Through Many-Valent Semantics Carolina Blasio IFCH/UNICAMP PhDs in - - PowerPoint PPT Presentation

Through Many-Valent Semantics

Carolina Blasio

IFCH/UNICAMP

PhD’s in Logic May 3rd, BOCHUM

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 1 / 20

Introduction

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 2 / 20

Suszko’s Thesis

"Obviously, any multiplication of logical values is a mad idea."

(Roman Suszko, 1977)

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 3 / 20

Suszko’s Thesis

"Obviously, any multiplication of logical values is a mad idea."

(Roman Suszko, 1977)

Every logic can be characterized by bivalent semantics.

(Malinowski, 1994; Wansing & Shramko, 2008; Caleiro, Marcos & Volpe, 2015)

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 3 / 20

Suszko’s Thesis

"Obviously, any multiplication of logical values is a mad idea."

(Roman Suszko, 1977)

Every logic can be characterized by bivalent semantics.

(Malinowski, 1994; Wansing & Shramko, 2008; Caleiro, Marcos & Volpe, 2015)

Many-valued semantics could be used as a tool.

(Avron, 2009)

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 3 / 20

Suszko’s Thesis

"Obviously, any multiplication of logical values is a mad idea."

(Roman Suszko, 1977)

Every logic can be characterized by bivalent semantics.

(Malinowski, 1994; Wansing & Shramko, 2008; Caleiro, Marcos & Volpe, 2015)

Many-valued semantics could be used as a tool.

(Avron, 2009)

Two kinds of truth-values

referential truth-values:

make up many-valued semantics

inferential truth-values:

consequence relation validity

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 3 / 20

Many-valent Semantic, but Bivalent Logics

Let S be a propositional language.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 4 / 20

Many-valent Semantic, but Bivalent Logics

Let S be a propositional language. Standard valuation matrix: M = V, D, O V := Truth-values, D := D ⊆ V, the designated values, O := Truth-functions for each connective of S. V D

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 4 / 20

Many-valent Semantic, but Bivalent Logics

Let S be a propositional language. Standard valuation matrix: M = V, D, O V := Truth-values, D := D ⊆ V, the designated values, O := Truth-functions for each connective of S. V D

Entailment relation based on M

v(Φ) = {v(φ)|φ ∈ Φ}

Γ | = ∆ iff there is no v such that v(Γ) ⊆ D and v(∆) ⊆ V − D

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 4 / 20

Many-valent Semantic, but Bivalent Logics

Let S be a propositional language. Standard valuation matrix: M = V, D, O V := Truth-values, D := D ⊆ V, the designated values, O := Truth-functions for each connective of S. V D

Entailment relation based on M

v(Φ) = {v(φ)|φ ∈ Φ}

Γ | = ∆ iff there is no v such that v(Γ) ⊆ D and v(∆) ⊆ V − D

Proposition (The following holds in a standard logic:)

Reflexivity α α Monotonicity If Γ′ | = ∆′, then Γ′, Γ′′ | = ∆′, ∆′′ Transitivity If Σ, Γ | = ∆, Π for every quasi-partition* Σ, Π of a Θ⊆S, then Γ | = ∆.

* Σ ∪ Π = Θ and Σ ∩ Π = ∅

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 4 / 20

Many-valent Semantic, but Bivalent Logics

Let S be a propositional language. Standard valuation matrix: M = V, Y, O V := Truth-values, Y := Y ⊆ V, the accepted values, O := Truth-functions for each connective of S. V Y

Entailment relation based on M

v(Φ) = {v(φ)|φ ∈ Φ}

Γ | = ∆ iff there is no v such that v(Γ) ⊆ Y and v(∆) ⊆ V − Y

Proposition (The following holds in a standard logic:)

Reflexivity α α Monotonicity If Γ′ | = ∆′, then Γ′, Γ′′ | = ∆′, ∆′′ Transitivity If Σ, Γ | = ∆, Π for every quasi-partition* Σ, Π of a Θ⊆S, then Γ | = ∆.

