Abstract
Priest argued in Fusion and Confusion (Priest in Topoi 34(1):55–61, 2015a) for a new concept of logical consequence over the relevant logic B, one where premises my be “confused” together. This paper develops Priest’s idea. Whereas Priest uses a substructural proof calculus, this paper provides a Hilbert proof calculus for it. Using this it is shown that Priest’s consequence relation is weaker than the standard Hilbert consequence relation for B, but strictly stronger than Anderson and Belnap’s original relevant notion of consequence. Unlike the latter, however, Priest’s consequence relation does not satisfy a variant of the variable sharing property. This paper shows that how it can be modified so as to do so. Priest’s consequence relation turns out to be surprisingly weak in some respects. The prospects of strengthening it is raised and discussed in a broader philosophical context.
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Notes
Anderson and Belnap used ‘&’ for extensional conjunction; I’ll use ‘\(\wedge \)’.
These latter two terms are used interchangably in Priest (2015a).
See Øgaard (2021c) for a discussion on how to extend the variable sharing property to formulas with truth-constants.
The same holds for any logic for which the C-entailment theorem applies to. This is a very large class; see (Øgaard (2021a, § 3) for details.
See Øgaard (2021b) for more on this result.
Since this paper is primarily on Priest’s new notion of logical consequence, I will follow him and write sequent where Restall uses consecution. Furthermore, Restall uses ‘\(\vdash \)’ as the sequent symbol, whereas Priest uses ‘\(\rhd \)’.
Again, I follow Priest’s notation; Restall uses ‘0’ to stand for \({\mathbf {t}}\), the comma to stand for extensional conjunction and the semicolon to stand for intensional conjunction.
Note that every structural rule is assumed for \(\oplus \), and so we may ignore order in structure such as \(A_1 \oplus \ldots \oplus A_n\). There is another detail, however, that needs to be commented upon a bit more carefully, namely the behaviour of the Church constant \(\bot \). \(\bot \) is intuitively the conjunction of every proposition. In Routley-Meyer semantics, \(\bot \) is demanded to fail at every evaluation point. A consequence of this is that \(C \rightarrow (\bot \rightarrow \bot )\) is valid in the semantics. This is a theorem of D W—B with (R5) replaced by its axiomatic version—but not of B as here defined. In other semantics, such as the algebraic semantics of Meyer and Routley (1972), however, \(C \rightarrow (\bot \rightarrow \bot )\) can be made to fail. Since also Priest (2015a)’s rule for \(\bot \) only yields (A8), I’ve chosen to stick to the more common way of adding \(\bot \) in the case of B. Note, then, that Restall’s rule for \(\bot \),
, does yield \(C \rightarrow (\bot \rightarrow \bot )\). However, by replacing the rule with 
—or equivalently as the axiom \(\bot \rhd A\) which is Priest (2015a)’s rule for \(\bot \)—will allow Restall’s soundness and completeness result to hold true with regards to how B is defined here. For more on this, see Øgaard (2021b).Note that adding \(C \rightarrow (\bot \rightarrow \bot )\) is not without its consequences: it was shown in Øgaard (2016, Theorem 10) that if added to \({{\mathbf {B}}}{{\mathbf {X}}}^d\)—B as here defined where it includes both \(\circ \) and \({\mathbf {t}}\), but with excluded middle and the meta-rule of reasoning by cases added—then the naïve theory of truth is trivialized. Whether or not it trivializes without the added \(\bot \) axiom is currently unknown.
See Restall (2000, Lemma 4.17) for a proof to the effect that there is no loss in generality involved in thus restricting our attention to formulas.
The notion of a confusion was, to my knowledge, first introduced in Restall (2000) (def. 4.26). Restall allows \(\top \) to be a member of \({\mathcal {C}}(\Sigma )\), but since this is inconsequential and Priest does not, I’ve chosen to go with the stricter notion.
\(\mathfrak{B} (\Sigma)\) is the set of bunches corresponding to the set of confusions over the set of formulas \(\Sigma \).
Again I will have to refer the reader to Restall’s book for details.
This follows using Restall’s deduction theorem for RW (Restall 2000, p. 87).
See Humberstone (2010) for a nice discussion of Smiley’s distinction.
For instance, let \(B_i\) be a logical axiom, \(\Gamma = \{B_i\}\), and \(\Delta = \varnothing \).
I would like to thank the reviewer for insisting that a proof beyond a general remark on cutting and pasting proofs was needed and for in essense suggesting the proof given here.
