Bounds for the set of LFI extensible with a transparent truth predicate
Bounds for the set of LFI extensible with a transparent truth predicate
D.E. Szmuc References
Dami´an E. Szmuc (joint work with Eduardo Barrio and Federico Pailos) CONICET - UNIVERSIDAD DE BUENOS AIRES
2nd Workshop CLE-Buenos Aires Logic Group CLE-Unicamp, Brazil. March 31 to April 03, 2015
Introduction Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
Aim of this paper: to begin the investigation on the minimal and maximal LFIs that can be non-trivially (v.g paradox-free) extended with a transparent truth predicate. Starting Point: One paraconsistent logic usually employed to deal with semantic paradoxes is Priest’s Logic LP. It would be interesting and also desirable to extend it with a truth predicate Tr and a consistency operator ◦ to have a paraconsistent theory of truth and inconsistency. ⇒ Sadly, the resulting theory of truth is trivial. We know this because its paracomplete dual, Strong Kleene Logic K3 augmented with Tr and a consistency operator is known to be trivial in the presence of a truth predicate. ([3], [2], ch. 2)
Introduction Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
Hence, some interesting questions arise: is there a set of LFIs meaningfully extensible with a transparent truth predicate? Carnielli and Rodrigues’ [1] theory Verum proves that this collection is, in fact, non-empty. ⇒ Thus, the question we want to address in this paper is: which are the minimal and maximal LFIs that belong to that set?
Sketch of this talk Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc
1
We discuss the case of PWK+ , the paraconsistent version of Weak Kleene Logic Kw 3 , augmented with Tr and a consistency operator.
2
This logic is proven to be an interesting logic of both formal inconsistency and truth, since it is not subject to trivialization neither via The Liar Paradox (as was independently hinted in [4]), nor via Curry’s Paradox, as we show in this paper.
3
We point out the existence of some weaker LFIs that might handle well the notions of consistency and truth, which indicates that PWK+ is not the smallest of the set of paraconsistent logics in question.
4
We compare how these logics fare with regard to principles and axioms for both truth and consistency
References
Diagnosis of LP’s failure Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc
One of the reasons of the failure of LP’s extension with ◦ was the possibility to define a classical negation . ∼, simply as ∼ φ =def ¬φ ∨ ¬ ◦ φ or, alternatively,
References
∼ φ =def ¬(φ ∧ ◦φ) With the following corresponding truth table φ 1 1 2
0
∼φ 0 1 1
Notice that, in this context, JustTrueφ =def ¬ ∼ φ
Diagnosis of LP’s failure Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
With this negation, it is possible to define a (strengthened) Liar sentence in a theory Th with enough power to represent its own syntax, by `Th λ∗ ↔∼ Tr (pλ∗ q) It is easy to see that λ∗ cannot have a stable valuation and that, therefore, there is no model for the theory. ⇒ Hence, we conclude that we need to work with weaker LFIs, in which it isn’t possible to define such a strong (v.g. classical) negation. ⇒ Equivalently, LFIs that are as strong as K3 or LP which are able to define a similar consistency operator, cannot handle Tr non-trivially.
Weaker paraconsistent logics: FDE Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
It can be proven that FDE is not extensible to a LFI, call it FDE+ that can consistently handle a truth predicate. FDE is the meet of K3 and LP, both logics that can define a classical negation when the consistency operator is around. Hence, FDE also can define a classical negation when the operator ◦ is added with a reasonable interpretation. Therefore, strengthened Liars arise in FDE and so it cannot be extended to a logic of formal inconsistency and truth. (*) There are some other weaker logics: the paraconsistent versions of the asymmetrical logics Lisp and its right-to-left dual. But since they are intermediate, there are some workarounds to define the problematic connectives anyway.