* Σ ∪ Π = Θ and Σ ∩ Π = ∅

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 4 / 20

Many-valent Semantic, but Bivalent Logics

Let S be a propositional language. Standard valuation matrix: M = V, N, O V := Truth-values, N := N ⊆ V, the rejected values, O := Truth-functions for each connective of S. V N

Entailment relation based on M

v(Φ) = {v(φ)|φ ∈ Φ}

Γ | = ∆ iff there is no v such that v(Γ) ⊆ V − N and v(∆) ⊆ N

Proposition (The following holds in a standard logic:)

Reflexivity α α Monotonicity If Γ′ | = ∆′, then Γ′, Γ′′ | = ∆′, ∆′′ Transitivity If Σ, Γ | = ∆, Π for every quasi-partition* Σ, Π of a Θ⊆S, then Γ | = ∆.

* Σ ∪ Π = Θ and Σ ∩ Π = ∅

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 4 / 20

Logical Bivalence into Question

Before Suszko’s Thesis

non-determinism, probability, predictions and uncertainty issues; 1920’s: Łukaziewicz’s Ł3; referential truth-values; Suszko Reduction: bivalent.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 5 / 20

Logical Bivalence into Question

Before Suszko’s Thesis

non-determinism, probability, predictions and uncertainty issues; 1920’s: Łukaziewicz’s Ł3; referential truth-values; Suszko Reduction: bivalent.

After Suszko’s Thesis

non-determinism, probability, predictions and uncertainty again!; 1990’s: Malinowski q-entailment; many-valent;

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 5 / 20

Logical Bivalence into Question

Before Suszko’s Thesis

non-determinism, probability, predictions and uncertainty issues; 1920’s: Łukaziewicz’s Ł3; referential truth-values; Suszko Reduction: bivalent.

After Suszko’s Thesis

non-determinism, probability, predictions and uncertainty again!; 1990’s: Malinowski q-entailment; many-valent; non-reflexive/ non-transitive entailments!!!

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 5 / 20

Section 1 Trivalent Logics

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 6 / 20

q-entailment (G. Malinowski, 1990)

Related to the reasoning by hypotheses; If no statement of the conclusion is accepted then some of the premisses should be rejected.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 7 / 20

q-entailment (G. Malinowski, 1990)

Related to the reasoning by hypotheses; If no statement of the conclusion is accepted then some of the premisses should be rejected. Lq = S, q

q-matrix: Q = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S.

V Y N

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 7 / 20

q-entailment (G. Malinowski, 1990)

Lq = S, q

q-matrix: Q = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S.

V Y N

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 7 / 20

q-entailment (G. Malinowski, 1990)

Lq = S, q

q-matrix: Q = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S.

V Y N

q-entailment relation based on Q

Γ | =q ∆ iff there is no v such that v(Γ) ⊆ N and v(∆) ⊆ Y

where Y := V − Y and N := V − N.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 7 / 20

q-entailment (G. Malinowski, 1990)

Lq = S, q

q-matrix: Q = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S.

V Y N

q-entailment relation based on Q

Γ | =q ∆ iff there is no v such that v(Γ) ⊆ N and v(∆) ⊆ Y

where Y := V − Y and N := V − N.

Proposition (The following holds in a q-logic:)

Monotonicity If Γ′ | = ∆′, then Γ′, Γ′′ | = ∆′, ∆′′ Transitivity If Σ, Γ | = ∆, Π for every q-partition* Σ, Π of a Θ⊆S, then Γ | = ∆.

* Σ ∪ Π ⊆ Θ and Σ ∩ Π = ∅

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 7 / 20

q-entailment (G. Malinowski, 1990)

Lq = S, q

q-matrix: Q = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S.

V Y N

q-entailment relation based on Q

Γ | =q ∆ iff there is no v such that v(Γ) ⊆ N and v(∆) ⊆ Y

where Y := V − Y and N := V − N.

Proposition (The following holds in a q-logic:)

Monotonicity If Γ′ | = ∆′, then Γ′, Γ′′ | = ∆′, ∆′′ Transitivity If Σ, Γ | = ∆, Π for every q-partition* Σ, Π of a Θ⊆S, then Γ | = ∆.

* Σ ∪ Π ⊆ Θ and Σ ∩ Π = ∅

Reflexivity does not always hold!

