Abstract
In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic (ETL), an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof.
Highlights
In Belnap and Dunn’s well-known four-valued semantics for the logic of first-degree entailment FDE (Belnap [3, 4], Dunn [5]) the classical principles of Bivalence and Noncontradiction are given up
A more compact way to characterise the semantics of conjunction and disjunction in the Belnap-Dunn logic is to say that they correspond to meet and join in the following lattice, called L4 in [3, 4]
Since it is easy to see that iff {1 : γ | γ ∈ } ∪ {1 : φ} is irrefutable, it follows from Theorem 1 that the syntactic and semantic entailment relations of FDE
Summary
In Belnap and Dunn’s well-known four-valued semantics for the logic of first-degree entailment FDE (Belnap [3, 4], Dunn [5]) the classical principles of Bivalence (every sentence is true or false) and Noncontradiction (no sentence is both true and false) are given up. Regarding this somewhat unusual feature, which they call anti-primeness, P&R make the following remark This will not make it easy to find a nice sequent calculus for this logic. The following is an example of a sequent proof obtained using the PLt4 calculus. The following definition makes the informal interpretation of the four signs given above explicit by connecting sequents and the valuations refuting them. Inspection of the rules shows that in this case follows from a sequent 1 or from a pair of sequents 1 and 2, each containing fewer than n connectives One of these top sequents must be unprovable and by induction, refuted by some valuation V. Since it is easy to see that iff {1 : γ | γ ∈ } ∪ {1 : φ} is irrefutable, it follows from Theorem 1 that the syntactic and semantic entailment relations of FDE correspond. We like to argue, that is solely responsible for the difference in behaviour.
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