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Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

A Paraconsistent Sub-Logic of Intuitionistic Propositional Logic

Sankha S. Basu

Department of Mathematics, Indraprastha Institute of Information Technology, New Delhi, India

8th Indian Conference on Logic and its Applications (ICLA), 2019 Indian Institute of Technology Delhi, India March 05, 2019

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Scheme

1 Paraconsistency 2 Paraconsistent intuitionistic logic 3 Axiomatization 4 Soundness and Completeness 5 Conclusion

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Scheme

1 Paraconsistency 2 Paraconsistent intuitionistic logic 3 Axiomatization 4 Soundness and Completeness 5 Conclusion

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Paraconsistency

Definition A theory (i.e., a set of sentences closed under some deductive relation) is said to be (negation) consistent if for no sentence α is both α and ¬α provable, else inconsistent. A theory is said to be non-trivial if not every formula is provable, else trivial. Definition A logic is said to be paraconsistent or inconsistency tolerant if it admits inconsistent but non-trivial theories. In other words, a paraconsistent logic is a logic where it is not always possible to derive everything from a contradiction.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Paraconsistency

Classical logic, and also many non-classical logics, such as intuitionistic logic, fail in this because of the so-called principle of ‘explosion’ by which, for any sentence α, {α, ¬α} ⊢ β (ECQ : Ex contradictione quodlibet) Thus a necessary condition for a logic to be paraconsistent is that its consequence relation be not explosive, thus invalidating ECQ.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Scheme

1 Paraconsistency 2 Paraconsistent intuitionistic logic 3 Axiomatization 4 Soundness and Completeness 5 Conclusion

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Stage Setting

Definition Given a similarity type ν, the absolutely free algebra Fm of type ν

• ver a countably infinite set X of generators is called the formula

algebra of type ν; its underlying set will be denoted by Fm. The elements of Fm are called ν-terms or ν-formulas and referred to by the symbols t, s, . . . or α, β, ϕ, . . .. Members of X are called (propositional) variables and denoted by the symbols x, y, . . . or p, q, . . .. Definition A logic of type ν is a pair L = Fm, ⊢L, where Fm is the formula algebra of type ν, and ⊢L is the substitution-invariant consequence relation over Fm.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Paraconsistent Intuitionistic Logic (PIL)

Paraconsistent intuitionistic logic PIL = Fm, | =# can be semantically specified as follows. Fm is the formula algebra of type (2, 2, 2, 1, 0, 0), namely, of the type containing the connectives ∧, ∨, − →, ¬ , 0, 1. For any Heyting algebra A, we define its extension A# = A ∪ {ω}, where ω / ∈ A for all Heyting algebras A. Let S = {A# | A is a Heyting algebra}. The additional element ω satisfies the so-called contamination priniciple, that is, ¬ ω = ω and if A is any Heyting algebra, then for any a ∈ A, a ◦ ω = ω, where ◦ denotes any of the binary connectives ∧, ∨, − →. Finally, for any Σ ∪ {α} ⊆ Fm, we define Σ | =# α ⇐ ⇒ for every A# ∈ S and for every valuation v# : Fm → A#, v#[Σ] ⊆ {1, ω} implies v#(α) ∈ {1, ω}.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Relating | =# and | =IL

Theorem For all Σ ∪ {α} ⊆ Fm, we have that Σ | =# α if and only if there is a ∆ ⊆ Σ such that var(∆) ⊆ var(α) and ∆ | =IL α. Moreover, since IL (intuitionistic propositional logic) is finitary, we can find such a ∆ ⊆ Σ that is finite. Proof Outline Suppose Σ | =# α. Let ∆ = {ϕ ∈ Σ | var(ϕ) ⊆ var(α)}. Suppose ∆ | =IL α. So there exists a Heyting algebra A and a valuation v : Fm → A such that v[∆] = {1} but v(α) = 1. We construct the valuation v# : Fm → A# as follows. v#(p) = v(p) if p ∈ var(α) ω if p / ∈ var(α) For any ϕ ∈ Σ, either var(ϕ) ⊆ var(α) or var(ϕ) ∩ (var(Σ) \ var(α)) = ∅. In the first case, v#(ϕ) = 1, and in the second case, v#(ϕ) = ω. Thus v#[Σ] ⊆ {1, ω}. Then v#(α) ∈ {1, ω}. Now, v#(α) = v(α) ∈ A and v(α) = 1. So v#(α) / ∈ {1, ω}. This is a contradiction, which proves that ∆ | =IL α.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Relating | =# and | =IL