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 7 / 20

p-entailment (Frankowski, 2004)

Decreasing of certainty from the premisses to the conclusion; If all premisses are accepted, then some statements of the conclusion are not rejected.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 8 / 20

p-entailment (Frankowski, 2004)

Decreasing of certainty from the premisses to the conclusion; If all premisses are accepted, then some statements of the conclusion are not rejected. Lp = S, p

q-matrix: Q = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S.

V Y N

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 8 / 20

p-entailment (Frankowski, 2004)

Lp = S, p

q-matrix: Q = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S.

V Y N

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 8 / 20

p-entailment (Frankowski, 2004)

Lp = S, p

q-matrix: Q = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S.

V Y N

p-entailment relation based on Q

Γ | =p ∆ iff there is no v such that v(Γ) ⊆ Y and v(∆) ⊆ N

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 8 / 20

p-entailment (Frankowski, 2004)

Lp = S, p

q-matrix: Q = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S.

V Y N

p-entailment relation based on Q

Γ | =p ∆ iff there is no v such that v(Γ) ⊆ Y and v(∆) ⊆ N

Proposition (The following holds in a p-logic:)

Reflexivity α p α Monotonicity If Γ′ | = ∆′, then Γ′, Γ′′ | = ∆′, ∆′′

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 8 / 20

p-entailment (Frankowski, 2004)

Lp = S, p

q-matrix: Q = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N ⊆ V and Y ∩ N = ∅ O := Truth-functions for each connective of S.

V Y N

p-entailment relation based on Q

Γ | =p ∆ iff there is no v such that v(Γ) ⊆ Y and v(∆) ⊆ N

Proposition (The following holds in a p-logic:)

Reflexivity α p α Monotonicity If Γ′ | = ∆′, then Γ′, Γ′′ | = ∆′, ∆′′

The p-entailment is not transitive: given ∆ p ϕ and, for every ψ ∈ ∆, Γ p ψ, it is not the case that Γ p ϕ always holds.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 8 / 20

Other kind of trivalence

If we allow glutty reasoning...

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 9 / 20

Other kind of trivalence

dual q-matrix: D = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N = V O := Truth-functions for each connective of S.

V Y N

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 9 / 20

Other kind of trivalence

dual q-matrix: D = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N = V O := Truth-functions for each connective of S.

V Y N

dual q-entailment: | =d (Malinowski, 2000)

Γ | =d ∆ iff there is no v such that v(Γ) ⊆ N and v(∆) ⊆ Y Has the same properties of p-entailment: Reflexivity and Monotonicity, but Transitivity does not always hold.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 9 / 20

Other kind of trivalence

dual q-matrix: D = V, Y, N, O

V := Truth-values; Y := the accepted values; N:= the rejected values; Y ∪ N = V O := Truth-functions for each connective of S.

V Y N

dual q-entailment: | =d (Malinowski, 2000)

Γ | =d ∆ iff there is no v such that v(Γ) ⊆ N and v(∆) ⊆ Y Has the same properties of p-entailment: Reflexivity and Monotonicity, but Transitivity does not always hold.

dual p-entailment: | =b

Γ | =b ∆ iff there is no v such that v(Γ) ⊆ Y and v(∆) ⊆ N Has the same properties of q-entailment: Monotonicity and Transitivity, but Reflexivity does not always hold.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 9 / 20

Understanding Many-valence

Are q- and p-entailment and their duals legitimate entailments?

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 10 / 20

Understanding Many-valence

Are q- and p-entailment and their duals legitimate entailments? All entailment notions related to trivalent logics lack Reflexivity or Transitivity.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 10 / 20

Understanding Many-valence

Are q- and p-entailment and their duals legitimate entailments? All entailment notions related to trivalent logics lack Reflexivity or Transitivity. Nonetheless, those non-standard notions of entailment contain the standard reasoning in some sense.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 10 / 20

Understanding Many-valence

Are q- and p-entailment and their duals legitimate entailments? All entailment notions related to trivalent logics lack Reflexivity or Transitivity. Nonetheless, those non-standard notions of entailment contain the standard reasoning in some sense. If Y ∪ N = V, then q-matrix turns out to be a standard matrix.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 10 / 20

