Download CS 512, Spring 2017, Handout 05 [1ex] Semantics of Classical

CS 512, Spring 2017, Handout 05
Semantics of Classical Propositional Logic
(Continued)
Soundness, Completeness, Compactness
Assaf Kfoury
January 25, 2017
Assaf Kfoury, CS 512, Spring 2017, Handout 05
page 1 of 18
soundness
Assaf Kfoury, CS 512, Spring 2017, Handout 05
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soundness
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Let Γ a (possibly infinite) set of propositional WFF’s.
If, for every model/interpretation/valuation
(i.e., assignment of truth values to prop atoms), it holds that:
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whenever all the WFF’s in Γ evaluate to T ,
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it is also the case that ψ evaluates to T ,
then we write:
Γ |= ψ
in words, “Γ semantically entails ψ ”
Assaf Kfoury, CS 512, Spring 2017, Handout 05
page 3 of 18
soundness
I
Let Γ a (possibly infinite) set of propositional WFF’s.
If, for every model/interpretation/valuation
(i.e., assignment of truth values to prop atoms), it holds that:
I
whenever all the WFF’s in Γ evaluate to T ,
I
it is also the case that ψ evaluates to T ,
then we write:
Γ |= ψ
I
in words, “Γ semantically entails ψ ”
Theorem (Soundness):
If Γ ` ψ then Γ |= ψ .
(Slightly stronger than the statement of Soundness in [LCS, Theorem 1.35, p 46]:
If ϕ1 , ϕ2 , . . . , ϕn ` ψ then ϕ1 , ϕ2 , . . . , ϕn |= ψ .)
Assaf Kfoury, CS 512, Spring 2017, Handout 05
page 4 of 18
soundness
I
Let Γ a (possibly infinite) set of propositional WFF’s.
If, for every model/interpretation/valuation
(i.e., assignment of truth values to prop atoms), it holds that:
I
whenever all the WFF’s in Γ evaluate to T ,
I
it is also the case that ψ evaluates to T ,
then we write:
Γ |= ψ
I
in words, “Γ semantically entails ψ ”
Theorem (Soundness):
If Γ ` ψ then Γ |= ψ .
(Slightly stronger than the statement of Soundness in [LCS, Theorem 1.35, p 46]:
If ϕ1 , ϕ2 , . . . , ϕn ` ψ then ϕ1 , ϕ2 , . . . , ϕn |= ψ .)
I
Proof idea: “Course-of-values” induction on n > 1
(in a later Handout 06).
Assaf Kfoury, CS 512, Spring 2017, Handout 05
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completeness
Assaf Kfoury, CS 512, Spring 2017, Handout 05
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completeness
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Theorem (Completeness): If Γ |= ψ then Γ ` ψ .
(Stronger than the statement of Completeness in [LCS, Corollary 1.39, p 53]:
If ϕ1 , ϕ2 , . . . , ϕn |= ψ then ϕ1 , ϕ2 , . . . , ϕn ` ψ .)
Assaf Kfoury, CS 512, Spring 2017, Handout 05
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completeness
I
Theorem (Completeness): If Γ |= ψ then Γ ` ψ .
(Stronger than the statement of Completeness in [LCS, Corollary 1.39, p 53]:
If ϕ1 , ϕ2 , . . . , ϕn |= ψ then ϕ1 , ϕ2 , . . . , ϕn ` ψ .)
I
Proof idea in [LCS] (which works if Γ is a finite set {ϕ1 , . . . , ϕn }):
Establish 3 preliminary results. From ϕ1 , . . . , ϕn |= ψ , show that:
Assaf Kfoury, CS 512, Spring 2017, Handout 05
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completeness
I
Theorem (Completeness): If Γ |= ψ then Γ ` ψ .
(Stronger than the statement of Completeness in [LCS, Corollary 1.39, p 53]:
If ϕ1 , ϕ2 , . . . , ϕn |= ψ then ϕ1 , ϕ2 , . . . , ϕn ` ψ .)
I
Proof idea in [LCS] (which works if Γ is a finite set {ϕ1 , . . . , ϕn }):
Establish 3 preliminary results. From ϕ1 , . . . , ϕn |= ψ , show that:
1. |= ϕ1 → (ϕ2 → (ϕ3 → (· · · (ϕn → ψ) · · · ))) holds.
