csr2011 june18 15_45_avron
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A Multiple-Conclusion Calculus forFirst-Order Gödel Logic
Arnon Avron Ori Lahav
Tel Aviv University
CSR June 2011
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Gödel Logic
In 1933, Gödel introduced a sequence {Gn} of n-valuedmatrices and used them to show some importantproperties of intuitionistic logic.In 1959, Dummett embedded all the Gns in aninfinite-valued matrix Gω.Gω is equivalent to G[0,1], a natural matrix for thetruth-values [0,1].The logic of G[0,1] is called Gödel logic.Gödel logic is perhaps the most important intermediatelogic.Nowadays, Gödel logic is also recognized as one of thethree most basic fuzzy logics.
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Many-Valued Semantics
A structure M consists of:Non-empty domain DAn interpretation I:
I[c] ∈ D for every constantI[f ] ∈ Dn → D for every n-ary functionI[p] ∈ Dn → [0, 1] for every n-ary predicate
An M-evaluation is a function e:Assigning an element of D for every free variableNaturally extended to all terms (according to I)
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Many-Valued Semantics
A structure M consists of:Non-empty domain DAn interpretation I:
I[c] ∈ D for every constantI[f ] ∈ Dn → D for every n-ary functionI[p] ∈ Dn → [0, 1] for every n-ary predicate
An M-evaluation is a function e:Assigning an element of D for every free variableNaturally extended to all terms (according to I)
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Many-Valued Semantics
‖ • ‖Me is defined as follows:
‖p(t1, . . . , tn)‖Me = I[p][e[t1], . . . ,e[tn]]
‖⊥‖Me = 0
‖ψ1 ∨ ψ2‖Me = max{‖ψ1‖M
e , ‖ψ2‖Me }
‖ψ1 ∧ ψ2‖Me = min{‖ψ1‖M
e , ‖ψ2‖Me }
‖ψ1 ⊃ ψ2‖Me =
{1 ‖ψ1‖M
e ≤ ‖ψ2‖Me
‖ψ2‖Me otherwise
‖∀xψ‖Me = inf{‖ψ‖M
e[x :=d ]| d ∈ D}
‖∃xψ‖Me = sup{‖ψ‖M
e[x :=d ]| d ∈ D}
M is a model of a formula if ϕ if ‖ϕ‖Me = 1 for every e
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Many-Valued Semantics
‖ • ‖Me is defined as follows:
‖p(t1, . . . , tn)‖Me = I[p][e[t1], . . . ,e[tn]]
‖⊥‖Me = 0
‖ψ1 ∨ ψ2‖Me = max{‖ψ1‖M
e , ‖ψ2‖Me }
‖ψ1 ∧ ψ2‖Me = min{‖ψ1‖M
e , ‖ψ2‖Me }
‖ψ1 ⊃ ψ2‖Me =
{1 ‖ψ1‖M
e ≤ ‖ψ2‖Me
‖ψ2‖Me otherwise
‖∀xψ‖Me = inf{‖ψ‖M
e[x :=d ]| d ∈ D}
‖∃xψ‖Me = sup{‖ψ‖M
e[x :=d ]| d ∈ D}
M is a model of a formula if ϕ if ‖ϕ‖Me = 1 for every e
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Many-Valued Semantics
‖ • ‖Me is defined as follows:
‖p(t1, . . . , tn)‖Me = I[p][e[t1], . . . ,e[tn]]
‖⊥‖Me = 0
