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lecture 13: propositional logic – part II

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propositional logicGentzen system

PROP_G

design to be simplesyntax and vocabulary the same as PROP_Hit has , , as standard operators,and a much larger set of inference rules for

introducing and eliminating the operators

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propositional logicGentzen system

PROP_G

a different name – a natural deduction system

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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

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propositional logicGentzen system

if all formulae of a set is true then one of formulae of a set is true

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propositional logicGentzen system

either or can be derivedfrom and

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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

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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)

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propositional logicGentzen system

there are two groups of inference rules for introducing logical operators for rearranging sequent

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propositional logicGentzen system – introduction rules

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propositional logicGentzen system – introduction rules

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propositional logicGentzen system – introduction rules

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propositional logicGentzen system – introduction rules

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propositional logicGentzen system – structural rules

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reordering

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propositional logicGentzen system – structural rules

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weakening

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propositional logicGentzen system – structural rules

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contraction

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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

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propositional logicGentzen system – proof of PH1

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A (B A)

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Gentzen system – proof of PH2

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propositional logicGentzen system – proof of PH3

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A A

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propositional logicGentzen system – …

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Modus Ponens

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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

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propositional logicGentzen system

there are also theorems for soundness, consistency, decidability – both systems are equivalent

additional: cut elimination theoremany theorem which can be proved in PROP_G has a proof which does not contain a use of the CUT rule

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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

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propositional logictableau system

PROP_B has the same syntax and vocabulary as PROP_G

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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

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propositional logictableau system

a tableau is a tree of formulae, built up according to the following five rules:

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propositional logictableau system

(rule i) if A1, A2, … An are the premises of a proof, then

A1

A2

…An

is a tableau

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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

qr

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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

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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

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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

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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

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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)

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propositional logictableau system – proof 1

to show that r follows from p, q, (p [q r])

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propositional logictableau system – proof 2 (PH2)

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propositional logictableau system – proof 2 (PH2) cont.

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propositional logictableau system – proof 3

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