 A A

of 38 propositional logic Gentzen system – … ece 720, winter ‘1221 Modus Ponens

of 38 propositional logic Gentzen 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 ece 720, winter ‘1222

of 38 propositional logic Gentzen 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 ece 720, winter ‘1223

of 38 propositional logic tableau 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 ece 720, winter ‘1224

of 38 propositional logic tableau system PROP_B has the same syntax and vocabulary as PROP_G ece 720, winter ‘1225

of 38 propositional logic tableau 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 ece 720, winter ‘1226

of 38 propositional logic tableau system a tableau is a tree of formulae, built up according to the following five rules: ece 720, winter ‘1227

of 38 propositional logic tableau system (rule i) if A 1, A 2, … A n are the premises of a proof, then A 1 A 2 … A n is a tableau ece 720, winter ‘1228

of 38 propositional logic tableau system (rule ii) if some branch contains a formula A i which is of the form B i C i then the tree formed by adding Bi and Ci on the end is a tableaup q r q r ece 720, winter ‘1229

of 38 propositional logic tableau system (rule iii) if some branch contains a formula A i which is of the form B i C i then the tree formed by adding B i and C i on the end is a tableaup p q p q ece 720, winter ‘1230

of 38 propositional logic tableau system (rule iv) if A i is B i C i then the tree is extended by adding new branches B i and C i so thatr p q p q ece 720, winter ‘1231

of 38 propositional logic tableau system (rule v) if A i is B i for some non-atomic B i, then the tree is extended by adding C i D i when B i is C i D i C i when B i is C i ece 720, winter ‘1232

of 38 propositional logic tableau 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 ece 720, winter ‘1233

of 38 propositional logic tableau 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) ece 720, winter ‘1234

of 38 propositional logic tableau system – proof 1 to show that r follows from p, q, (p [q r]) ece 720, winter ‘1235

of 38 propositional logic tableau system – proof 2 (PH2) ece 720, winter ‘1236

of 38 propositional logic tableau system – proof 2 (PH2) cont. ece 720, winter ‘1237

of 38 propositional logic tableau system – proof 3 ece 720, winter ‘1238