SLIDE 1
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model Some Results
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model Some Results
A New Model for the Liar Some Results Luca Castaldo - - PowerPoint PPT Presentation
A New Model for the Liar Some Results Luca Castaldo - - PowerPoint PPT Presentation
A New Model for the Liar L. Castaldo What, and Why? The New Model A New Model for the Liar Some Results Luca Castaldo luca.castaldo@bristol.ac.uk University of Bristol PhDs in Logic X 1st-4th May 2018, Prague, Czech Republic A New Model
SLIDE 2
SLIDE 3
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
Section 1 What, and Why?
SLIDE 4
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
Preliminaries
The formal language I am going to interpret is the language Lpa of Peano arithmetic extended by a unary truth predicate T.
SLIDE 5
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
Preliminaries
The formal language I am going to interpret is the language Lpa of Peano arithmetic extended by a unary truth predicate
- T. The addition of a truth predicate T to Lpa delivers two
classes of pathological sentences:
SLIDE 6
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
Preliminaries
The formal language I am going to interpret is the language Lpa of Peano arithmetic extended by a unary truth predicate
- T. The addition of a truth predicate T to Lpa delivers two
classes of pathological sentences:
- Truth-tellers:
Θ := {τ | PA ⊢ τ ↔ Tτ}
- Liars
Λ := {λ | PA ⊢ λ ↔ ¬Tλ}
SLIDE 7
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
Questions
Two diabolical questions:
- Are Liars true or untrue?
- Are Truth-tellers true or untrue?
SLIDE 8
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
Liars and Truth-tellers
Liars are paradoxical, as they cannot be consistently declared true, or untrue. Truth-tellers are not paradoxical, as they can.
SLIDE 9
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
Kripke’s Idea
Following Kripke, one defines a “minimal fixed point model” M, (E∞, A∞) for L t
pa, such that:
- M is the standard interpretation of Lpa.
- The interpretation of T is a pair (E∞, A∞) of disjoint
subsets of N.
SLIDE 10
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
Kripke’s Idea
Following Kripke, one defines a “minimal fixed point model” M, (E∞, A∞) for L t
pa, such that:
- M is the standard interpretation of Lpa.
- The interpretation of T is a pair (E∞, A∞) of disjoint
subsets of N.
- E ∪ A = N.
- Liars and Truth-tellers are outside E∞ ∪ A∞.
- Underlying logic 3-valued, e.g. Kleene’s K3.
SLIDE 11
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
Dialetheic Idea
- r Liberalisation of Kripke’s construction
A dialetheic model is a structure M, (E, A), such that:
- M is the standard interpretation of Lpa.
- The interpretation of T is a pair (E, A) of overlapping
subsets of N.
SLIDE 12
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
Dialetheic Idea
- r Liberalisation of Kripke’s construction
A dialetheic model is a structure M, (E, A), such that:
- M is the standard interpretation of Lpa.
- The interpretation of T is a pair (E, A) of overlapping
subsets of N.
- Liars are element of E ∩ A.
- Truth-tellers are outside E ∪ A.
- Underlying logic 4-valued, e.g. Belnap-Dunn’s FDE.
SLIDE 13
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
The Gap
No theory of truth satisfies the two following desiderata:
1 It differentiates between Liars and Truth-tellers. 2 It does not admit true contradictions.
Kripkean theories fall short of 1. Liberalisations of Kripke’s construction admit inconsistent interpretations.
SLIDE 14
A New Model for the Liar
- L. Castaldo
What, and Why?
Preliminaries Liars and Truth-Tellers The Gap
The New Model Some Results
Purpose
Without arguing for or against dialetheism, I propose a theory which fills the gap: a consistent interpretation of L t
pa
that sees the difference between Θ and Λ.
SLIDE 15
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
Section 2 The New Model
SLIDE 16
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
Model Outline
M, (E, A, X) will be the interpretation of L t
pa with:
- M is the standard interpretation of Lpa.
SLIDE 17
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
Model Outline
M, (E, A, X) will be the interpretation of L t
pa with:
- M is the standard interpretation of Lpa.