Weaker paraconsistent logics: PWK, PWK+ Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc
∧PWK 1
1 1
References
1 2
1 2
0
0
1 2 1 2 1 2 1 2
0 0
∨PWK 1
1 1
1 2
1 2
1 2
0
0
0
φ 1 1 2
0
◦φ 1 0 1
1 2 1 2 1 2 1 2
0 1
⊃PWK 1
1 1
1 2
1 2
1 2
0
0
1
φ 1
¬φ 1
1 2
1 2
0
1
1 2 1 2 1 2 1 2
0 0 1 2
0
Weaker paraconsistent logics: PWK, PWK+ Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
It is known from the literature in paradoxes and semantic closure that Kw 3 + ◦ + Tr is a non-trivial logic and, in fact, has many models (maximal, minimal and intrinsic). Hence, taking its paraconsistent dual renders also a consistent and non-trivial theory. More importantly, PWK+ = PWK + ◦ + Tr is a logic where Explosion is invalid, but the Principle of Gentle Explosion is valid. V.g. φ, ¬φ 0 ψ but, nevertheless ◦φ, φ, ¬φ ` ψ ⇒ PWK+ is a genuine LFI which is extensible with a truth predicate.
PWK+ is non-trivial Bounds for the set of LFI extensible with a transparent truth predicate
It is interesting to remark some facts about PWK+ classical negation is not definable in it; (!) Notice the case where ∼ 1
D.E. Szmuc References
φ
= ×
1 2
(¬φ
∨
1 2
1 2
¬ ◦ φ) 1
the definable conditional doesn’t validate Modus Ponens irrestrictedly (!) Notice the case where φ,
φ⊃ψ
1 2
1 2
2
ψ 0
⇒ Therefore, PWK+ isn’t prone to trivialization neither via the (strengthened) Liar Paradox1 , nor via Curry’s Paradox. 1 Independently
hinted by H. Omori and R. Ciuni
Beyond PWK+ Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
Given PWK+ ’s main features for being able to handle both truth and consistency in the object language are its expressive impossibilities, one might ask if there are other paraconsistent logics which fulfill this description. And indeed there are. We consider three additional logics with truth-functional semantics, and one logic with non-deterministic semantics. However, in this talk we will focus on the former logics. The first three are all sublogics of FDE FDEwk = the meet of LP and Kw3 FDEpwk = the meet of PWK and K3 WFDE = the meet of PWK and Kw3
Beyond PWK+ Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
The last one is a sublogic of PWK, we call it PWKND and is exactly like PWK restricted to V = {0, 1}, but with the following non-deterministic negation ¬φ {1, 0} {1}
φ 1 0
To all these logics, we will add a consistency operator as a primitive (and, indeed, non-definable) operator, with the following truth-table φ 1 1 2
0
◦φ 1 0 1
We will denote the resulting systems by the addition of an name of the logic in question.
+
to the
FDEw Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
(*) Take the usual lattice ordering of the values ∧FDE w 1 b n 0
1 1 b n 0
b b b n 0
n n n n n
⊃FDE w 1 b n 0
∨FDE w 1 b n 0
0 0 0 n 0 1 1 b n 1
b b b n b
n n n n n
1 1 1 n 1 0 0 b n 0
b 1 b n b
n n n n n
0 1 b n 0
FDEpwk Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
(*) Take the usual lattice ordering of the values ∧FDE pwk 1 b n 0
1 1 b n 0
b b b b b
n n b n 0
⊃FDE pwk 1 b n 0
∨FDE pwk 1 b n 0
0 0 b 0 0 1 1 b 1 1
b b b b b
n n b n 1
1 1 b 1 1 0 0 b n 1
b b b b b
n 1 b n n
0 1 b n 0
WFDE Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
(*) Take the usual lattice ordering of the values and add e as the infimum of that ordering. e behaves like n in Kw 3 , i.e. it is infectious. ∧WFDE 1 b n e 0