‖ψ1 ∨ ψ2‖Me = max{‖ψ1‖M
e , ‖ψ2‖Me }
‖ψ1 ∧ ψ2‖Me = min{‖ψ1‖M
e , ‖ψ2‖Me }
‖ψ1 ⊃ ψ2‖Me =
{1 ‖ψ1‖M
e ≤ ‖ψ2‖Me
‖ψ2‖Me otherwise
‖∀xψ‖Me = inf{‖ψ‖M
e[x :=d ]| d ∈ D}
‖∃xψ‖Me = sup{‖ψ‖M
e[x :=d ]| d ∈ D}
M is a model of a formula if ϕ if ‖ϕ‖Me = 1 for every e
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Many-Valued Semantics
‖ • ‖Me is defined as follows:
‖p(t1, . . . , tn)‖Me = I[p][e[t1], . . . ,e[tn]]
‖⊥‖Me = 0
‖ψ1 ∨ ψ2‖Me = max{‖ψ1‖M
e , ‖ψ2‖Me }
‖ψ1 ∧ ψ2‖Me = min{‖ψ1‖M
e , ‖ψ2‖Me }
‖ψ1 ⊃ ψ2‖Me =
{1 ‖ψ1‖M
e ≤ ‖ψ2‖Me
‖ψ2‖Me otherwise
‖∀xψ‖Me = inf{‖ψ‖M
e[x :=d ]| d ∈ D}
‖∃xψ‖Me = sup{‖ψ‖M
e[x :=d ]| d ∈ D}
M is a model of a formula if ϕ if ‖ϕ‖Me = 1 for every e
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Many-Valued Semantics
‖ • ‖Me is defined as follows:
‖p(t1, . . . , tn)‖Me = I[p][e[t1], . . . ,e[tn]]
‖⊥‖Me = 0
‖ψ1 ∨ ψ2‖Me = max{‖ψ1‖M
e , ‖ψ2‖Me }
‖ψ1 ∧ ψ2‖Me = min{‖ψ1‖M
e , ‖ψ2‖Me }
‖ψ1 ⊃ ψ2‖Me =
{1 ‖ψ1‖M
e ≤ ‖ψ2‖Me
‖ψ2‖Me otherwise
‖∀xψ‖Me = inf{‖ψ‖M
e[x :=d ]| d ∈ D}
‖∃xψ‖Me = sup{‖ψ‖M
e[x :=d ]| d ∈ D}
M is a model of a formula if ϕ if ‖ϕ‖Me = 1 for every e
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Kripke-Style Semantics
A frame is a tupleW = 〈W ,≤,D, I, {Iw}w∈W 〉 where:W - nonempty set of worlds≤ - linear order on WD - non-empty (constant) domainI - interpretation of constants and functions
Iw - is a predicate interpretation for every w ∈W :Iw [p] ⊆ Dn for every n-ary predicate pPersistence: Iu[p] ⊆ Iw [p] if u ≤ w
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Kripke-Style Semantics
A frame is a tupleW = 〈W ,≤,D, I, {Iw}w∈W 〉 where:W - nonempty set of worlds≤ - linear order on WD - non-empty (constant) domainI - interpretation of constants and functionsIw - is a predicate interpretation for every w ∈W :
Iw [p] ⊆ Dn for every n-ary predicate pPersistence: Iu[p] ⊆ Iw [p] if u ≤ w
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Kripke-Style Semantics
Satisfaction relation for a frame, a world, and an evaluationof the free variables:
W,w ,e � p(t1, . . . , tn) iff 〈e[t1], . . . ,e[tn]〉 ∈ Iw [p]W,w ,e 6� ⊥
W,w ,e � ψ1 ∨ ψ2 iffW,w ,e � ψ1 orW,w ,e � ψ2W,w ,e � ψ1 ∧ ψ2 iffW,w ,e � ψ1 andW,w ,e � ψ2W,w ,e � ψ1 ⊃ ψ2 iff for every u ≥ w :
W,u,e 6� ψ1 orW,u,e � ψ2W,w ,e � ∀xψ iffW,w ,e[x :=d ] � ψ for every d ∈ DW,w ,e � ∃xψ iffW,w ,e[x :=d ] � ψ for some d ∈ D
W is a model of a formula if ϕ ifW,w ,e � ϕ for every wand e
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Kripke-Style Semantics