Understanding Many-valence

Are q- and p-entailment and their duals legitimate entailments? All entailment notions related to trivalent logics lack Reflexivity or Transitivity. Nonetheless, those non-standard notions of entailment contain the standard reasoning in some sense. If Y ∪ N = V, then q-matrix turns out to be a standard matrix. If Y ∩ N = ∅, then dual q-matrix turns out to be a standard matrix.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 10 / 20

Understanding Many-valence

Are q- and p-entailment and their duals legitimate entailments? All entailment notions related to trivalent logics lack Reflexivity or Transitivity. Nonetheless, those non-standard notions of entailment contain the standard reasoning in some sense. If Y ∪ N = V, then q-matrix turns out to be a standard matrix. If Y ∩ N = ∅, then dual q-matrix turns out to be a standard matrix. How to compare those notions to each other and to the standard notion of entailment?

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 10 / 20

Section 2 Tetravalent Logics

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 11 / 20

Generalizing a little bit more

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 12 / 20

Generalizing a little bit more

A way to distinguish the kinds of reasoning using a single framework is given by a four-place consequence relation called B-entailment.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 12 / 20

Generalizing a little bit more

A way to distinguish the kinds of reasoning using a single framework is given by a four-place consequence relation called B-entailment. B-entailment presents different kinds of logical reasoning of standard and non-standard consequence relations in a single framework.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 12 / 20

Generalizing a little bit more

A way to distinguish the kinds of reasoning using a single framework is given by a four-place consequence relation called B-entailment. B-entailment presents different kinds of logical reasoning of standard and non-standard consequence relations in a single framework. LB = S, ..

• ..

Generalized q-matrix: B = V, Y, N, O

V := Truth-values; Y := Y ⊆ V, the accepted values; N := N ⊆ V, the rejected values; O := Truth-functions for each connective of S.

Y N V

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 12 / 20

Generalizing a little bit more

LB = S, ..

• ..

Generalized q-matrix: B = V, Y, N, O

V := Truth-values; Y := Y ⊆ V, the accepted values; N := N ⊆ V, the rejected values; O := Truth-functions for each connective of S.

Y N V

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 12 / 20

Generalizing a little bit more

LB = S, ..

• ..

Generalized q-matrix: B = V, Y, N, O

V := Truth-values; Y := Y ⊆ V, the accepted values; N := N ⊆ V, the rejected values; O := Truth-functions for each connective of S.

Y N V

B-entailment relation

Ψ Γ

• ∆

Φ iff there is no v ∈ SEMB such that v(Γ) ⊆ Y and v(∆) ⊆

Y and v(Φ) ⊆ N and v(Ψ) ⊆ N .

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 12 / 20

Generalizing a little bit more

LB = S, ..

• ..

Generalized q-matrix: B = V, Y, N, O

V := Truth-values; Y := Y ⊆ V, the accepted values; N := N ⊆ V, the rejected values; O := Truth-functions for each connective of S.

Y N V

B-entailment relation

Ψ Γ

• ∆

Φ iff there is no v ∈ SEMB such that v(Γ) ⊆ Y and v(∆) ⊆

Y and v(Φ) ⊆ N and v(Ψ) ⊆ N .

Proposition (The following holds in a B-logic:)

B-reflexivities (t) α

• α

and (f ) α

• α

B-monotonicity If Ψ′

Γ′

• ∆′

Φ′ , then Ψ′,Ψ′′ Γ′,Γ′′

• ∆′,∆′′

Φ′,Φ′′

B-transitivities (t) If

Ψ Σ,Γ

• ∆,Π

Φ

for every quasi-partition Σ, Π of a Θ⊆S, then Ψ

Γ

• ∆

Φ .

(f ) If

Σ,Ψ Γ

• ∆

Φ,Π for every quasi-partition Σ, Π of a Θ⊆S, then Ψ Γ

• ∆

Φ .

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 12 / 20

Generalizing a little bit more

LB = S, ..

• ..

Generalized q-matrix: B = V, Y, N, O

V := Truth-values; Y := Y ⊆ V, the accepted values; N := N ⊆ V, the rejected values; O := Truth-functions for each connective of S.

Y N V

B-entailment relation

Ψ Γ

• ∆

Φ iff there is no v ∈ SEMB such that v(Γ) ⊆ Y and v(∆) ⊆

Y and v(Φ) ⊆ N and v(Ψ) ⊆ N . The B-entailment has versions of all desirable properties of a consequence relation!

