TY - JOUR
AU1 - Standefer,, Shawn
AB - Abstract In relevant logics, necessary truths need not imply each other. In justification logic, necessary truths need not all be justified by the same reason. There is an affinity to these two approaches that suggests their pairing will provide good logics for tracking reasons in a fine-grained way. In this paper, I will show how to extend relevant logics with some of the basic operators of justification logic in order to track justifications or reasons. I will define and study three kinds of frames for these logics. For the first kind of frame, I show soundness and highlight a difficulty in proving completeness. This motivates two alternative kinds of frames, with respect to which completeness results are obtained. Axioms to strengthen the justification logic portions of these logics are considered. I close by developing an analogy between the dot operator of justification logic and theory fusion in relevant logics. Two and two is four, and my cat is on the porch. That my cat is on the porch does not imply that two and two is four. One of the motivating ideas for relevant logics, supplied by Anderson and Belnap [1, 18], is that the necessary truth, or even logical truth, of a proposition does not suffice for that proposition to be implied by another proposition. The relevant logician would, similarly, not want to treat a reason for one necessary truth as a reason for all necessary truths. After all, a proof that two and two is four provides a good reason to believe that arithmetic fact, whereas seeing my cat on the porch does not. Justification logic provides ways to represent and manipulate justifications and reasons that fit naturally with motivations for relevant implications. In the remainder of this paper, I will develop the pairing of relevant logic with some logical machinery from justification logic. Relevant logics are non-classical logics that require a tight connection between the antecedent and consequent of their conditionals.1 They have an epistemic interpretation in terms of information.2 They are also discerning, in the sense that not all theorems are equivalent. Justification logics are modal logics for fine-grained tracking of justifications and reasons. They originate with the work of Artemov [3, 4], who introduced the logic of proofs as an explicit counterpart to provability logic, replacing the single modality |$\Box$| for ‘there is a proof that’ with multiple modalities |$[\![ t ]\!]$| for ‘|$t$| is a proof that’. This interpretation was developed into a family of justification logics by Artemov, Fitting and others.3 It is a natural thought that relevant and justification logics should go together. A key intuition behind justification logic, namely that justifications are highly discriminating, is in harmony with many of the intuitions behind relevant logic, e.g. that necessary truth does not suffice to make a claim relevant to all other claims. However, they have not yet been brought into contact.4 This paper will attempt to do so, albeit with a focus on the aspects of interest to the relevant logician. Justification logic is a rich topic, but for the purposes of this paper, I will focus on a limited portion of it, namely the dot operator on justification terms, which extracts information from its arguments to yield a new justification.5 The characteristic axiom of justification logic, featuring this operator, is \begin{equation*} {\textsf{K}}\qquad [\![ t ]\!](A\mathbin{\rightarrow} B)\mathbin{\rightarrow} ([\![ s ]\!]\ A\mathbin{\rightarrow} [\![ t\cdot s ]\!]\ B). \end{equation*} This axiom captures the way in which evidence composition works in justification logic. To the relevant logician, it may be reminiscent of two things from models for relevant logics. One is Urquhart’s operational models for relevant logics, in particular the clause for the implication.6 \begin{equation*} x\Vdash A\mathbin{\rightarrow} B \;\; \textrm{iff} \;\; \forall y(\,y\Vdash A\mathbin{\Rightarrow}x\cdot y\Vdash B) \end{equation*} The other is Fine’s operation of theory composition, \begin{equation*} t\circ s=\{ B: \exists A((A\mathbin{\rightarrow} B)\in t \,\&\, A\in s)\}, \end{equation*} where |$t$| and |$s$| are theories, sets of formulas closed under conjunction introduction and provable implication.7 In both of the latter two cases, the composition operation is intimately connected to the logical behaviour of the relevant conditional. Part of the purpose of this paper is to see whether that idea is correct and to what extent it can be made so. While the naive idea that the justification dot operator is closely connected to theory fusion does not turn out not to be correct, there is much for the relevant logician in the area of justification logic. One can view the justification logics, at least the limited fragment of the justification-logical machinery used here, as tracking reasons in a fine-grained way that extends what one can do with the basic vocabulary of relevant logic. The independence of the justification dot from theory fusion means that one