Satisfaction relation for a frame, a world, and an evaluationof the free variables:
W,w ,e � p(t1, . . . , tn) iff 〈e[t1], . . . ,e[tn]〉 ∈ Iw [p]W,w ,e 6� ⊥W,w ,e � ψ1 ∨ ψ2 iffW,w ,e � ψ1 orW,w ,e � ψ2W,w ,e � ψ1 ∧ ψ2 iffW,w ,e � ψ1 andW,w ,e � ψ2
W,w ,e � ψ1 ⊃ ψ2 iff for every u ≥ w :W,u,e 6� ψ1 orW,u,e � ψ2
W,w ,e � ∀xψ iffW,w ,e[x :=d ] � ψ for every d ∈ DW,w ,e � ∃xψ iffW,w ,e[x :=d ] � ψ for some d ∈ D
W is a model of a formula if ϕ ifW,w ,e � ϕ for every wand e
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Kripke-Style Semantics
Satisfaction relation for a frame, a world, and an evaluationof the free variables:
W,w ,e � p(t1, . . . , tn) iff 〈e[t1], . . . ,e[tn]〉 ∈ Iw [p]W,w ,e 6� ⊥W,w ,e � ψ1 ∨ ψ2 iffW,w ,e � ψ1 orW,w ,e � ψ2W,w ,e � ψ1 ∧ ψ2 iffW,w ,e � ψ1 andW,w ,e � ψ2W,w ,e � ψ1 ⊃ ψ2 iff for every u ≥ w :
W,u,e 6� ψ1 orW,u,e � ψ2
W,w ,e � ∀xψ iffW,w ,e[x :=d ] � ψ for every d ∈ DW,w ,e � ∃xψ iffW,w ,e[x :=d ] � ψ for some d ∈ D
W is a model of a formula if ϕ ifW,w ,e � ϕ for every wand e
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Kripke-Style Semantics
Satisfaction relation for a frame, a world, and an evaluationof the free variables:
W,w ,e � p(t1, . . . , tn) iff 〈e[t1], . . . ,e[tn]〉 ∈ Iw [p]W,w ,e 6� ⊥W,w ,e � ψ1 ∨ ψ2 iffW,w ,e � ψ1 orW,w ,e � ψ2W,w ,e � ψ1 ∧ ψ2 iffW,w ,e � ψ1 andW,w ,e � ψ2W,w ,e � ψ1 ⊃ ψ2 iff for every u ≥ w :
W,u,e 6� ψ1 orW,u,e � ψ2W,w ,e � ∀xψ iffW,w ,e[x :=d ] � ψ for every d ∈ DW,w ,e � ∃xψ iffW,w ,e[x :=d ] � ψ for some d ∈ D
W is a model of a formula if ϕ ifW,w ,e � ϕ for every wand e
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Kripke-Style Semantics
Satisfaction relation for a frame, a world, and an evaluationof the free variables:
W,w ,e � p(t1, . . . , tn) iff 〈e[t1], . . . ,e[tn]〉 ∈ Iw [p]W,w ,e 6� ⊥W,w ,e � ψ1 ∨ ψ2 iffW,w ,e � ψ1 orW,w ,e � ψ2W,w ,e � ψ1 ∧ ψ2 iffW,w ,e � ψ1 andW,w ,e � ψ2W,w ,e � ψ1 ⊃ ψ2 iff for every u ≥ w :
W,u,e 6� ψ1 orW,u,e � ψ2W,w ,e � ∀xψ iffW,w ,e[x :=d ] � ψ for every d ∈ DW,w ,e � ∃xψ iffW,w ,e[x :=d ] � ψ for some d ∈ D
W is a model of a formula if ϕ ifW,w ,e � ϕ for every wand e
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Proof Theory
Sonobe 1975 - first cut-free Gentzen-type sequent calculusOther calculi have been proposed later by Corsi, Avelloneet al., Dyckhoff and othersAll of them use some ad-hoc rules of a nonstandard formA. 1991 - hypersequent calculus with standard rules: HGExtended to first-order Gödel logic by Baaz and Zach in2000: HIF