Proof Outline Conversely, suppose ∆ | =IL α for some ∆ ⊆ Σ with var(∆) ⊆ var(α). Suppose v# : Fm → A# for some A# ∈ S be a valuation such that v#[Σ] ⊆ {1, ω}. The possible cases are v#(p) = ω for some p ∈ var(α) or v#(p) = ω for all p ∈ var(α). In the first case, v#(α) = ω. In the second case, let A be the Heyting algebra corresponding to A# ∈ S and a0 ∈ A (fixed). We construct a valuation v : Fm → A as follows. v(p) =

• v#(p)

if p ∈ var(α) a0

• therwise

Then, since var(∆) ⊆ var(α), v[∆] = v#[∆] ⊆ v#[Σ] ⊆ {1, ω}. So v[∆] ⊆ {1, ω} ∩ A = {1}. Thus v#(α) = v(α) = 1 ∈ {1, ω}. This proves that Σ | =# α.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Scheme

1 Paraconsistency 2 Paraconsistent intuitionistic logic 3 Axiomatization 4 Soundness and Completeness 5 Conclusion

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

An axiomatization of PIL

Definition (HPIL) HPIL is the logic Fm, ⊢#, where ⊢# is the consequence relation of the deductive system with the following axioms and inference rule.

1 α −

→ (β − → α);

2 (α −

→ (β − → γ)) − → ((α − → β) − → (α − → γ));

3 α −

→ (β − → α ∧ β);

4 α ∧ β −

→ α;

5 α ∧ β −

→ β;

6 α −

→ α ∨ β;

7 β −

→ α ∨ β;

8 (α −

→ γ) − → ((β − → γ) − → (α ∨ β − → γ));

9 (α −

→ β) − → ((α − → ¬ β) − → ¬ α);

10 0 −

→ α; [RMP] α, α − → β β provided that var(α) ⊆ var(β).

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Connections with IL

Remark Axioms (1)–(10) along with unrestricted modus ponens gives a Hilbert system for intuitionistic propositional logic (IL). It is easy to check that PIL fails to satisfy modus ponens (MP). However, it satisfies RMP as we will see later. Because of the restricted nature of the inference relation, HPIL is weaker than IL. However, the following result shows that both logics have the same theorems. Theorem For any ϕ ∈ Fm, ⊢# ϕ if and only if ⊢IL ϕ.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Connections with IL

Proof Outline (= ⇒) Immediate, since the axioms (1)–(10) are axioms of IL and RMP is also an instance of MP. (⇐ =) Suppose ⊢IL ϕ and D = ϕ1, . . . , ϕn is a proof of ϕ in IL. We use induction on the length n of D. In the inductive step, suppose ϕ = ϕl is obtained by MP from ϕi and ϕj = ϕi − → ϕ, where i, j < l. By the induction hypothesis, there are proofs of ϕi, ϕj in HPIL. Let ψ1, . . . , ψm be the result of gluing these HPIL proofs so that ψk = ϕi(k < m) and ψm = ϕj = ϕi − → ϕ.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Connections with IL

Proof Outline We use a substitution of variables: σ(p) = p if p ∈ var(ϕ) a

• therwise

, where a is some fixed variable in ϕ or 0 if var(ϕ) = ∅. Note that σ(ψ1), . . . , σ(ψm) is still a proof in HPIL. Now, σ(ψm) = σ(ψk − → ϕ) = σ(ψk) − → σ(ϕ) = σ(ψk) − → ϕ. Moreover, var(σ(ψk)) ⊆ var(ϕ), by definition of σ. Thus ϕ follows from σ(ψk) and σ(ψm) by an application of RMP, and σ(ψ1), . . . , σ(ψk), . . . , σ(ψm), ϕ is a proof of ϕ in HPIL.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Restricted Deduction Theorem

Theorem (Restricted Deduction Theorem) For any Σ ∪ {α, β} ⊆ Fm, we have the following. (i) Σ ∪ {α} ⊢# β implies Σ ⊢# α − → β. In particular, if α ⊢# β then ⊢# α − → β. (ii) If Σ ⊢# α − → β and var(α) ⊆ var(β), then Σ ∪ {α} ⊢# β. In particular, if ⊢# α − → β and var(α) ⊆ var(β), then α ⊢# β.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

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1 Paraconsistency 2 Paraconsistent intuitionistic logic 3 Axiomatization 4 Soundness and Completeness 5 Conclusion

