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 627, winter ‘13 2 ¬ ∧ ∨ of 38 propositional logic Gentzen system PROP_G a different name – a natural deduction system ece 627, winter ‘13 3 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 627, winter ‘13 4 ⇒ ⇒ of 38 propositional logic Gentzen system Γ Δ if all formulae of a set Γ is true then one of formulae of a set Δ is true ece 627, winter ‘13 5 ⇒ of 38 propositional logic Gentzen system Γ1, Γ2 Δ1, Δ2 either Δ1 or Δ2 can be derived from Γ1 and Γ2 ece 627, winter ‘13 6 ⇒
Transcript
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lecture 13: propositional logic – part II
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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
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¬
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∧
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∨
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propositional logic Gentzen system
PROP_G a different name
– a natural deduction system
ece#627,#winter#‘13# 3# 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
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⇒
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⇒
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propositional logic Gentzen system
Γ Δ#
if all formulae of a set Γ is true then one of formulae of a set Δ is true
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⇒
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propositional logic Gentzen system
Γ1, Γ2 Δ1, Δ2#
either Δ1 or Δ2 can be derived from Γ1 and Γ2
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⇒
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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
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⇒
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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)
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propositional logic Gentzen system
there are two groups of inference rules
! for introducing logical operators ! for rearranging sequent
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propositional logic Gentzen system – introduction rules
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Γ, A⇒ΔΓ, A∧B⇒Δ
(L ∧)
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Γ1⇒ A,Δ1 Γ2⇒ B,Δ2Γ1,Γ2⇒ A∧B,Δ1,Δ2
(R∧)
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propositional logic Gentzen system – introduction rules
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Γ1, A⇒Δ1 Γ2,B⇒Δ2Γ1,Γ2, A∨B⇒Δ1,Δ2
(L ∨)
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Γ⇒ A,ΔΓ⇒ A∨B,Δ
(R∨)
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propositional logic Gentzen system – introduction rules
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Γ1⇒ A,Δ1 Γ2,B⇒Δ2Γ1,Γ2, A→ B⇒Δ1,Δ2
(L→)
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Γ, A⇒ B,ΔΓ⇒ A→ B,Δ
(R→)
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propositional logic Gentzen system – introduction rules
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Γ⇒ A,ΔΓ,¬A⇒Δ
(L¬)
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Γ, A⇒ΔΓ⇒¬A,Δ
(R¬)
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propositional logic Gentzen system – structural rules
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Γ, A,B⇒ΔΓ,B, A⇒Δ
(L R)
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Γ⇒ A,B,ΔΓ⇒ B, A,Δ
(R R)
reordering
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propositional logic Gentzen system – structural rules
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Γ⇒ΔΓ, A⇒Δ
(L W )
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Γ⇒ΔΓ⇒ A,Δ
(RW )
weakening
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propositional logic Gentzen system – structural rules
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Γ, A, A⇒ΔΓ, A⇒Δ
(L C)
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Γ⇒ A, A,ΔΓ⇒ A,Δ
(R C)
contraction
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Γ1⇒ A,Δ1 Γ2, A⇒Δ2Γ1,Γ2⇒Δ1,Δ2
(CUT )
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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
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⇒
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propositional logic Gentzen system – proof of PH1
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A⇒ AA,B⇒ A
(L W )
A⇒ (B→ A)(R→)
⇒ A→ (B→ A)(R→)
A (B A)
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→
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→
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Gentzen system – proof of PH2
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B⇒ B C⇒ CB, B→C⇒ C
A⇒ A B⇒ BA, A→ B⇒ B
A⇒ AA, B, [A→ (B→C)]⇒ C
A, A→ B, [A→ (B→C)]⇒ C
A→ B, [A→ (B→C)]⇒ A→C
[A→ (B→C)]⇒ (A→ B)→ (A→C)
⇒ [A→ (B→C)]→ [(A→ B)→ (A→C)]
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propositional logic Gentzen system – proof of PH3
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A⇒ A
A A
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→
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propositional logic Gentzen system – …
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A⇒ A B⇒ BA, A→ B⇒ B
(L→)
Modus Ponens
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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
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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
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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
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propositional logic tableau system
PROP_B has the same syntax and vocabulary as PROP_G
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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
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propositional logic tableau system
a tableau is a tree of formulae, built up according to the following five rules:
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propositional logic tableau 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 logic tableau 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
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propositional logic tableau 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 logic tableau 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 logic tableau 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 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
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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)
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propositional logic tableau system – proof 1
to show that r follows from p, q, (p [q r])
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1 ¬r2 p3 q4 p→ (q→ r))
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5 ¬pCLOSED
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7 ¬qCLOSED
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8 rCLOSED
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6 q→ r
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propositional logic tableau system – proof 2 (PH2)
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1 ¬([A→ (B→C)]→ [(A→ B)→ (A→C)]
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2 [A→ (B→C)]∧¬[(A→ B)→ (A→C)]
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3 [A→ (B→C)]4 ¬[(A→ B)→ (A→C)]
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5 (A→ B)∧¬(A→C)]
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6 A→ B7 ¬(A→C)
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8 A∧¬C
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9 A10 ¬C
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propositional logic tableau system – proof 2 (PH2) cont.