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Hypersequents
A sequent (Gentzen 1934) is an object of the formϕ1, . . . , ϕn ⇒ ψ1, . . . , ψm
Intuition: ϕ1 ∧ . . . ∧ ϕn ⊃ ψ1 ∨ . . . ∨ ψmSingle-conclusion sequent: m ≤ 1
A hypersequent is an object of the form s1 | . . . | sn wherethe si ’s are sequents
Intuition: s1 ∨ . . . ∨ snSingle-conclusion hypersequent consists ofsingle-conclusion sequents only
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Hypersequents
A sequent (Gentzen 1934) is an object of the formϕ1, . . . , ϕn ⇒ ψ1, . . . , ψm
Intuition: ϕ1 ∧ . . . ∧ ϕn ⊃ ψ1 ∨ . . . ∨ ψmSingle-conclusion sequent: m ≤ 1
A hypersequent is an object of the form s1 | . . . | sn wherethe si ’s are sequents
Intuition: s1 ∨ . . . ∨ snSingle-conclusion hypersequent consists ofsingle-conclusion sequents only
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HG Single-Conclusion Hypersequent SystemStructural Rules
ϕ⇒ ϕ (EW )H
H | Γ⇒ E
(IW ⇒)H | Γ⇒ E
H | Γ, ψ ⇒ E(⇒ IW )
H | Γ⇒H | Γ⇒ ψ
(cut)H1 | Γ1 ⇒ ϕ H2 | Γ2, ϕ⇒ E
H1 | H2 | Γ1, Γ2 ⇒ E
(com)H1 | Γ1, Γ
′1 ⇒ E1 H2 | Γ2, Γ
′2 ⇒ E2
H1 | H2 | Γ1, Γ′2 ⇒ E1 | Γ2, Γ
′1 ⇒ E2
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HG Single-Conclusion Hypersequent SystemStructural Rules
ϕ⇒ ϕ (EW )H
H | Γ⇒ E
(IW ⇒)H | Γ⇒ E
H | Γ, ψ ⇒ E(⇒ IW )
H | Γ⇒H | Γ⇒ ψ
(cut)H1 | Γ1 ⇒ ϕ H2 | Γ2, ϕ⇒ E
H1 | H2 | Γ1, Γ2 ⇒ E
(com)H1 | Γ1, Γ
′1 ⇒ E1 H2 | Γ2, Γ
′2 ⇒ E2
H1 | H2 | Γ1, Γ′2 ⇒ E1 | Γ2, Γ
′1 ⇒ E2
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HG Single-Conclusion Hypersequent SystemStructural Rules
ϕ⇒ ϕ (EW )H
H | Γ⇒ E
(IW ⇒)H | Γ⇒ E
H | Γ, ψ ⇒ E(⇒ IW )
H | Γ⇒H | Γ⇒ ψ
(cut)H1 | Γ1 ⇒ ϕ H2 | Γ2, ϕ⇒ E
H1 | H2 | Γ1, Γ2 ⇒ E
(com)H1 | Γ1, Γ
′1 ⇒ E1 H2 | Γ2, Γ
′2 ⇒ E2
H1 | H2 | Γ1, Γ′2 ⇒ E1 | Γ2, Γ
′1 ⇒ E2
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HG Single-Conclusion Hypersequent SystemLogical Rules
(∨ ⇒)H1 | Γ1, ψ1 ⇒ E H2 | Γ2, ψ2 ⇒ E
H1 | H2 | Γ1, Γ2, ψ1 ∨ ψ2 ⇒ E
(⇒ ∨1)H | Γ⇒ ψ1
H | Γ⇒ ψ1 ∨ ψ2(⇒ ∨2)
H | Γ⇒ ψ2H | Γ⇒ ψ1 ∨ ψ2
(⊃⇒)H1 | Γ1 ⇒ ψ1 H2 | Γ2, ψ2 ⇒ EH1 | H2 | Γ1, Γ2, ψ1 ⊃ ψ2 ⇒ E
(⇒⊃)H | Γ, ψ1 ⇒ ψ2
H | Γ⇒ ψ1 ⊃ ψ2
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HG Single-Conclusion Hypersequent SystemLogical Rules
(∨ ⇒)H1 | Γ1, ψ1 ⇒ E H2 | Γ2, ψ2 ⇒ E
H1 | H2 | Γ1, Γ2, ψ1 ∨ ψ2 ⇒ E
(⇒ ∨1)H | Γ⇒ ψ1
H | Γ⇒ ψ1 ∨ ψ2(⇒ ∨2)
H | Γ⇒ ψ2H | Γ⇒ ψ1 ∨ ψ2
(⊃⇒)H1 | Γ1 ⇒ ψ1 H2 | Γ2, ψ2 ⇒ EH1 | H2 | Γ1, Γ2, ψ1 ⊃ ψ2 ⇒ E
(⇒⊃)H | Γ, ψ1 ⇒ ψ2
H | Γ⇒ ψ1 ⊃ ψ2
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HIF Single-Conclusion Hypersequent System
