Abstract
We introduce a general notion of semi-implication which generalizes both the implications used in classical, intuitionistic, and many other logics, as well as those used in relevance logics. It is mainly based on the relevant deduction property (RDP)—a weak form of the classical-intuitionistic deduction theorem which has motivated the design of the intensional fragments of the relevance logic R. However, CL ↔, the pure equivalential fragment of classical logic, also enjoys the RDP with respect to ↔. We show that in the language of → this is the only exception. This observation leads to an adequate definition of semi-implication, according to which a finitary logic L has a semi-implication → iff L has a strongly sound and complete Hilbert-type system which is an extension by axiom schemas of HR → (the standard Hilbert-type system for the implicational fragment of R). We also show that in the presence of a conjunction, or a disjunction, or an implication, a connective → of a logic L is a semi-implication iff it is an implication (i.e. it satisfies the classical-intuitionistic deduction theorem), and the same is true if L is induced by a matrix which has a single designated value or a single non-designated value.
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Notes
- 1.
Here ψ 1∧⋯∧ψ n might be taken to stand, e.g. for (⋯((ψ 1∧ψ 2)∧ψ 3)⋯)∧ψ n .
- 2.
A connective → of a propositional logic L has the variable sharing property (VSP) if Atoms(φ)∩Atoms(ψ)≠∅ whenever ⊢ L φ→ψ.
- 3.
This theorem has already been proved in [7]. We reproduce here the proof in order to make this paper self-contained.
- 4.
For the extensions of HR → this theorem was first proved in [8].
- 5.
This is particularly easy using Leśniewski famous criterion (see, e.g. Corollary 7.31.7 in [12]), according to which a formula built up using ↔ as the sole connective is a classical tautology iff every propositional atom occurs in the formula an even number of times.
- 6.
- 7.
This is a slight generalization of a theorem of Prior, who showed in [14] (p. 307) that CL ↔ is Post-complete in the sense that one cannot add any new axiom to it in its language.
- 8.
Theorem 6.4 is about finitary logics, but its proof shows that being an extension of R → is a property of every logic with a semi-implication. It is also worth noting that by a theorem of Shoesmith and Smiley (see [15, 16]), \(\mathbf{L}_{ \mathcal {M}}\) is finitary whenever \(\mathcal {M}\) is finite.
- 9.
This is essentially proved in [9], which strengthened Meyer’s result about the weak soundness and completeness of RM for the original ℵ0-Sugihara matrix.
- 10.
This means that a sentence φ is valid in Sobociński’s matrix iff \(\vdash _{\mathbf{RM}_{\stackrel {\neg }{\rightarrow }}}\varphi \).
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Avron, A. (2015). Semi-implication: A Chapter in Universal Logic. In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-10193-4_3
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