- The interpretation of T is a triple (E, A, X) of pairwise
disjoint subsets of N.
SLIDE 18
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
Model Outline
M, (E, A, X) will be the interpretation of L t
pa with:
- M is the standard interpretation of Lpa.
- The interpretation of T is a triple (E, A, X) of pairwise
disjoint subsets of N.
- Liars are element of X.
- Truth-tellers are outside E ∪ A ∪ X.
- Underlying logic 4-valued, but not FDE.
SLIDE 19
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
The new Logic
Truth Values
The new 4-valued logic is defined by the structure: W = {1, 0, p, u} D = {1} C = {f¬, f∨, f∧} where:
- W = set of truth values (true, false, paradoxical,
undefined);
- D = set of the sole designated value;
- C = set of truth functions.
SLIDE 20
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
The new Logic
Truth Tables
¬ 1 1 p p u u ∨ 1 p u 1 1 1 1 1 1 p u p 1 p p u u 1 u u u ∧ 1 p u 1 1 p u p p p u u u u u
SLIDE 21
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
Valuation Function
Part 1
The valuation function VM,(E,A,X) : L t
pa −
→ {1, 0, p, u}, assigning to each sentence ϕ ∈ L t
pa a truth value in the
model M, (E, A, X), is defined as follows: (a) For atomic Lpa-sentences: (b) For atomic L t
pa-sentences T(n):
SLIDE 22
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
Valuation Function
Part 1
The valuation function VM,(E,A,X) : L t
pa −
→ {1, 0, p, u}, assigning to each sentence ϕ ∈ L t
pa a truth value in the
model M, (E, A, X), is defined as follows: (a) For atomic Lpa-sentences: V(ϕ) =
- 1
if M | = ϕ if M | = ¬ϕ (b) For atomic L t
pa-sentences T(n):
SLIDE 23
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
Valuation Function
Part 1
The valuation function VM,(E,A,X) : L t
pa −
→ {1, 0, p, u}, assigning to each sentence ϕ ∈ L t
pa a truth value in the
model M, (E, A, X), is defined as follows: (a) For atomic Lpa-sentences: V(ϕ) =
- 1
if M | = ϕ if M | = ¬ϕ (b) For atomic L t
pa-sentences T(n):
V(T(n)) = 1 if n ∈ E if n ∈ A p if n ∈ X u if n / ∈ E ∪ A ∪ X
SLIDE 24
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
Valuation Function
Part 2
(c) V(∃viϕ(vi)) = 1 if for some n ∈ N
- V
- ϕ(n)
- = 1
- if for all n ∈ N
- V
- ϕ(n)
- = 0
- p
if for some n ∈ N
- V
- ϕ(n)
- = p
- and
for all m ∈ N
- V
- ϕ(m)
- ∈ {0, p}
- u
if for some n ∈ N
- V
- ϕ(n)
- = u
- and
for no n ∈ N
- V
- ϕ(n)
- = 1
SLIDE 25
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
The new Logic
Consequence Relation
Definition
An inference from Σ to ϕ is valid iff if V(ψ) = 1 (the sole designated value), for any ψ ∈ Σ, then V(ϕ) = 1.
SLIDE 26
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
Remarks
Remark
The new logic is not paraconsistent.
Remark
Disjunctive Syllogism is a valid rule of inference.
Remark
Ex Contradictione Quodlibet is a valid rule of inference.
SLIDE 27
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
The Interpretation of T
The interpretation (E, A, X) of T is defined thus: E := E∞ = the extension of T in Mfp. A := A∞ = the anti-extension of T in Mfp. X := the minimal fixed point of Γ.