1 1 b n e 0
b b b b e b
n n b n e 0
e e e e e e
0 0 b 0 e 0
⊃WFDE 1 b n e 0
1 1 b 1 e 1
∨WFDE 1 b n e 0 b b b b e b
n n b n e 1
e e e e e e
1 1 b 1 e 1 0 0 b n e 0
b b b b e b
n 1 b n e n
e e e e e e
0 1 b n e 0
Weaker Paraconsistent logics are non-trivial Bounds for the set of LFI extensible with a transparent truth predicate
classical negation is not definable any of them; (!) Notice the following cases
D.E. Szmuc References
+ FDEw + , WFDE pwk + PWK , FDE+ , WFDE+
∼ 1 1
φ n (e) b
= × ×
¬φ n (e) b
∨ n (e) b
in each of these logics, the definable conditional doesn’t validate Modus Ponens irrestrictedly (!) Notice the following cases + PWK+ , FDEw + , WFDE pwk + PWK , FDE+ , WFDE+
φ b b
φ⊃ψ b b
2
ψ 0 n (e)
¬◦φ 1 1
Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc
Finally, this is how the ordering looks like for these non-classical logics. CL K3
LP FDE
References
Kw 3
PWK
FDEw
FDEpwk
WFDE
Comparing Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
We want, nevertheless, to evaluate these logics and to see how well they do compared to some other proposed logics of this sort, e.g. Carnielli & Rodrigues’ theory Verum. We will focus on some requirements that are somehow traditional nowadays, for theories of truth: Irrestricted validity of the T-Scheme Irrestricted validity of the T-Rules Problematic Instances of the Diagonal Lemma
T-Scheme and T-Rules Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
For which L do the following hold? T-Scheme ∀φ ∈ L, L φ ↔ Tr (pφq), i.e. ∀v , v (φ ↔ Tr (pφq)) ∈ DL T-Rules ∀φ ∈ L : φ `L Tr (pφq), Tr (pφq) `L φ, ¬φ `L ¬Tr (pφq) and ¬Tr (pφq) `L ¬φ Transparency ∀φ ∈ L : ∀v , v (φ) = v (Tr (pφq)) Remark: Transparency implies irrestricted iteration of Tr . The answer is the following
T-Scheme and T-Rules Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
For which L do the following hold? T-Scheme ∀φ ∈ L, L φ ↔ Tr (pφq), i.e. ∀v , v (φ ↔ Tr (pφq)) ∈ DL T-Rules ∀φ ∈ L : φ `L Tr (pφq), Tr (pφq) `L φ, ¬φ `L ¬Tr (pφq) and ¬Tr (pφq) `L ¬φ Transparency ∀φ ∈ L : ∀v , v (φ) = v (Tr (pφq)) Remark: Transparency implies irrestricted iteration of Tr . The answer is the following PWK+ FDEw + FDEpwk + WFDE+ Verum
T-Scheme X × × × X
T-Rules X X X X X
Transparency X X X X ?
Iteration of Tr X X X X ×
Problematic instances of Diagonal Lemma Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
Let, e.g. λ be the Liar sentence of the theory in question. Then the question is, for which L it the case that...? L λ ↔ ¬Tr (pλq), i.e. ∀v , v (λ ↔ ¬Tr (pλq)) ∈ DL The answer is the following
Problematic instances of Diagonal Lemma Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
Let, e.g. λ be the Liar sentence of the theory in question. Then the question is, for which L it the case that...? L λ ↔ ¬Tr (pλq), i.e. ∀v , v (λ ↔ ¬Tr (pλq)) ∈ DL The answer is the following
PWK+ FDEw + FDEpwk + WFDE+ Verum
Satisfaction of all instances of the Diagonal Lemma X × Xonly if ¬∃φ ∈ L, v (φ) = n × Xonly if ¬∃φ ∈ L, v (φ) = n × Xonly if ¬∃φ ∈ L, v (φ) = n(e) X
Comparing LFIs Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