can adjust properties of the former without altering the base logic. By contrast, one might try to define a justification-like operator in the basic vocabulary of relevant logic, with ‘|$[\![ \,p ]\!] q\mathbin{=_{Df}} p\mathbin{\rightarrow } q$|’, saying that |$p$| is a reason for |$q$|⁠.8 The analog of K, taking the dot to be either conjunction or (object language) fusion, would be invalid in many of the usual relevant logics considered. Further, extensions with some axioms, such as the modal T axiom, would be undesirable.9 It seems better, then, to separate out the reason-tracking aspect from the base logic.10 There is some prior work in relevant modal and epistemic logic that I will build upon. Relevant modal logics, modal logics whose base logic is a relevant logic, have not been studied as much as classical modal logics.11 Routley and Meyer [51] presented a model-theoretic study of an S4-ish extension of R, and this logic and its extensions were further studied by [43], [45] and [36, 37]. Fuhrmann [26, 27] studied frames for monotone, conjunctively regular relevant modal logics, among others, and [38] and [41] studied relevant modal logics in connection with conditional logics. Modals for substructural logics more generally have been studied by [39], [18], [50] and [55–61]. Wansing [65] studied relevant epistemic logic in the context of the knowability paradox.12 Bilková et al. [11] and Bílková et al. [12] studied relevant epistemic logics and Sedlár [53, 54] studied substructural epistemic logics, including ones based on relevant logics. Punčochář and Sedlár [49] present relevant logics for pooling information.13 This paper will proceed as follows. In §1, I will present the axiomatics (§1.1) and models (§1.2) for the relevant logics to be considered. I will then introduce evidence models (§2) for justification logics. A difficult proving completeness via the canonical model method is elaborated into a proof that the canonical frames for certain logics from this section do not satisfy the required frame conditions that is proved in an appendix. In §3, I will motivate an alternate kind of model, called justification models, and explore some of their features. The justification logics arising from these models are still fairly weak, so I present some natural ways of strengthening them (§4). In §5, I motivate a third kind of model, neighbourhood models, as a satisfying alternative to justification models. Finally, I will close with some discussion of the similarities between the term-forming dot of justification logic and theory fusion of relevant logics (§6). 1 Background Frame methods for modal logics on Routley–Meyer ternary relational frames, or RM-frames, add to the frames a binary accessibility relation, |$S$|⁠.14 In this section, I will define the basic concepts from RM-frames used throughout the paper. I will use the language |$\mathscr{L}$| defined as follows. First, there is a set of justification terms built from the following grammar, with a countable set of variables, |$x_1,x_2, y_1, \ldots ,$| and a binary term-forming operator:15 \begin{equation*} t::= x \;|\; (t\cdot t). \end{equation*} The set of terms will be denoted by Terms. Next, the formulas are built up as follows, where At is a countable set of atoms and |$p\in{\textsf{At}}$|⁠:16 \begin{equation*} A::= p \;|\; \mathord{\sim} A \;|\; (A\,\&\, A) \;|\; (A\lor A) \;|\; (A\mathbin{\rightarrow} A) \;|\; [\![ t ]\!]\ A. \end{equation*} I will take |$A\mathbin{\leftrightarrow } B$| to be defined as |$(A\mathbin{\rightarrow } B)\,\&\,(B\mathbin{\rightarrow } A)$|⁠. To cut down on parentheses, unary connectives will bind tightest, followed by conjunction and disjunction, followed by the conditional and the defined biconditional. The justification modals will drop out until the end of this section while the background for the base relevant logics is put into place. 1.1 Axiomatics All the logics under consideration in this paper will be in the framework FMLA, namely as sets of formulas.17 All the relevant logics considered will contain the following axioms and rules, where ‘|$\mathbin{\Rightarrow }$|’ in the rules is used to mark off the premises from the conclusion in the rules: A1 |$A\mathbin{\rightarrow } A$| A2 |$A\,\&\, B\mathbin{\rightarrow } A$| A3 |$A\,\&\, B\mathbin{\rightarrow } B$| A4 |$(A\mathbin{\rightarrow } B)\,\&\,(A\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\mathbin{\rightarrow } B\,\&\, C)$| A5 |$A\mathbin{\rightarrow } A\lor B$| A6 |$B\mathbin{\rightarrow } A\lor B$| A7 |$(A\mathbin{\rightarrow } C)\,\&\,(B\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\lor B\mathbin{\rightarrow } C)$| A8 |$A\,\&\,(B\lor C)\mathbin{\rightarrow } (A\,\&\, B)\lor (A\,\&\, C)$| A9 |$\mathord{\sim }\mathord{\sim } A\mathbin{\rightarrow } A$| R1 |$A, A\mathbin{\rightarrow } B \mathbin{\Rightarrow } B$| R2 |$A, B\mathbin{\Rightarrow } A\,\&\, B$| R3 |$A\mathbin{\rightarrow } \mathord{\sim } B\mathbin{\Rightarrow } B\mathbin{\rightarrow } \mathord{\sim } A$| R4 |$A\mathbin{\rightarrow } B, C\mathbin{\rightarrow } D\mathbin{\Rightarrow } (B\mathbin{\rightarrow } C)\mathbin{\rightarrow }(A\mathbin{\rightarrow } D)$|⁠. The logics will differ in which of the following axioms are adopted: C1 |$(A\mathbin{\rightarrow } B)\,\&\,(B\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\mathbin{\rightarrow } C)$| C2 |$(A\mathbin{\rightarrow } B)\mathbin{\rightarrow } ((B\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\mathbin{\rightarrow } C))$| C3 |$(A\mathbin{\rightarrow } B)\mathbin{\rightarrow } ((C\mathbin{\rightarrow } A)\mathbin{\rightarrow } (C\mathbin{\rightarrow } B))$| C4 |$(A\mathbin{\rightarrow }(A\mathbin{\rightarrow } B))\mathbin{\rightarrow } (A\mathbin{\rightarrow } B)$| C5 |$(A\mathbin{\rightarrow }(B\mathbin{\rightarrow } C))\mathbin{\rightarrow }(B\mathbin{\rightarrow }(A\mathbin{\rightarrow } C))$| C6 |$(A\mathbin{\rightarrow } \mathord{\sim } B)\mathbin{\rightarrow } (B\mathbin{\rightarrow } \mathord{\sim } A)$| C7 |$(A\mathbin{\rightarrow } \mathord{\sim } A)\mathbin{\rightarrow } \mathord{\sim } A$|⁠. I can now state the make up of the different relevant logics that are of interest for this paper. Each logic is the least set of formulas containing the indicated axioms and closed under the indicated rules.18 BA1-9, R1-4 DJB + C1 + C6 TWB + C2-3 + C6 TTW + C4 + C7 RWTW + C5 RRW + C4 There are other well-studied relevant logics in the literature, but I will focus on the above selection of relatively well-known logics.19 Before turning to the frames, a comment about the logics is under consideration. Little in this paper turns on the choice of base logic, provided it is between B and R. An exception is the argument of the appendix, which works for T and weaker logics, although not for R and RW. 1.2 Frames In this section, I will present the frames for the relevant logics listed above. The modal elements will be added in later sections. I will use the following common definitions, where there is a given non-empty set |$K$|⁠, |$P\subseteq K$|⁠, and |$R\subseteq K\times K\times K$|⁠: |$a\leq b\mathbin{=_{Df}} \exists x\in P, Rxab$|⁠, |$Rabcd\mathbin{=_{Df}} \exists x(Rabx\,\&\, Rxcd)$|⁠, and |$Ra(bc)d\mathbin{=_{Df}} \exists x(Raxd\,\&\, Rbcx )$|⁠. Definition 1.1 A Routley–Meyer frame, RM-frame, |$F$| is a quadruple |$\langle K,P,^*,R \rangle$|⁠, where |$K\neq \emptyset$|⁠, |$P\subseteq K$|⁠, |$^*:K\mapsto K$| such that |${{{a}^{**}}}=a$|⁠, and |$R\subseteq K^3$|⁠, satisfying the following conditions: |$\leq$| is reflexive and transitive. |$a\leq b\mathbin{\Rightarrow } b^*\leq \ a^*$| |$d\leq a \,\&\, Rabc\mathbin{\Rightarrow } Rdbc$|⁠; |$c\leq d\,\&\, Rabc\mathbin{\Rightarrow } Rabd$| Definition 1.2 An RM-model|$M$| is a pair |$\langle F, V \rangle$| of an RM-frame |$F$| and a valuation function |$V:{\textsf{At}}\times K\mapsto \{ 0,1\}$| such that if |$V(\,p,a)=1$| and |$a\leq b$|⁠, then |$V(\,p,b)=1$|⁠. For a model |$M(=\langle F,V \rangle )$|⁠, |$M$| is built on the frame |$F$|⁠, and |$F$| is the frame underlying|$M$|⁠. The valuation can be extended to an interpretation on the whole language, |$\mathbin{\Vdash }$|⁠, according to the following clauses: |$x\Vdash p$| iff |$V(\,p,x)=1$| |$x\Vdash \mathord{\sim } A$| iff |$x^*\not \Vdash A$| |$x\Vdash A\,\&\, B$| iff |$x\Vdash A$| and |$x\Vdash B$| |$x\Vdash A\lor B$| iff |$x\Vdash A$| or |$x\Vdash B$| |$x\Vdash A\mathbin{\rightarrow } B$| iff |$\forall yz (Rxyz \,\&\, y\Vdash A \mathbin{\Rightarrow } z\Vdash B)$| Definition 1.3 (Validity). A formula |$A$|holds in a model |$M$|⁠, |$\models _{M}{A}$|⁠, iff |$\forall x\in P, x\Vdash A$|⁠. A formula |$A$| is valid on an RM-frame |$F$|⁠, |$\models _{F}{A}$| iff |$A$| holds in all models |$M$| built on |$F$|⁠. A formula |$A$| is valid in a class of frames|$\mathscr{F}$|⁠, |$\models _{\mathscr{F}}{A}$|⁠, iff for all |$F\in \mathscr{F}$|⁠, |$A$| is valid on |$F$|⁠. There are two standard lemmas for the meta-theory of relevant logics I will now record, the heredity and verification lemmas. Lemma 1.4 (Heredity). For all formulas |$A$|⁠, if |$a\Vdash A$| and |$a\leq b$|⁠, then |$b\Vdash A$|⁠. Lemma 1.5 (Verification). In a given model |$M$|⁠, |$A\mathbin{\rightarrow } B$| holds in |$M$| iff for all |$a\in K$|⁠, |$a\Vdash A\mathbin{\Rightarrow }a\Vdash B$|⁠. The verification lemma streamlines proofs of soundness. Analogs of both lemmas are provable for the models to be studied in subsequent sections. The frames for the different relevant logics are those that satisfy the well-known frame conditions in Table 1. The frames for a given relevant logic L containing some axioms from among C1–C7 will be those that satisfy the corresponding conditions in Table 1. Table 1. RM-frame conditions C1 |$(A\mathbin{\rightarrow } B)\,\&\,(B\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\mathbin{\rightarrow } C)$| |$Rabc\mathbin{\Rightarrow } Ra(ab)c$| C2 |$(A\mathbin{\rightarrow } B)\mathbin{\rightarrow } ((B\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\mathbin{\rightarrow } C))$| |$Rabcd\mathbin{\Rightarrow } Ra(bc)d$| C3 |$(A\mathbin{\rightarrow } B)\mathbin{\rightarrow } ((C\mathbin{\rightarrow } A)\mathbin{\rightarrow } (C\mathbin{\rightarrow } B))$| |$Rabcd\mathbin{\Rightarrow } Rb(ac)d$| C4 |$(A\mathbin{\rightarrow }(A\mathbin{\rightarrow } B))\mathbin{\rightarrow } (A\mathbin{\rightarrow } B)$| |$Rabc\mathbin{\Rightarrow } Rabbc$| C5 |$(A\mathbin{\rightarrow }(B\mathbin{\rightarrow } C))\mathbin{\rightarrow }(B\mathbin{\rightarrow }(A\mathbin{\rightarrow } C))$| |$Rabcd\mathbin{\Rightarrow } Racbd$| C6 |$(A\mathbin{\rightarrow } \mathord{\sim } B)\mathbin{\rightarrow } (B\mathbin{\rightarrow } \mathord{\sim } A)$| |$Rabc\mathbin{\Rightarrow } Rac^*b^*$| C7 |$(A\mathbin{\rightarrow } \mathord{\sim } A)\mathbin{\rightarrow } \mathord{\sim } A$| |$Raa^*a$| C1 |$(A\mathbin{\rightarrow } B)\,\&\,(B\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\mathbin{\rightarrow } C)$| |$Rabc\mathbin{\Rightarrow } Ra(ab)c$| C2 |$(A\mathbin{\rightarrow } B)\mathbin{\rightarrow } ((B\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\mathbin{\rightarrow } C))$| |$Rabcd\mathbin{\Rightarrow } Ra(bc)d$| C3 |$(A\mathbin{\rightarrow } B)\mathbin{\rightarrow } ((C\mathbin{\rightarrow } A)\mathbin{\rightarrow } (C\mathbin{\rightarrow } B))$| |$Rabcd\mathbin{\Rightarrow } Rb(ac)d$| C4 |$(A\mathbin{\rightarrow }(A\mathbin{\rightarrow } B))\mathbin{\rightarrow } (A\mathbin{\rightarrow } B)$| |$Rabc\mathbin{\Rightarrow } Rabbc$| C5 |$(A\mathbin{\rightarrow }(B\mathbin{\rightarrow } C))\mathbin{\rightarrow }(B\mathbin{\rightarrow }(A\mathbin{\rightarrow } C))$| |$Rabcd\mathbin{\Rightarrow } Racbd$| C6 |$(A\mathbin{\rightarrow } \mathord{\sim } B)\mathbin{\rightarrow } (B\mathbin{\rightarrow } \mathord{\sim } A)$| |$Rabc\mathbin{\Rightarrow } Rac^*b^*$| C7 |$(A\mathbin{\rightarrow } \mathord{\sim } A)\mathbin{\rightarrow } \mathord{\sim } A$| |$Raa^*a$| View Large Table 1. RM-frame conditions C1 |$(A\mathbin{\rightarrow } B)\,\&\,(B\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\mathbin{\rightarrow } C)$| |$Rabc\mathbin{\Rightarrow } Ra(ab)c$| C2 |$(A\mathbin{\rightarrow } B)\mathbin{\rightarrow } ((B\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\mathbin{\rightarrow } C))$| |$Rabcd\mathbin{\Rightarrow } Ra(bc)d$| C3 |$(A\mathbin{\rightarrow } B)\mathbin{\rightarrow } ((C\mathbin{\rightarrow } A)\mathbin{\rightarrow } (C\mathbin{\rightarrow } B))$| |$Rabcd\mathbin{\Rightarrow } Rb(ac)d$| C4 |$(A\mathbin{\rightarrow }(A\mathbin{\rightarrow } B))\mathbin{\rightarrow } (A\mathbin{\rightarrow } B)$| |$Rabc\mathbin{\Rightarrow } Rabbc$| C5 |$(A\mathbin{\rightarrow }(B\mathbin{\rightarrow } C))\mathbin{\rightarrow }(B\mathbin{\rightarrow }(A\mathbin{\rightarrow } C))$| |$Rabcd\mathbin{\Rightarrow } Racbd$| C6 |$(A\mathbin{\rightarrow } \mathord{\sim } B)\mathbin{\rightarrow } (B\mathbin{\rightarrow } \mathord{\sim } A)$| |$Rabc\mathbin{\Rightarrow } Rac^*b^*$| C7 |$(A\mathbin{\rightarrow } \mathord{\sim } A)\mathbin{\rightarrow } \mathord{\sim } A$| |$Raa^*a$| C1 |$(A\mathbin{\rightarrow } B)\,\&\,(B\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\mathbin{\rightarrow } C)$| |$Rabc\mathbin{\Rightarrow } Ra(ab)c$| C2 |$(A\mathbin{\rightarrow } B)\mathbin{\rightarrow } ((B\mathbin{\rightarrow } C)\mathbin{\rightarrow } (A\mathbin{\rightarrow } C))$| |$Rabcd\mathbin{\Rightarrow } Ra(bc)d$| C3 |$(A\mathbin{\rightarrow } B)\mathbin{\rightarrow } ((C\mathbin{\rightarrow } A)\mathbin{\rightarrow } (C\mathbin{\rightarrow } B))$| |$Rabcd\mathbin{\Rightarrow } Rb(ac)d$| C4 |$(A\mathbin{\rightarrow }(A\mathbin{\rightarrow } B))\mathbin{\rightarrow } (A\mathbin{\rightarrow } B)$| |$Rabc\mathbin{\Rightarrow } Rabbc$| C5 |$(A\mathbin{\rightarrow }(B\mathbin{\rightarrow } C))\mathbin{\rightarrow }(B\mathbin{\rightarrow }(A\mathbin{\rightarrow } C))$| |$Rabcd\mathbin{\Rightarrow } Racbd$| C6 |$(A\mathbin{\rightarrow } \mathord{\sim } B)\mathbin{\rightarrow } (B\mathbin{\rightarrow } \mathord{\sim } A)$| |$Rabc\mathbin{\Rightarrow } Rac^*b^*$| C7 |$(A\mathbin{\rightarrow } \mathord{\sim } A)\mathbin{\rightarrow } \mathord{\sim } A$| |$Raa^*a$| View Large Say that a formula |$A$| is valid in all frames for |${\textsf{L}}$|⁠, |$\models _{{\textsf{L}}}{A}$|⁠, iff it is valid on the class of RM-frames satisfying the frame conditions for L. Say that a formula |$A$| is a theorem of |${\textsf{L}}$|⁠, |$\vdash _{{\textsf{L}}}{A}$|⁠, iff there is a proof of |$A$| from the axioms and rules of |${\textsf{L}}$|⁠. There are some standard definitions for the canonical model construction I will record here. Definition 1.6 A set of sentences |$X$| is an L-theory iff both (i) if |$\vdash _{{\textsf{L}}}{A\mathbin{\rightarrow } B}$| and |$A\in X$|⁠, then |$B\in X$|⁠, and (ii) if |$A\in X$| and |$B\in X$|⁠, then |$A\,\&\, B\in X$|⁠. A set |$X$| of formulas is prime iff |$A\lor B\in X$| only if |$A\in X$| or |$B\in X$|⁠. A theory is L-regular provided it contains all theorems of |${\textsf{L}}$| and is an L-theory. A pair |$\langle X,Y \rangle$| of sets of formulas is L-inconsistent iff there are formulas |$A_1,\ldots ,A_n\in X$| and |$B_1,\ldots ,B_m\in Y$| such that |$\vdash _{L}{(A_1\,\&\,\ldots \,\&\, A_n)\mathbin{\rightarrow }(B_1\lor \ldots \lor