A Note on Suszko’s Reduction and Suszko’s Thesis

Rossella Marrano

Scuola Normale Superiore, Pisa

Joint work with Hykel Hosni

April 25, 2014

The problem

Roman Suszko (1919-1979)

Obviously, any multiplication of logicalvalues is a mad idea. (1977)

Suszko’s Reduction (SR)Every Tarskian logic has anadequate bivalent semantics.

Suszko’s Thesis (ST)True and false are the only logicalvalues.

Łukasiewicz is the chief perpetrator of a magnificent conceptual deceitlasting out in mathematical logic to the present day.

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Abstract logic and algebraic semantics

I 〈L,`〉 where ` ⊆ P(L)× L

I ` is Tarskian if satisfies the following:

(REF) θ ∈ Γ⇒ Γ ` θ,

(MON) Γ ⊆ ∆, Γ ` θ ⇒ ∆ ` θ,

(TR) Γ ` θ, Γ, θ ` φ⇒ Γ ` φ.

I FM = 〈For , C〉

I 〈FM,`〉

I A = 〈A, { fc | c ∈ C }〉

I h : For → A with h ∈ Hom(FM,A)

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Logical Matrices: the set of designated values

I M = 〈A,D〉 with D ⊂ A.

Validity

γ ∈ For is valid under h ∈ Hom(FM,A) iff h(γ) ∈ D

Having a model (or satisfiability)

Γ ⊆ For has a model iff there exists h ∈ Hom(FM,A) such that ∀γ ∈ Γ h(γ) ∈ D

Tautology

γ ∈ For is a tautology iff for all h ∈ Hom(FM,A) h(γ) ∈ D

Logical Consequence

Γ |=M φ⇐⇒ ∀h ∈ Hom(FM,A) if ∀γ ∈ Γ h(γ) ∈ D then h(φ) ∈ D.

Lemma (Wójcicki’s Theorem)

Every structural Tarskian logic has an adequate n-valued matrix semantics, forn ≤ |For |.

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Example: Ł3

“To me, personally, the principle of bivalence does not appear to beself-evident. Therefore I am entitled not to recognize it, and to accept theview that besides truth and falsehood there exist other truth-values,including at least one more, the third truth-value.” (Łukasiewicz, 1922)

I FM = 〈For ,¬,∧,∨,→〉

I A = 〈{0, 12 , 1},F¬,F∧,F∨,F→〉

I h : For → {0, 12 , 1}, with h ∈ Hom(FM,A)

I D = {1}I M = 〈A, {1}〉

I Γ |=M φ⇐⇒ ∀h ∈ Hom(FM,A) if v(Γ) = 1 then v(φ) = 1.

a conclusion follows logically from some premises if and only if, whenever thepremises are true, the conclusion is also true. (Tarski, 1936)

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Suszko’s reduction: the standard presentation

Theorem (Suszko, 1977)

Every Tarskian logic has an adequate bivalent semantics.

I L = 〈FM,`〉

I By Wójcicki’s Theorem, there exists M = 〈A,D〉 s.t.I |A| is countable,I |=M = `

I for any h ∈ Hom(FM,A) and for any φ ∈ For define h2 : For → {0, 1}:

h2(φ) =

{1, if h(φ) ∈ D;0, if h(φ) /∈ D.

I There exists M2 = 〈A,D〉 s.t.I |A| = 2,I |=2 = |=M

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Alternative proof (Tsuji, 1998)

Theorem (Suszko, 1977)

Every Tarskian logic has an adequate bivalent semantics.

I L = 〈FM,`〉

I CL ={

Γ̄ ⊆ For∣∣ Γ̄ ` φ implies φ ∈ Γ̄

}I for all Γ̄ ⊆ For

vΓ̄(φ) =

{1, if φ ∈ Γ̄;0, if φ /∈ Γ̄.

I |= defined on{vΓ̄ : For → {0, 1}

∣∣ Γ̄ ∈ CL}

I |= = `

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Our proposal: a two-fold result

SR1 Every Tarskian logic has an adequate bivalent semantics.

SR2 Every n-valued matrix semantics can be reduced to a bivalentsemantics.

Is this a semantics?

I syntactical nature

I truth-functionality

Is this a reduction?

I ontological reduction

I truth-functionality

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What is left?

The logical content

SR1 Tarskian axioms on logical consequence fully characterise a‘1-preserving’ notion of consequence.

SR2 the distinction between designed and undesigned values restoresbivalence.

Intrinsic bivalence of the Tarskian notion of logical consequence.

A philosophical content? Against Suszko’s thesis

SST True and false are the only logical values.

WST Every Tarskian logic is logically two-valued.

No direct philosophical implications on the nature of truth-values and on thestatus of many-valuedness.

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Philosophical ‘feedback’

Logical consequence

I meta-level bivalence

I different notions of logical consequence

Degrees of truth

I more than one notion of truth in the model

I ...or ‘degrees of falsity’?

Methodological lesson

I mathematical theorem/philosohical issues

I formalisation in philosophy

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References

I J.Łukasiewicz. Selected works.L. Borkowski (ed.), North-Holland Pub. Co.,Amsterdam, 1970.

I J. M. Font. Taking Degrees of Truth Seriously. Studia Logica, 91(3):383–406,2009.

I R. Suszko. The Fregean Axiom and Polish Mathematical Logic in the 1920s,Studia Logica, XXXVI (4), 1977.

I A. Tarski. On the concept of following logically. 1936

I M. Tsuji. Many-Valued Logics and Suszko’s Thesis Revisited. Studia Logica,60:299–309, 1998.

I H. Wansing and Y. Shramko. Suszko’s Thesis, Inferential Many-valuedness, andthe Notion of a Logical System. Studia Logica, 88(3):405–429, 2008.

I R. Wójcicki. Some Remarks on the Consequence Operation in Sentential Logics.Fundamenta Mathematicae, 8:269–279, 1970.

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Truth-functionality

[. . . ] logical valuations are morphism (of formulas to the zero-one model) insome exceptional cases only. (Suszko, 1977)

Łukasiewicz three-valued logic

I h : For → {0, 12 , 1}

I D = {1}

I SR: ∀h ∀φ ∃h2 : For → {0, 1}

h2(φ) =

{1, if h(φ) = 1;0, otherwise.

h2 is not compositional

I if h(φ) = 12 then h2(φ) = h2(¬φ) = 0

I if h(φ) = 1 then h2(φ) = 1 and h2(¬φ) = 0

Back

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