(∀ ⇒)H | Γ, ϕ{t/x} ⇒ E
H | Γ, ∀xϕ⇒ E
(⇒ ∀)H | Γ⇒ ϕ{y/x}
H | Γ⇒ ∀xϕ
where y doesn’t occur free in any component of the conclusion.
similar rules for ∃
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HIF Single-Conclusion Hypersequent System
(∀ ⇒)H | Γ, ϕ{t/x} ⇒ E
H | Γ, ∀xϕ⇒ E
(⇒ ∀)H | Γ⇒ ϕ{y/x}
H | Γ⇒ ∀xϕ
where y doesn’t occur free in any component of the conclusion.
similar rules for ∃
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Cut-Admissibility
(cut)H1 | Γ1 ⇒ ϕ H2 | Γ2, ϕ⇒ E
H1 | H2 | Γ1, Γ2 ⇒ E
One of the most important properties of a (hyper)sequentcalculus, provides the key for proof-searchTraditional syntactic cut-admissibility proofs are notoriouslyprone to errors, especially (but certainly not only) in thecase of hypersequent systems
the first proof of cut-elimination for HIF was erroneous
A semantic proof is usually more reliable and easier tocheckA semantic proof usually provides also a proof ofcompleteness as well as strong cut-admissibility
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Cut-Admissibility
(cut)H1 | Γ1 ⇒ ϕ H2 | Γ2, ϕ⇒ E
H1 | H2 | Γ1, Γ2 ⇒ E
One of the most important properties of a (hyper)sequentcalculus, provides the key for proof-searchTraditional syntactic cut-admissibility proofs are notoriouslyprone to errors, especially (but certainly not only) in thecase of hypersequent systems
the first proof of cut-elimination for HIF was erroneous
A semantic proof is usually more reliable and easier tocheckA semantic proof usually provides also a proof ofcompleteness as well as strong cut-admissibility
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MCG Multiple-Conclusion Hypersequent SystemStructural Rules
ϕ⇒ ϕ (EW )H
H | Γ⇒ ∆
(IW ⇒)H | Γ⇒ ∆
H | Γ, ψ ⇒ ∆(⇒ IW )
H | Γ⇒ ∆
H | Γ⇒ ∆, ψ
(cut)H1 | Γ1 ⇒ ∆1, ϕ H2 | Γ2, ϕ⇒ ∆2
H1 | H2 | Γ1, Γ2 ⇒ ∆1,∆2
(com)H1 | Γ1, Γ
′1 ⇒ ∆1 H2 | Γ2, Γ
′2 ⇒ ∆2
H1 | H2 | Γ1, Γ′2 ⇒ ∆1 | Γ2, Γ
′1 ⇒ ∆2
(split)H | Γ⇒ ∆1,∆2
H | Γ⇒ ∆1 | Γ⇒ ∆2
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MCG Multiple-Conclusion Hypersequent SystemStructural Rules
ϕ⇒ ϕ (EW )H
H | Γ⇒ ∆
(IW ⇒)H | Γ⇒ ∆
H | Γ, ψ ⇒ ∆(⇒ IW )
H | Γ⇒ ∆
H | Γ⇒ ∆, ψ
(cut)H1 | Γ1 ⇒ ∆1, ϕ H2 | Γ2, ϕ⇒ ∆2
H1 | H2 | Γ1, Γ2 ⇒ ∆1,∆2
(com)H1 | Γ1, Γ
′1 ⇒ ∆1 H2 | Γ2, Γ
′2 ⇒ ∆2
H1 | H2 | Γ1, Γ′2 ⇒ ∆1 | Γ2, Γ
′1 ⇒ ∆2
(split)H | Γ⇒ ∆1,∆2
H | Γ⇒ ∆1 | Γ⇒ ∆2
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MCG Multiple-Conclusion Hypersequent SystemLogical Rules
(⊃⇒)H1 | Γ1 ⇒ ∆1, ψ1 H2 | Γ2, ψ2 ⇒ ∆2