SLIDE 28
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
The Paradox Operator
Definition
Definition
The paradox operator Γ : P(N) − → P(N) is defined thus: Γ(S) = {n | ζ(n, S, A)} where ζ(n, S, A) abbreviates the disjunction of (i)-(v) below: (see next slide)
SLIDE 29
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
The Paradox Operator
Definition
(i) n = gn(ϕ) and ϕ is an “atomic Liar” (ii) n = gn(¬ϕ) and gn(ϕ) ∈ S (iii) n = gn(ϕ ∨ ψ) and (gn(ϕ) ∈ S or gn(ψ) ∈ S) and
- (gn(ϕ) ∈ S ⇒ gn(ψ) ∈ S ∪ A) and
(gn(ψ) ∈ S ⇒ gn(ϕ) ∈ S ∪ A)
- (iv)
n = gn(∃viϕ(vi)) and for some m ∈ N
- gn(ϕ(m)) ∈ S
- and
for all k ∈ N
- gn(ϕ(k)) ∈ S ∪ A
- (v)
n = gn(T(m)) and m ∈ S
SLIDE 30
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model
4-valued logic, Truth-Tables 4-valued Logic, Valuation Function 4-valued Logic, Consequence Relation Interpretation of T
Some Results
The Paradox Operator
Properties
Lemma
Γ is monotone, i.e. Sα ⊆ Sβ ⇒ Γ(Sα) ⊆ Γ(Sβ)
Lemma
Γ has a minimal fixed point, i.e. For some Sκ, Sκ = Γ(Sκ) and for all Sξ Sξ = Γ(Sξ) ⇒ Sκ ⊆ Sξ
SLIDE 31
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model Some Results
Consistency Adequacy Transparency
Section 3 Some Results
SLIDE 32
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model Some Results
Consistency Adequacy Transparency
Consistency
The new interpretation is consistent in the following sense:
Lemma
X ∩ (E ∪ A) = ∅
SLIDE 33
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model Some Results
Consistency Adequacy Transparency
Consistency
The new interpretation is consistent in the following sense:
Lemma
X ∩ (E ∪ A) = ∅
Idea of Proof.
A transfinite sequence of members of P(N) indexed by
- rdinals is defined as follows:
S0 := ∅ Sα+1 := Γ(S) Sλ :=
- α<λ
Sα The proof divides into two steps: One shows that (i) the sequence is non-decreasing, in the sense that Sα ⊆ Sα+1 for all α and that (ii) each member of the sequence does not
- verlap with E or A. Since X is the least fixed point of Γ, it
follows that X ∩ (E ∪ A) = ∅.
SLIDE 34
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model Some Results
Consistency Adequacy Transparency
Adequacy
The new interpretation is Liars- and Truth-tellers-adequate in the following sense:
Lemma
(1) λ ∈ Λ∗ ⇒ (i) V(λ) = p and (ii) gn(λ) ∈ X (2) τ ∈ Θ∗ ⇒ (i) V(τ) = u and (ii) gn(τ) / ∈ X
SLIDE 35
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model Some Results
Consistency Adequacy Transparency
Adequacy
The new interpretation is Liars- and Truth-tellers-adequate in the following sense:
Lemma
(1) λ ∈ Λ∗ ⇒ (i) V(λ) = p and (ii) gn(λ) ∈ X (2) τ ∈ Θ∗ ⇒ (i) V(τ) = u and (ii) gn(τ) / ∈ X
Idea of Proof.
(1)-(ii) is trivial. For (2)-(ii), one shows by contraposition that if gn(ϕ) ∈ X, then ϕ is undefined in all consistent Kripkean fixed points. Since Truth-tellers have a truth value in some consistent fixed point, they are outside X. (1)-(ii) and (2)-(ii) imply, by definitions of Λ and Θ, (1)-(i) and (2)-(i).
SLIDE 36
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model Some Results
Consistency Adequacy Transparency
Transparency
The new interpretation is transparent in the following sense:
Theorem
V(ϕ) = V(Tϕ)
SLIDE 37
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model Some Results
Consistency Adequacy Transparency
Transparency
The new interpretation is transparent in the following sense:
Theorem
V(ϕ) = V(Tϕ)
Idea of Proof.
That V(ϕ) = 1(0) iff V(Tϕ) = 1(0) follows immediately by definitions of E and A, the new logic, and the work of
- Kripke. By induction on the complexity of ϕ, one then
shows that V(ϕ) = p iff V(Tϕ) = p.
SLIDE 38
A New Model for the Liar
- L. Castaldo
What, and Why? The New Model Some Results
Consistency Adequacy Transparency