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Soundness

Lemma For any ϕ ∈ Fm, if ⊢# ϕ, then | =# ϕ. Proof Outline Let A# ∈ S and v# : Fm → A# be any valuation. If for some variable p ∈ var(ϕ), v#(p) = ω, then v#(ϕ) = ω, and so | =# ϕ. If on the other hand, for all p ∈ var(ϕ), v#(p) = ω, then v#(p) ∈ A, the Heyting algebra corresponding to A#, for all p ∈ var(ϕ). This implies that v#(ϕ) ∈ A. We construct the valuation v : Fm → A as follows. v(p) =

• v#(p)

if p ∈ var(ϕ) a0

• therwise,

, where a0 ∈ A (fixed) Then v(ϕ) = v#(ϕ). Now, since ⊢# ϕ, ⊢IL ϕ which implies that v#(ϕ) = v(ϕ) = 1. Hence | =# ϕ.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Soundness

Lemma Suppose ϕ, ψ ∈ Fm with var(ϕ) ⊆ var(ψ). Then for any A# ∈ S, and any valuation v# : Fm → A#, if v#[{ϕ, ϕ − → ψ}] ⊆ {1, ω}, then v#(ψ) ∈ {1, ω}. That is, RMP preserves validity. Proof Outline Suppose not. Then v#[{ϕ, ϕ − → ψ}] ⊆ {1, ω}, but v#(ψ) / ∈ {1, ω}. So v#(p) = ω for all p ∈ var(ψ). Then v#(ψ) ∈ A, the Heyting algebra corresponding to A#. Let v#(ψ) = a ∈ A, where a < 1. Since var(ϕ) ⊆ var(ψ), v#(p) = ω for all p ∈ var(ϕ), and hence v#(ϕ), v#(ϕ − → ψ) = 1. Then v#(ϕ − → ψ) = 1 − → a. For any a ∈ A, 1 − → a = a, v#(ϕ − → ψ) = a < 1. This is a contradiction.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Soundness

Theorem (Soundness) For all Σ ∪ {α} ⊆ Fm, if Σ ⊢# α, then Σ | =# α. Proof Straightforward from the two lemmas.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Completeness

Theorem For any ϕ ∈ Fm, if | =# ϕ then ⊢# ϕ. Proof. Suppose | =# ϕ. Then | =IL ϕ. We know that IL is complete with respect to valuations in Heyting algebras. So ⊢IL ϕ. Hence ⊢# ϕ.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Completeness

Theorem (Completeness) For all Σ ∪ {α} ⊆ Fm, if Σ | =# α, then Σ ⊢# α. Proof. Suppose Σ | =# α and Σ = ∅. Then there exists a finite ∆ ⊆ Σ with var(∆) ⊆ var(α) such that ∆ | =IL α. Now, by the completeness of IL with respect to valuations in Heyting algebras, ∆ ⊢IL α. Let ∆ = {ϕ1, . . . , ϕn}. Then by applying the deduction theorem in IL n times, we have ⊢IL (ϕ1 − → (. . . − → (ϕn − → α) . . .)). Then ⊢# (ϕ1 − → (. . . − → (ϕn − → α) . . .)). Now, since var(∆) ⊆ var(α), by applying the converse of the Deduction theorem n times, we have ∆ ⊢# α. Finally, since ∆ ⊆ Σ, we have Σ ⊢# α.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

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1 Paraconsistency 2 Paraconsistent intuitionistic logic 3 Axiomatization 4 Soundness and Completeness 5 Conclusion

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Concluding Remarks

ECQ fails in HPIL. To see this, consider α = p and β = q where p, q are variables, and a valuation v# : Fm → A#, where A# ∈ S, such that v#(p) = ω and v#(q) = 0. Then {α, ¬α} | =# β. Then, by the Soundness Theorem, {α, ¬α} ⊢# β. Thus HPIL is paraconsistent. Interestingly, however, 0 − → α is an axiom of HPIL. For any α ∈ Fm, ¬(α ∧ ¬α) is a theorem of IL and hence of

• HPIL. Thus the law of non-contradiction holds in HPIL.

However, it is noteworthy that the sentence α ∧ ¬α is not always false either. To see this, take α = p, a variable and a valuation v# such that v#(p) = ω.

Paraconsistency Paraconsistent intuitionistic logic Axiomatization Soundness and Completeness Conclusion

Thank You