H1 | H2 | Γ1, Γ2, ψ1 ⊃ ψ2 ⇒ ∆1,∆2(⇒⊃)
H | Γ, ψ1 ⇒ ψ2
H | Γ⇒ ψ1 ⊃ ψ2
(∀ ⇒)H | Γ, ϕ{t/x} ⇒ ∆
H | Γ, ∀xϕ⇒ ∆(⇒ ∀)
H | Γ⇒ ∆, ϕ{y/x}H | Γ⇒ ∆, ∀xϕ
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Results
MCG is strongly sound and complete with respect to theKripke semantics of the standard first-order Gödel logic
MCG admits strong cut-admissibility:for every set H of hypersequents closed under substitutionand a hypersequent H:H ` H iff there exists a proof of H from H in which thecut-formula of every application of the cut rule is in frm[H]
As a corollary, we obtain the same results for HIF, theoriginal single-conclusion calculus
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Results
MCG is strongly sound and complete with respect to theKripke semantics of the standard first-order Gödel logicMCG admits strong cut-admissibility:
for every set H of hypersequents closed under substitutionand a hypersequent H:H ` H iff there exists a proof of H from H in which thecut-formula of every application of the cut rule is in frm[H]
As a corollary, we obtain the same results for HIF, theoriginal single-conclusion calculus
![Page 34: Csr2011 june18 15_45_avron](https://reader033.vdocuments.site/reader033/viewer/2022051311/5445e6d1afaf9ff32a8b46d6/html5/thumbnails/34.jpg)
Results
MCG is strongly sound and complete with respect to theKripke semantics of the standard first-order Gödel logicMCG admits strong cut-admissibility:for every set H of hypersequents closed under substitutionand a hypersequent H:H ` H iff there exists a proof of H from H in which thecut-formula of every application of the cut rule is in frm[H]
As a corollary, we obtain the same results for HIF, theoriginal single-conclusion calculus
![Page 35: Csr2011 june18 15_45_avron](https://reader033.vdocuments.site/reader033/viewer/2022051311/5445e6d1afaf9ff32a8b46d6/html5/thumbnails/35.jpg)
Results
MCG is strongly sound and complete with respect to theKripke semantics of the standard first-order Gödel logicMCG admits strong cut-admissibility:for every set H of hypersequents closed under substitutionand a hypersequent H:H ` H iff there exists a proof of H from H in which thecut-formula of every application of the cut rule is in frm[H]
As a corollary, we obtain the same results for HIF, theoriginal single-conclusion calculus
![Page 36: Csr2011 june18 15_45_avron](https://reader033.vdocuments.site/reader033/viewer/2022051311/5445e6d1afaf9ff32a8b46d6/html5/thumbnails/36.jpg)
Thank you!