On the logic of pragmatic truth
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From these results, the logic LPT is a positive logic, given by the Axioms (A1) - (A8) plus the rule MP, adding the Axioms (A9)-(A16). The Axiom (A12) is equivalent to the formula (ϕ ∧ ¬ϕ ∧ ◦ϕ) → ψ. This emerges that if {ϕ, ¬ϕ, ◦ϕ} ⊆ Γ , then for every formula ψ, we have that Γ ` ψ. The Axiom (A13) is also essential for the paraconsistent character of the LPT, because this axiom indicates that a formula and its negation can occur in certain situations. Proposition 1.4. The Axiom (A16): (ϕ ∧ ¬ϕ ∧ ψ) → ¬(ϕ ∧ ψ) ∧ ¬(ψ ∧ ϕ) can be proved from the others. Proof. 1. 2. 3. 4. 5. 6. 7. 8. 9.
ϕ ∧ ¬ϕ ∧ ψ (ϕ ∧ ¬ϕ ∧ ψ) → ¬ϕ ¬ϕ ¬ϕ → (¬ϕ ∨ ¬ψ) (¬ϕ ∨ ¬ψ) → ¬(ϕ ∧ ψ) (¬ϕ ∨ ¬ψ) → (¬ψ ∨ ¬ϕ) (¬ψ ∨ ¬ϕ) → ¬(ψ ∧ ϕ) ¬ϕ → (¬(ϕ ∧ ψ) ∧ (¬(ψ ∧ ϕ)) A16
hypothesis A4 MP in 1, 2 A6 De Morgan Commutativity De Morgan LPT 4-7 LPT in 2 and 8
Thus, we suggest to change this axiom and include this one: (A16*) ◦ϕ → ◦¬ϕ. It is a immediate consequence from (A16*) that ` ◦ϕ ↔ ◦¬ϕ. The last three axioms just indicate that the operators ¬, ∧ and → preserve the consistency. In Coniglio and Silvestrini (2014), there is a proof in which the deductive system above is sound and complete according to the matricial semantics MLP T , besides it uses the concept of bivaluations. In the following sections, we give a proof straight from the matricial model MLP T , motivated by Epstein (1990).
2. Soundness For the soudness, we need to show that every Theorem of LPT is valid according to the model MLP T . This is known as weak soundness. We shall establish