We also want to evaluate this logics and to see how well they do with regard to axioms related to the notions of consistency, as they appear in the literature about LFIs. (!) This is not an attempt to axiomatize this theories, just brute testing, to see what comes out as valid and what doesn’t
Comparing LFIs Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
(bc0) (bc1) (bc2) (bc3) (bc4) (bc5) (cj1) (cj2) (cj3) (ca1) (ca2) (ca3) (co1) (co2) (co3)
◦α → (α → (¬α → ¬β)) ◦α → (α → (¬α → β)) ¬¬ ◦ α → ◦α ¬◦α→¬◦α ¬◦α→¬◦α ◦α → ¬¬ ◦ α ¬ ◦ (α ∧ β) ↔ ((¬ ◦ α ∧ β) ∨ (¬ ◦ β ∧ α)) ¬ ◦ (α ∨ β) ↔ ((¬ ◦ α ∧ ¬β) ∨ (¬ ◦ β ∧ ¬α)) ¬ ◦ (α → β) ↔ (α ∧ ¬ ◦ β) (◦α ∧ ◦β) → ◦(α ∧ β) (◦α ∧ ◦β) → ◦(α ∨ β) (◦α ∧ ◦β) → ◦(α → β) (◦α ∨ ◦β) → ◦(α ∧ β) (◦α ∨ ◦β) → ◦(α ∨ β) (◦α ∨ ◦β) → ◦(α → β)
Comparing LFIs Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
(cr1) (cr2) (cr3) (cv1) (cv2) (cv3) (ci) (cl) (cd) (cb) (cw) (cf) (ce)
(◦α ∧ ◦β) → ◦(α ∨ β) (◦α ∨ ◦β) → ◦(α ∨ β) (◦α → ◦β) → ◦(∨ → β) ◦(α ∧ β) ◦(α ∨ β) ◦(α → β) ¬ ◦ α → (α ∧ ¬α) ¬(α ∧ ¬α) → ◦α ¬(¬α ∧ α) → ◦α (¬(α ∧ ¬α) ∨ ¬(¬α ∧ α)) → ◦α ◦(¬α) ¬¬α → α α → ¬¬α
Which LFI should we choose? Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
bc0 bc1 bc2-bc5
PWK+ X X X
FDEw + × × X
FDEpwk + X X X
WFDE+ × × X
ci-cl-cd-cb ca1-c3 c01-c03 cr1-cr2 cv1-c3 cj1-cj3
X X × X × X
× X × X × ×
× X × X × ×
× X × X × ×
cw cf ce
× X X
× × ×
× × ×
× × ×
Which LFI should we choose? Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc
This is how the all-things-considered ordering looks like in the case of the extension of these paraconsistent logics with a truth predicate and a consistency operator. PWK+
References
FDEpwk +
WFDE+
FDEw +
Conclusions Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
We started to conduct an investigation on the minimal and maximal LFIs that can be interesting logics of formal inconsistency and truth This lead us to determine that paraconsistent logics usually employed to deal with paradoxes aren’t good options for this project We have found at least one reasonable logic for this purpose: PWK+ : paraconsistent weak Kleene logic augmented with Tr and ◦. Even if there exist non-trivial logics of formal inconsistency and truth weaker that PWK+ , they seem to be too weak. Moreover, PWK is better than them with regard to logical and consistency principles and axioms. Another option, incomparable to PWK+ is FDEw . However, this logic is not so good regarding consistency principles and axioms. Therefore, notwithstanding the fact that there PWK+ is not the minimal logic of formal inconsistency and truth, it seems to be one of the best options so far.
Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
Obrigado! Gracias!
References Bounds for the set of LFI extensible with a transparent truth predicate D.E. Szmuc References
[1] W. Carnielli and A. Rodrigues. Formal Plenitude and Curry s Paradox: Sketch of a Non-Hierarchical Theory for Arithmetical Truth. CLPS13: Congress on Logic and Philosophy of Science, 2013. [2] A. Gupta and N. Belnap. The Revision Theory of Truth. MIT Press, Cambridge, Mass, 1993. [3] A. Gupta and R. Martin. A Fixed Point Theorem for the Weak Kleene Valuation Scheme. Journal of Philosophical Logic, 13:131–135, 1984. [4] H. Omori and R. Ciuni. Consistency operator and ‘just-true’ operator in paraconsistent Weak Kleene logic. 5th WCP, 2014.