Assaf Kfoury, CS 512, Spring 2017, Handout 05
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completeness
I
Theorem (Completeness): If Γ |= ψ then Γ ` ψ .
(Stronger than the statement of Completeness in [LCS, Corollary 1.39, p 53]:
If ϕ1 , ϕ2 , . . . , ϕn |= ψ then ϕ1 , ϕ2 , . . . , ϕn ` ψ .)
I
Proof idea in [LCS] (which works if Γ is a finite set {ϕ1 , . . . , ϕn }):
Establish 3 preliminary results. From ϕ1 , . . . , ϕn |= ψ , show that:
1. |= ϕ1 → (ϕ2 → (ϕ3 → (· · · (ϕn → ψ) · · · ))) holds.
2. ` ϕ1 → (ϕ2 → (ϕ3 → (· · · (ϕn → ψ) · · · ))) is a valid sequent.
Assaf Kfoury, CS 512, Spring 2017, Handout 05
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completeness
I
Theorem (Completeness): If Γ |= ψ then Γ ` ψ .
(Stronger than the statement of Completeness in [LCS, Corollary 1.39, p 53]:
If ϕ1 , ϕ2 , . . . , ϕn |= ψ then ϕ1 , ϕ2 , . . . , ϕn ` ψ .)
I
Proof idea in [LCS] (which works if Γ is a finite set {ϕ1 , . . . , ϕn }):
Establish 3 preliminary results. From ϕ1 , . . . , ϕn |= ψ , show that:
1. |= ϕ1 → (ϕ2 → (ϕ3 → (· · · (ϕn → ψ) · · · ))) holds.
2. ` ϕ1 → (ϕ2 → (ϕ3 → (· · · (ϕn → ψ) · · · ))) is a valid sequent.
3. ϕ1 , ϕ2 , . . . , ϕn ` ψ is a valid sequent.
Assaf Kfoury, CS 512, Spring 2017, Handout 05
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completeness
I
Theorem (Completeness): If Γ |= ψ then Γ ` ψ .
(Stronger than the statement of Completeness in [LCS, Corollary 1.39, p 53]:
If ϕ1 , ϕ2 , . . . , ϕn |= ψ then ϕ1 , ϕ2 , . . . , ϕn ` ψ .)
I
Proof idea in [LCS] (which works if Γ is a finite set {ϕ1 , . . . , ϕn }):
Establish 3 preliminary results. From ϕ1 , . . . , ϕn |= ψ , show that:
1. |= ϕ1 → (ϕ2 → (ϕ3 → (· · · (ϕn → ψ) · · · ))) holds.
2. ` ϕ1 → (ϕ2 → (ϕ3 → (· · · (ϕn → ψ) · · · ))) is a valid sequent.
3. ϕ1 , ϕ2 , . . . , ϕn ` ψ is a valid sequent.
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If Γ is infinite, we need another preliminary result: Compactness .
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compactness
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compactness
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Γ is said to be satisfiable if there is a model/interpretation/valuation
which satisfies/makes true every ϕ in Γ.
Assaf Kfoury, CS 512, Spring 2017, Handout 05
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compactness
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Γ is said to be satisfiable if there is a model/interpretation/valuation
which satisfies/makes true every ϕ in Γ.
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Theorem (Compactness) (not in [LCS]):
Γ is satisfiable iff every finite subset of Γ is satisfiable.
Assaf Kfoury, CS 512, Spring 2017, Handout 05
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compactness
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Γ is said to be satisfiable if there is a model/interpretation/valuation
which satisfies/makes true every ϕ in Γ.
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Theorem (Compactness) (not in [LCS]):
Γ is satisfiable iff every finite subset of Γ is satisfiable.
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Corollary (not in [LCS]):
If Γ |= ψ then there is a finite subset Γ0 ⊆ Γ such that Γ0 |= ψ .
Assaf Kfoury, CS 512, Spring 2017, Handout 05
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compactness
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Γ is said to be satisfiable if there is a model/interpretation/valuation
which satisfies/makes true every ϕ in Γ.
I
Theorem (Compactness) (not in [LCS]):
Γ is satisfiable iff every finite subset of Γ is satisfiable.
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Corollary (not in [LCS]):
If Γ |= ψ then there is a finite subset Γ0 ⊆ Γ such that Γ0 |= ψ .
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For proofs of Compactness above and its corollary, click here .
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