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 12 / 20

Generalizing a little bit more

LB = S, ..

• ..

Generalized q-matrix: B = V, Y, N, O

V := Truth-values; Y := Y ⊆ V, the accepted values; N := N ⊆ V, the rejected values; O := Truth-functions for each connective of S.

Y N V

B-entailment relation

Ψ Γ

• ∆

Φ iff there is no v ∈ SEMB such that v(Γ) ⊆ Y and v(∆) ⊆

Y and v(Φ) ⊆ N and v(Ψ) ⊆ N .

Proposition (The following holds in a B-logic:)

B-reflexivities (t) α

• α

and (f ) α

• α

B-monotonicity If Ψ′

Γ′

• ∆′

Φ′ , then Ψ′,Ψ′′ Γ′,Γ′′

• ∆′,∆′′

Φ′,Φ′′

B-transitivities (t) If

Ψ Σ,Γ

• ∆,Π

Φ

for every quasi-partition Σ, Π of a Θ⊆S, then Ψ

Γ

• ∆

Φ .

(f ) If

Σ,Ψ Γ

• ∆

Φ,Π for every quasi-partition Σ, Π of a Θ⊆S, then Ψ Γ

• ∆

Φ .

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 12 / 20

Refining truth-values

B-matrix allows for gappy reasoning if Y ∩ N = ∅. B-matrix allows for glutty reasoning if Y ∪ N = V. Y N V

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 13 / 20

Refining truth-values

B-matrix allows for gappy reasoning if Y ∩ N = ∅. B-matrix allows for glutty reasoning if Y ∪ N = V. Y N V If glutty reasoning is not allowed, then ϕ

• ϕ.

V Y N

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 13 / 20

Refining truth-values

B-matrix allows for gappy reasoning if Y ∩ N = ∅. B-matrix allows for glutty reasoning if Y ∪ N = V. Y N V If glutty reasoning is not allowed, then ϕ

• ϕ.

V Y N If gappy reasoning is not allowed, then ϕ

ϕ.

V Y N

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 13 / 20

Refining truth-values

B-matrix allows for gappy reasoning if Y ∩ N = ∅. B-matrix allows for glutty reasoning if Y ∪ N = V. Y N V If glutty reasoning is not allowed, then ϕ

• ϕ.

V Y N If gappy reasoning is not allowed, then ϕ

ϕ.

V Y N If neither glutty nor gappy reasoning are allowed, then ϕ

• ϕ and ϕ

ϕ.

V Y N

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 13 / 20

Different kinds of logical reasoning I

Bivalent entailment: If ϕ

• ϕ and ϕ

ϕ holds, then ..

• .. turns out to be |

=. Γ ∆ iff there is no s such that v(Γ) ⊆ Y ∪ N and v(∆) ⊆ Y ∪ N iff Γ′′

Γ′

• ∆′

∆′′ where Γ′ ∪ Γ′′ = Γ and ∆′′ ∪ ∆′′ = ∆

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 14 / 20

Different kinds of logical reasoning I

Bivalent entailment: If ϕ

• ϕ and ϕ

ϕ holds, then ..

• .. turns out to be |

=. Γ ∆ iff there is no s such that v(Γ) ⊆ Y ∪ N and v(∆) ⊆ Y ∪ N iff Γ′′

Γ′

• ∆′

∆′′ where Γ′ ∪ Γ′′ = Γ and ∆′′ ∪ ∆′′ = ∆

Standard entailment generalized: Without restrictions we also can get standard entailments: Γ t ∆ iff there is no s such that v(Γ) ⊆ Y and v(∆) ⊆ Y iff Γ