The model was found with the help of MaGIC—an acronym for Matrix Generator for Implication Connectives—which is an open source computer program created by John K. Slaney (Slaney (1995)).
See Priest (2008, ch. 9) for a tableaux system and semantics for \({\mathbf {N}}_4\).
To see why, note that both \(A \wedge B \leftrightarrow B \wedge A\) and \((A \wedge B \rightarrow A) \rightarrow (A \wedge B \rightarrow A)\) are logical theorems of \({\mathbf {N}}_4\), but that \((A \wedge B \rightarrow A) \rightarrow (B \wedge A \rightarrow A)\) is not since \(\rightarrow \) formulas can be given arbitrary truth-values at “non-normal worlds”.
The result is due to Dunn and Slaney and can be found in Routley et al. (1982, pp. 366f).
The proof is almost identical to the mentioned proof by Dunn and Slaney, but uses the Curry-sentence \(\lambda \leftrightarrow (T\langle \lambda \rangle \circ (T\langle \lambda \rangle \wedge {\mathbf {t}}) \rightarrow \bot )\) instead of \(\lambda \leftrightarrow (T\langle \lambda \rangle \circ T\langle \lambda \rangle \rightarrow \bot )\) .
I should emphasize that Priest (2015a) only considers the positive fragment of B, and so the status of excluded middle is not touched in the paper currently under consideration.
B with (R5) replaced by \((A \rightarrow\, \thicksim\! B) \rightarrow (B \rightarrow \,\thicksim\!A)\) is called DW.
Brady showed in Brady (1989) that naïve set theory, and therefore also naïve truth theory, is non-trivial in a certain logic extending the \(\circ \)- and \({\mathbf {t}}\)-free fragment of DW augmented by both ConSyll and excluded middle. That the construction also allows for the Ackermann constant was to my knowledge first noted in Beall (2009, pp. 121ff). That it can be strengthened to the mentioned axiomatic version is first noted here. So is the fact that Brady’s construction validates the t-enthymematic versions of the pre- and suffixing axiom. The proofs are rather straight-forward, and so I leave them to the interested reader. Brady’s construction in Brady (2006, § 6.3) on the other hand, validates even the pre- and suffixing axioms, although does not allow for the Ackermann constant, not even in rule form.
B is a sublogic of infinitely-valued Łukasiewics logic and so the naïve theory of truth is non-trivial over B. See Hájek et al. (2000) for details on this. Note, however, that the theory has no \(\omega \)-model over this logic. Whether or not this extends to B is currently unknown.
I’ll come back to this fact in the next section.
That this is so is easily verified using Slaney’s MaGIC—use B augmented with the axiom \(A \leftrightarrow A \circ A\). I leave the details as an exercise for the reader.
I should emphasize, however, that whether or not naïve set theory is trivial in B\(^{\circ }\) is as of yet unknown. Note that Priest (2015a) is primarily a discussion of the theory of naïve validity which Priest discusses in the context of the naïve theory of truth. Note, then, that these theories are regarded as extending the logic, not as axiomatic theories. For instance, Priest notion of naïve truth theory is that it augments the proof system by in effect adding as logically true, every sequent \(A \rhd T\langle A \rangle \) and \(T\langle A \rangle \rhd A\). In terms of CF-entailment, this translates to allowing \(A \leftrightarrow T\langle A \rangle \) as well as any instance of the self-reference schema as a logical axiom, and so every rule may be applied to it unrestrictedly. Note, then, that the mentioned triviality proof for naïve set theory over B relies on an unrestricted rule of extensionality—that is a primitive rule \(\forall x(x \in a \leftrightarrow x \in b) \Vdash \forall y (a \in y \leftrightarrow b \in y)\) on par with, say modus ponens—as well as the abstraction schema on par with the logical axioms. The requirements of the triviality proof, then, are on line with Priest’s view of the naïve theories of truth and validity. See Øgaard (2021b) for more on ways of augmenting Restall’s consecution calculus.
, does yield 
—or equivalently as the axiom References
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I would very much like to thank the referee as well as my colleagues in the Bergen Logic Group for comments and suggestions which helped to improve the paper significantly.
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Øgaard, T.F. Confused Entailment. Topoi 41, 207–219 (2022). https://doi.org/10.1007/s11245-021-09758-x
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DOI: https://doi.org/10.1007/s11245-021-09758-x