1. A Note on Suszko’s Reduction and Suszko’s Thesis
Rossella Marrano
Scuola Normale Superiore, Pisa
Joint work with Hykel Hosni
April 25, 2014

2. The problem
Roman Suszko (1919-1979)
Obviously, any multiplication of logical
values is a mad idea. (1977)
Suszko’s Reduction (SR)
Every Tarskian logic has an
adequate bivalent semantics.
Suszko’s Thesis (ST)
True and false are the only logical
values.
Łukasiewicz is the chief perpetrator of a magnificent conceptual deceit
lasting out in mathematical logic to the present day.
Rossella Marrano (SNS) Suszko’s reduction and Suszko’s thesis 25/04/2014 2 / 12

3. Abstract logic and algebraic semantics
L, where ⊆ P(L) × L
is Tarskian if satisfies the following:
(REF) θ ∈ Γ ⇒ Γ θ,
(MON) Γ ⊆ ∆, Γ θ ⇒ ∆ θ,
(TR) Γ θ, Γ, θ φ ⇒ Γ φ.
FM = For, C
FM,
A = A, { fc | c ∈ C }
h: For → A with h ∈ Hom(FM, A)
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4. Logical Matrices: the set of designated values
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, for
n ≤ |For|.
Rossella Marrano (SNS) Suszko’s reduction and Suszko’s thesis 25/04/2014 4 / 12

5. Example: Ł3
“To me, personally, the principle of bivalence does not appear to be
self-evident. Therefore I am entitled not to recognize it, and to accept the
view that besides truth and falsehood there exist other truth-values,
including at least one more, the third truth-value.” (Łukasiewicz, 1922)
FM = For, ¬, ∧, ∨, →
A = {0, 1
2 , 1}, F¬, F∧, F∨, F→
h: For → {0, 1
2 , 1}, with h ∈ Hom(FM, A)
D = {1}
M = A, {1}
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 the
premises are true, the conclusion is also true. (Tarski, 1936)
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6. Suszko’s reduction: the standard presentation
Theorem (Suszko, 1977)
Every Tarskian logic has an adequate bivalent semantics.
L = FM,
By Wójcicki’s Theorem, there exists M = A, D s.t.
|A| is countable,
|=M =
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.
There exists M2 = A, D s.t.
|A| = 2,
|=2 = |=M
Rossella Marrano (SNS) Suszko’s reduction and Suszko’s thesis 25/04/2014 6 / 12

7. Alternative proof (Tsuji, 1998)
Theorem (Suszko, 1977)
Every Tarskian logic has an adequate bivalent semantics.
L = FM,
CL = ¯Γ ⊆ For ¯Γ φ implies φ ∈ ¯Γ
for all ¯Γ ⊆ For
v¯Γ(φ) =
1, if φ ∈ ¯Γ;
0, if φ /∈ ¯Γ.
|= defined on v¯Γ : For → {0, 1} ¯Γ ∈ CL
|= =
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8. 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 bivalent
semantics.
Is this a semantics?
syntactical nature
truth-functionality
Is this a reduction?
ontological reduction
truth-functionality
Rossella Marrano (SNS) Suszko’s reduction and Suszko’s thesis 25/04/2014 8 / 12

9. 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 restores
bivalence.
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 the
status of many-valuedness.
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10. Philosophical ‘feedback’
Logical consequence
meta-level bivalence
different notions of logical consequence
Degrees of truth
more than one notion of truth in the model
...or ‘degrees of falsity’?
Methodological lesson
mathematical theorem/philosohical issues
formalisation in philosophy
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11. References
J.Łukasiewicz. Selected works.L. Borkowski (ed.), North-Holland Pub. Co.,
Amsterdam, 1970.
J. M. Font. Taking Degrees of Truth Seriously. Studia Logica, 91(3):383–406,
2009.
R. Suszko. The Fregean Axiom and Polish Mathematical Logic in the 1920s,
Studia Logica, XXXVI (4), 1977.
A. Tarski. On the concept of following logically. 1936
M. Tsuji. Many-Valued Logics and Suszko’s Thesis Revisited. Studia Logica,
60:299–309, 1998.
H. Wansing and Y. Shramko. Suszko’s Thesis, Inferential Many-valuedness, and
the Notion of a Logical System. Studia Logica, 88(3):405–429, 2008.
R. Wójcicki. Some Remarks on the Consequence Operation in Sentential Logics.
Fundamenta Mathematicae, 8:269–279, 1970.
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12. Truth-functionality
[. . . ] logical valuations are morphism (of formulas to the zero-one model) in
some exceptional cases only. (Suszko, 1977)
Łukasiewicz three-valued logic
h: For → {0, 1
2 , 1}
D = {1}
SR: ∀h ∀φ ∃h2 : For → {0, 1}
h2(φ) =
1, if h(φ) = 1;
0, otherwise.
h2 is not compositional
if h(φ) = 1
2 then h2(φ) = h2(¬φ) = 0
if h(φ) = 1 then h2(φ) = 1 and h2(¬φ) = 0
Back
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