B_m)}$|⁠. A pair |$\langle X,Y \rangle$| of sets of formulas is L-consistent iff it is not L-inconsistent. The notion of L-inconsistency is used to prove one of the central lemmas for the construction of the canonical model. Lemma 1.7 (Prime extension). If a pair |$\langle X,Y \rangle$| is L-consistent, then there is a prime L-theory |$Z\supseteq X$| such that |$\langle Z,Y \rangle$| is L-consistent. See [50, Chapter 5.2] for the proof technique. Careful choice of the second member of the pair provides a way to inflate sets of formulas to prime theories while maintaining the appropriate relations in the canonical frame. This lemma holds for many logics extending B with additional axioms and rules and holds for all logics under consideration in this paper. Theorem 1.8 (Soundness and completeness). Where |${\textsf{L}}$| is one of the relevant logics above, |$\vdash _{{\textsf{L}}}{A}$| iff |$\models _{{\textsf{L}}}{A}$|⁠. For proofs, see [50, Chapter 11.3]. I will briefly sketch the construction of the canonical model |$M$| for the logic L here. The canonical model |$M$| is the pair of the following canonical frame and canonical valuation: |$K_M$| is the set of prime L-theories. |$P_M$| is the set of L-regular, prime L-theories. |$R_Mabc$| iff |$\forall B,C((B\mathbin{\rightarrow } C)\in a\,\&\, B\in b\mathbin{\Rightarrow } C\in c)$|⁠. |${a}^{*_M}$| is |$\{ A\in \mathscr{L}: \mathord{\sim } A\not \in a\}$|⁠. |$V_M(\,p,a)=1$| iff |$p\in a$|⁠. This definition has the consequence that |$a\leq b$| iff |$a \subseteq b$|⁠. The canonical frame satisfies the frame conditions for the logic |${\textsf{L}}$|⁠. With that background in place, I will turn to the task of adding justification modals to relevant logics. These logics result from adding to a basic relevant logic L the axiom \begin{equation*} {\textsf{K}} \qquad [\![ t ]\!](A\mathbin{\rightarrow} B)\mathbin{\rightarrow}([\![ s ]\!]\ A\mathbin{\rightarrow} [\![ t\cdot s ]\!]\ B) \end{equation*} and a selection of the following modal axioms and rules: RE|$A\mathbin{\leftrightarrow } B\mathbin{\Rightarrow } [\![ t ]\!] \ A\mathbin{\leftrightarrow } [\![ t ]\!] \ B$| RM|$A\mathbin{\rightarrow } B\mathbin{\Rightarrow } [\![ t ]\!] \ A\mathbin{\rightarrow } [\![ t ]\!] \ B$| RN|$A\mathbin{\Rightarrow } [\![ t ]\!] \ A$| M|$[\![ t ]\!] (A\,\&\, B)\mathbin{\rightarrow } [\![ t ]\!] \ A\,\&\,[\![ t ]\!] \ B$| C|$[\![ t ]\!] \ A\,\&\, [\![ t ]\!] \ B\mathbin{\rightarrow } [\![ t ]\!] (A\,\&\, B)$|⁠. The axiom M follows from RM, A2, A3 and A4. As such, I will not consider M separately for the rest of the paper. I will present different frames for candidate relevant justification logics.20 I will begin with evidence frames, which are a straightforward combination of RM-frames with evidence models for classical justification logics. There is an issue with proving completeness for these, so I will motivate a turn to multiple accessibility relation frames (§3) and then to neighbourhood frames (§5). 2 Evidence frames In this section, I will combine RM-frames with Fitting’s evidence models for justification logics to obtain RME-frames. The basic justification logic, to be considered in this section, will be |${\textsf{L.K}}$|⁠, whose axiom is K and whose relevant logic base is a logic |${\textsf{L}}$| from the options in §1.1. The logic L.K is sound with respect to the class of RME-frames, but there is an obstacle to proving completeness, which arises from the fact that the evidence sets need not be theories, in the sense of §1.2, namely closed under conjunction introduction and provable implications. This obstacle will motivate the adoption of a different class of frames and subsequent strengthening of the logic in the following section. The frames for classical justification logics are Kripke frames equipped with evidence functions.21 An evidence function |$\mathcal{E}$| is a function from |$W\times{\textsf{Terms}}$| to |$\mathcal{P}(\mathscr{L})$|⁠. The intuitive interpretation is that the evidence sets specify what formulas are justified by a term at a given world. The classical truth condition for the justification modal is \begin{equation*} w\mathbin{\Vdash} [\![ t ]\!]\ A \; \textrm{iff} \; \forall u(Swu\mathbin{\Rightarrow} u\mathbin{\Vdash} A)\ \textrm{and}\ A\in{\mathcal{E}}(w,t), \end{equation*} where |$S$| is the binary modal accessibility relation.22 To obtain the first kind of models for justification logics, I will adapt Fitting’s evidence models to the current setting. I will use the following definitions, where |$S\subseteq K\times K$|⁠: |$Sa\mathbin{=_{Df}} \{ b: Sab\}$| |$S(Rab)c\mathbin{=_{Df}} \exists x(Rabx\,\&\, Sxc)$| |$R(Sa)(Sb)c \mathbin{=_{Df}} \exists x\exists y(Sax\,\&\, Sby\,\&\, Rxyc)$| Definition 2.1 An RM evidence frame, RME-frame is an RM-frame |$F$| with an evidence function |$\mathcal{E}:K\times{\textsf{Terms}}\mapsto \mathcal{P}(\mathscr{L})$| and relation |$S\subseteq K\times K$|⁠, satisfying the following conditions: |$a\leq b\mathbin{\Rightarrow } Sb\subseteq Sa$| |$a\leq b\mathbin{\Rightarrow } {\mathcal{E}}(a,t)\subseteq{\mathcal{E}}(b,t)$| |$S(Rab)c\mathbin{\Rightarrow } R(Sa)(Sb)c$| |$Rabc \,\&\, (A\mathbin{\rightarrow } B)\in{\mathcal{E}}(a,t) \,\&\, A\in{\mathcal{E}}(b,s)\mathbin{\Rightarrow } B\in{\mathcal{E}}(c,t\cdot s)$| I will note that the first two conditions are used in the proof of the heredity lemma below and the second two are used to prove the soundness of K. Definition 2.2 An RME-model|$M$| is a pair |$\langle F, V \rangle$| of an RME-frame |$F$| and a valuation |$V:{\textsf{At}}\times K\mapsto \{ 0,1\}$| such that if |$V(\,p,a)=1$| and |$a\leq b$|⁠, then |$V(\,p,b)=1$|⁠. The valuation can be extended to an interpretation |$\mathbin{\Vdash }$|⁠, as with RM-models, with the following additional clause: |$x\Vdash [\![ t ]\!] \ A$| iff |$\forall y(Sxy\mathbin{\Rightarrow } y\Vdash A)$| and |$A\in{\mathcal{E}}(x,t)$| The heredity lemma is provable in the present setting. Lemma 2.3 (Heredity). For all formulas |$A$|⁠, if |$a\Vdash A$| and |$a\leq b$|⁠, then |$b\Vdash A$|⁠. Proof. Most of this proof is routine for relevant logics. The new case is when |$A$| is |$[\![ t ]\!] \ B$|⁠. Suppose |$a\leq b$| and |$a\Vdash [\![ t ]\!] \ B$|⁠. Then for all |$x$| such that |$Sax$|⁠, |$x\Vdash B$| and |$B\in{\mathcal{E}}(a,t)$|⁠. Since |$Sb\subseteq Sa$|⁠, it follows that for all |$y$| such that |$Sby$|⁠, |$Say$|⁠. Suppose |$c$| is such that |$Sbc$|⁠. Then |$Sac$|⁠, whence |$c\Vdash B$|⁠. By assumption |${\mathcal{E}}(a,t)\subseteq{\mathcal{E}}(b,t)$|⁠, so |$B\in{\mathcal{E}}(b,t)$|⁠. Therefore, |$b\Vdash [\![ t ]\!] \ B$|⁠, as desired. The verification lemma is provable as well, although it does not appeal to any of the new conditions. The justification logic L.K will be obtained from the rules and axioms of L together with the axiom K. The class of frames for L.K will be the RME-frames that satisfy the conditions for L. Theorem (Soundness). If |$\vdash _{{\textsf{L.K}}}{A}$|⁠, then |$\models _{{\textsf{L.K}}}{A}$|⁠. Proof. The proof is by induction on the length of the proof of |$A$|⁠. The main case for us is the axiom K, as the other cases are standard. Suppose |$M$| is a model. From the verification lemma, it is enough to show that for all |$a\in K$|⁠, if |$a\Vdash [\![ t ]\!] (B\mathbin{\rightarrow } C)$| then |$a\Vdash [\![ s ]\!] \ B\mathbin{\rightarrow }[\![ t\cdot s ]\!] \ C$|⁠. To that end, let |$a$| be an arbitrary point such that |$a\Vdash [\![ t ]\!] (B\mathbin{\rightarrow } C)$|⁠. Suppose |$a\not \Vdash [\![ s ]\!] \ B\mathbin{\rightarrow }[\![ t\cdot s ]\!] \ C$|⁠. Then there are |$b,c$| such that |$Rabc$|⁠, |$b\Vdash [\![ t ]\!] \ B$|⁠, and |$c\not \Vdash [\![ t\cdot s ]\!] \ C$|⁠. It follows that for all |$x$| such that |$Sax$|⁠, |$x\Vdash B\mathbin{\rightarrow } C$| and |$B\mathbin{\rightarrow } C\in{\mathcal{E}}(a,t)$|⁠, and for all |$y$| such that |$Sby$|⁠, |$y\Vdash B$| and |$B\in{\mathcal{E}}(b,s)$|⁠. From |$c\not \Vdash [\![ t\cdot s ]\!] \ C$|⁠, it follows that either there is a point |$d$| such that |$Scd$| and |$d\not \Vdash C$| or |$C\not \in{\mathcal{E}}(c,t\cdot s)$|⁠. Case: Suppose there is a point |$d$| such that |$Scd$| and |$d\not \Vdash C$|⁠. Then |$S(Rab)d$|⁠, which implies |$R(Sa)(Sb)d$|⁠. Let |$g$| and |$h$| be points witnessing |$R(Sa)(Sb)d$|⁠, so |$Sag$|⁠, |$Sbh$| and |$Rghd$|⁠. Since |$Sag$|⁠, |$g\Vdash B\mathbin{\rightarrow } C$|⁠, and since |$Sbh$|⁠, |$h\Vdash B$|⁠. As |$Rghd$|⁠, |$d\Vdash C$|⁠, contradicting the assumption. Case: Suppose |$C\not \in{\mathcal{E}}(c,t\cdot s)$|⁠. Since |$B\mathbin{\rightarrow } C\in{\mathcal{E}}(a,t)$|⁠, |$B\in{\mathcal{E}}(b,s)$|⁠, and |$Rabc$|⁠, |$C\in{\mathcal{E}}(c,t\cdot s)$|⁠, contradicting the assumption. Therefore, K is valid. After proving soundness, it would be natural to prove the converse, completeness. There is, however, a problem with doing so using the usual sort of canonical model construction. I will briefly explain what it is. The canonical model |$M$| for L.K is obtained by adapting the definitions from §1.2, with the following additions:23 |$S_Mab$| iff |$\{ A: \exists t\in{\textsf{Terms}}\ [\![ t ]\!] \ A\in a\}\subseteq b$|⁠. |$\mathcal{E}_M(a,t)=\{ A: [\![ t ]\!] \ A\in a\}$|⁠. The problem is showing that the resulting model satisfies the condition \begin{equation*} S(Rab)c\mathbin{\Rightarrow} R(Sa)(Sb)c, \end{equation*} which is required for the canonical model proof. In the appendix, I will prove that the canonical frame for T.K does not satisfy this condition. The counterexample works where the base logic is weaker than T. The canonical model construction runs into serious difficulty, and there does not appear to be a way to salvage it.24 The issue creating trouble for this construction is one that does not occur in classical justification logic, since there is no ternary accessibility relation to coordinate with the binary modal accessibility relation. Rather than further pursue models with evidence functions, I will turn to models based on other sorts of frames. I will provide some philosophical motivation for these frames and then demonstrate that they have some technical pay off as well. 3 Alternative frames The RME-frames take