• ∆

Ψ f Φ iff there is no s such that v(Ψ) ⊆ N and v(Φ) ⊆ N iff Ψ

• Φ

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 14 / 20

Different kinds of logical reasoning II

Trivalent “gappy” entailments: When ϕ

• ϕ holds,

Ψ q ∆ iff there is no s such that v(Ψ) ⊆ N and v(∆) ⊆ Y iff Ψ

∆

Γ p Φ iff there is no s such that v(Γ) ⊆ Y and v(Φ) ⊆ N iff Γ

• Φ

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 15 / 20

Different kinds of logical reasoning II

Trivalent “gappy” entailments: When ϕ

• ϕ holds,

Ψ q ∆ iff there is no s such that v(Ψ) ⊆ N and v(∆) ⊆ Y iff Ψ

∆

Γ p Φ iff there is no s such that v(Γ) ⊆ Y and v(Φ) ⊆ N iff Γ

• Φ

Trivalent “glutty” entailments: When ϕ

ϕ holds,

Ψ d ∆ iff there is no s such that v(Ψ) ⊆ N and v(∆) ⊆ Y iff Ψ

∆

Γ b Φ iff there is no s such that v(Γ) ⊆ Y and v(Φ) ⊆ N iff Γ

• Φ

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 15 / 20

Different kinds of logical reasoning II

Trivalent “gappy” entailments: When ϕ

• ϕ holds,

Ψ q ∆ iff there is no s such that v(Ψ) ⊆ N and v(∆) ⊆ Y iff Ψ

∆

Γ p Φ iff there is no s such that v(Γ) ⊆ Y and v(Φ) ⊆ N iff Γ

• Φ

Trivalent “glutty” entailments: When ϕ

ϕ holds,

Ψ d ∆ iff there is no s such that v(Ψ) ⊆ N and v(∆) ⊆ Y iff Ψ

∆

Γ b Φ iff there is no s such that v(Γ) ⊆ Y and v(Φ) ⊆ N iff Γ

• Φ

How to reason with both gap and glut inferential values?

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 15 / 20

Entailments for Tetravalent Logics

(Blasio,Marcos,Wansing, 2017) If we allow gappy and glutty reasoning...

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 16 / 20

Entailments for Tetravalent Logics

B-matrix: B = V, Y, N, O

V := Truth-values; Y := Y ⊆ V, the accepted values; N := N ⊆ V, the rejected values; O := Truth-functions for each connective of S.

Y N V

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 16 / 20

Entailments for Tetravalent Logics

B-matrix: B = V, Y, N, O

V := Truth-values; Y := Y ⊆ V, the accepted values; N := N ⊆ V, the rejected values; O := Truth-functions for each connective of S.

Y N V

generalized q-entailment: | =q

Γ | =q ∆ iff there is no v such that v(Γ) ⊆ N and v(∆) ⊆ Y Monotonicity holds, but Reflexivity and Transitivity do not always hold.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 16 / 20

Entailments for Tetravalent Logics

B-matrix: B = V, Y, N, O

V := Truth-values; Y := Y ⊆ V, the accepted values; N := N ⊆ V, the rejected values; O := Truth-functions for each connective of S.

Y N V

generalized q-entailment: | =q

Γ | =q ∆ iff there is no v such that v(Γ) ⊆ N and v(∆) ⊆ Y Monotonicity holds, but Reflexivity and Transitivity do not always hold.

generalized p-entailment: | =p

Γ | =p ∆ iff there is no v such that v(Γ) ⊆ Y and v(∆) ⊆ N Monotonicity holds, but Reflexivity and Transitivity do not always hold.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 16 / 20

Summary

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 17 / 20

Summary

B-entailment provides a uniform framework for studying standard and non-standard notions of entailment.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 18 / 20

Summary

B-entailment provides a uniform framework for studying standard and non-standard notions of entailment. The two dimensions has standard-like properties that do not appear in non-standard consequence relations.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 18 / 20

Summary

B-entailment provides a uniform framework for studying standard and non-standard notions of entailment. The two dimensions has standard-like properties that do not appear in non-standard consequence relations. Generalized reduction results

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 18 / 20

Summary

B-entailment provides a uniform framework for studying standard and non-standard notions of entailment. The two dimensions has standard-like properties that do not appear in non-standard consequence relations. Generalized reduction results Contribution to the discussion about the concept of logical entailment.

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 18 / 20

Summary

B-entailment provides a uniform framework for studying standard and non-standard notions of entailment. The two dimensions has standard-like properties that do not appear in non-standard consequence relations. Generalized reduction results Contribution to the discussion about the concept of logical entailment.

Thank you!

Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 18 / 20

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Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 19 / 20

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Carolina Blasio (IFCH/UNICAMP) Through Many-Valent Semantics Logic and Epistemology 20 / 20