the usual RM-frames and add a binary accessibility relation and an evidence function. The evidence function is a kind of syntactic filter, as there are almost no closure conditions on it. If one takes the terms to denote formal proofs in Peano arithmetic, this makes a lot of sense.25 Whether |$\pi$| is a proof of |$A$| or of |$A\,\&\, A$| depends on which of those formulas is the conclusion of the proof; on the common view taking proofs to be sequences of formulas, this would amount to which is the final element of the sequence. Further, justification logic delivers hyperintensional justifications, permitting one to distinguish justifications for distinct yet classically equivalent formulas.26 It is plausible that something could justify |$p\lor \mathord{\sim } p$| but not justify |$q\lor \mathord{\sim } q$|⁠. A bit of evidence could serve a reason for one but not the other. Imposing closure conditions, such as closure under classical entailment, on the evidence sets erases the hyperintensionality. Relevant logics, as opposed to classical logic, are polythetic, in the sense that a particular relevant logic has multiple non-equivalent logical truths.27 None of the relevant logics canvassed above have |$p\lor \mathord{\sim } p\mathbin{\rightarrow } q\lor \mathord{\sim } q$| as a theorem, and none take arbitrary theorems to be equivalent. It seems plausible that one can impose some substantive closure conditions on the evidence sets without losing the hyperintensionality of the justification terms. Indeed, this idea will be borne out below. In classical logic, by contrast, the addition of RM, or even RE, results in a justification for one tautology being a justification for all tautologies, which is, from the point of view of this paper, a very bad thing.28 There are some options for closure conditions that one could adopt. Rather than proceed by placing conditions on the evidence functions, I will take a slightly different tack, dropping them entirely and adding different accessibility relations, |$S_t$|⁠, for each justification term, |$t\in{\textsf{Terms}}$|⁠. The reason is the following. If one takes the logic L.K and adds the axiom C, as well as the rule |${\textsf{RM}}$|⁠, one strengthens the justification logic. One might think that the properties of the conditional in L would transfer to the evidence composition, much as fusion and the conditional are related in L.29 For a logic like R that has axiom C5, permutation, \begin{equation*} (A\mathbin{\rightarrow} (B\mathbin{\rightarrow} C))\mathbin{\rightarrow}(B\mathbin{\rightarrow}(A\mathbin{\rightarrow} C)), \end{equation*} this would lead one to expect that the evidence composition operation should commute, i.e. |$[\![ t\cdot s ]\!] \ A\mathbin{\rightarrow } [\![ s\cdot t ]\!] \ A$|⁠. This, however, does not follow. For a counterexample, I will use a three-valued model, whose values are |$0< {\frac{1}{2}}<1$|⁠, with 0 the only undesignated value. A valuation |$v$| maps |${\textsf{At}}$| to |$\{ 0, {\frac{1}{2}},1\}$|⁠. Conjunction and disjunction are numerical |$\min$| and |$\max$|⁠, respectively. The conditional, negation and the justification modalities are interpreted according to Table 2, where |$s$| does not contain |$x\cdot y$| subterm. Table 2. Three-valued algebra |$\mathbin{\rightarrow }$| 1 |${\frac{1}{2}}$| 0 |$\mathord{\sim }$| |$[\![ t(x\cdot y) ]\!]$| |$[\![ s ]\!]$| 1 1 0 0 0 1 1 |${\frac{1}{2}}$| 1 |${\frac{1}{2}}$| 0 |${\frac{1}{2}}$| 1 |${\frac{1}{2}}$| 0 1 1 1 1 0 0 |$\mathbin{\rightarrow }$| 1 |${\frac{1}{2}}$| 0 |$\mathord{\sim }$| |$[\![ t(x\cdot y) ]\!]$| |$[\![ s ]\!]$| 1 1 0 0 0 1 1 |${\frac{1}{2}}$| 1 |${\frac{1}{2}}$| 0 |${\frac{1}{2}}$| 1 |${\frac{1}{2}}$| 0 1 1 1 1 0 0 View Large Table 2. Three-valued algebra |$\mathbin{\rightarrow }$| 1 |${\frac{1}{2}}$| 0 |$\mathord{\sim }$| |$[\![ t(x\cdot y) ]\!]$| |$[\![ s ]\!]$| 1 1 0 0 0 1 1 |${\frac{1}{2}}$| 1 |${\frac{1}{2}}$| 0 |${\frac{1}{2}}$| 1 |${\frac{1}{2}}$| 0 1 1 1 1 0 0 |$\mathbin{\rightarrow }$| 1 |${\frac{1}{2}}$| 0 |$\mathord{\sim }$| |$[\![ t(x\cdot y) ]\!]$| |$[\![ s ]\!]$| 1 1 0 0 0 1 1 |${\frac{1}{2}}$| 1 |${\frac{1}{2}}$| 0 |${\frac{1}{2}}$| 1 |${\frac{1}{2}}$| 0 1 1 1 1 0 0 View Large The justification modals are interpreted as the identity function, except for those modals whose terms, |$t(x\cdot y)$|⁠, contain |$x\cdot y$| as a subterm. A valuation |$v$|satisfies a formula |$A$| iff |$v(A)\in \{ {\frac{1}{2}},1\}$|⁠. A formula |$A$| is valid iff all valuations |$v$| satisfy |$A$|⁠. It was shown by Meyer that all axioms of R are valid on the table above and the rules preserve validity.30 It is straightforward to see that the additional modal axioms are valid and the additional rule preserves validity. Consider |$[\![ x\cdot y ]\!] \ p\mathbin{\rightarrow } [\![ y\cdot x ]\!] \ p$| and the valuation |$v$| such that |$v(\,p)= {\frac{1}{2}}$|⁠. \begin{equation*} v\left([\![ x\cdot y ]\!]\ p\mathbin{\rightarrow} [\![\, y\cdot x ]\!]\ p\right)=[\![ x\cdot y ]\!]{\frac{1}{2}}\mathbin{\rightarrow} [\![ \, y\cdot x ]\!]{\frac{1}{2}}=1\mathbin{\rightarrow} {\frac{1}{2}}=0 \end{equation*} Thus, the strengthening of R.K with C and RM does not ensure that the dot commutes. A similar counterexample can be constructed to show that the dot is not associative. These facts are, perhaps, not too surprising, since the axiom introducing the dot, K, does not offer a way to manipulate the dots once introduced. The different justification modalities are, more or less, independent of each other, and even adoption of the relatively strong logic R does not affect this.31 Nonetheless, in RME-models, the justification modalities are interpreted using the same accessibility relation, with the evidence function differentiating them. When classical logic and Kripke models are being used, the evidence functions are the key to preventing an undesired result: closing the justification modals under classical consequence and the rule of necessitation, RN. Since the relevant logics under consideration are hyperintensional, there is less danger that closing justification modals under provable implications will collapse all of the distinctions that the justification modals let us draw. There is certainly not the near total collapse we see when closing the justification modals under classical consequence. This means that we can take RM on board while also maintaining a lot of the fine-grained distinctions of justification logic. To flesh out the ideas of the preceding paragraph formally, I will define new frames that drop the evidence function but include distinct accessibility relations for each justification modal. These frames will ensure the validity of RM and C, but they do not validate RN. Rejecting RN preserves a lot of the fine-grained distinctions that we might want to draw. For example, it will not be the case that every theorem is justified by every justification term, a point to which I will return below. With that in mind, we shall define new frames. Definition 3.1 A justification RM-frame, RMJ-frame, is an RM-frame |$F$| with the addition of a binary relation |$S_t\subseteq K\times K$|⁠, for each |$t\in{\textsf{Terms}}$|⁠, that obeys the conditions of an RM-frame as well as the following: |$a\leq b\mathbin{\Rightarrow } S_tb\subseteq S_ta$| |$S_{t\cdot s}(Rab)c\mathbin{\Rightarrow } R(S_ta)(S_sb)c$| The definition of a model is adapted straightforwardly, as are the definitions of holding in a model and being valid in a frame or class of frames. The truth condition for the modals is replaced by \begin{equation*} a\Vdash [\![ t ]\!]\ A \ \textrm{ iff }\ \forall x(S_tax \mathbin{\Rightarrow} x\Vdash A). \end{equation*} We next show that these models obey heredity. Lemma 3.2 (Heredity). For all formulas |$A$|⁠, if |$a\Vdash A$| and |$a\leq b$|⁠, then |$b\Vdash A$|⁠. Proof. The new case is when |$A$| is |$[\![ t ]\!] \ B$|⁠. Suppose |$a\leq b$| and |$a\Vdash [\![ t ]\!] \ B$|⁠. Then for all |$x$| such that |$S_tax$|⁠, |$x\Vdash B$|⁠. By assumption |$S_tb\subseteq S_ta$|⁠. Let |$y$| be such that |$S_tby$|⁠. Then |$S_tay$|⁠, so |$y\Vdash B$|⁠. Therefore, |$b\Vdash [\![ t ]\!] \ B$|⁠, as desired. The verification lemma carries over as with RME-models. I will call the logic extending L with the axioms K and |${\textsf{C}}$| and the rule |${\textsf{RM}}$|⁠, L.KCR. I will now demonstrate the soundness of the axioms K and |${\textsf{C}}$| and the rule |${\textsf{RM}}$|⁠. Theorem 3.3 (Soundness). If |$\vdash _{{\textsf{L.KCR}}}{A}$|⁠, then |$\models _{{\textsf{L.KCR}}}{A}$|⁠. Proof. I will focus on the axioms K and |${\textsf{C}}$| and the rule |${\textsf{RM}}$|⁠, appealing to the verification lemma in each case. K Suppose |$M$| is a model and |$a$| a point such that |$a\Vdash [\![ t ]\!] (B\mathbin{\rightarrow } C)$|⁠. To show that |$a\Vdash [\![ s ]\!] \ B\mathbin{\rightarrow }[\![ t\cdot s ]\!] \ C$|⁠, assume for reductio that there are |$b$| and |$c$| such that |$Rabc$|⁠, |$b\Vdash [\![ s ]\!] \ B$| and |$c\not \Vdash [\![ t\cdot s ]\!] \ C$|⁠. So, there is a point |$d$| such that |$S_{t\cdot s}cd$| and |$d\not \Vdash C$|⁠. This implies that |$S_{t\cdot s}(Rab)d$|⁠, so |$R(S_ta)(S_sb)d$|⁠. Let |$e$| and |$g$| be witnesses so that |$S_{t}ae$|⁠, |$S_{s}bg$| and |$Regd$|⁠. It follows that |$e\Vdash B\mathbin{\rightarrow } C$| and |$g\Vdash B$|⁠. As |$Regd$|⁠, |$d\Vdash C$|⁠, contradicting the assumption. C \begin{equation*} \begin{array}{ccc} a\Vdash [\![ t ]\!]\ B\,\&\, [\![ t ]\!]\ C & \textrm{iff} &a\Vdash [\![ t ]\!]\ B \ \textrm{and}\ a\Vdash [\![ t ]\!]\ C \\ & \textrm{iff} & \forall x(S_tax\mathbin{\Rightarrow} x\Vdash B) \ \textrm{and} \ \forall x(S_tax\mathbin{\Rightarrow} x\Vdash C)\\ & \textrm{iff} & \forall x(S_tax\mathbin{\Rightarrow} x\Vdash B \ \textrm{and}\ x\Vdash C) \\ & \textrm{iff} & \forall x(S_tax\mathbin{\Rightarrow} x\Vdash B\,\&\, C)\\ & \textrm{iff} & a\Vdash [\![ t ]\!](B\,\&\, C) \\ \end{array} \end{equation*} RM Suppose |$\models _{{\textsf{L.KCR}}}{B\mathbin{\rightarrow } C}$|⁠. To show |$\models _{{\textsf{L.KCR}}}{[\![ t ]\!] \ B\mathbin{\rightarrow }[\![ t ]\!] \ C}$|⁠, suppose for reduction that |$M$| is a model with a point |$a\in K$| such that |$a\Vdash [\![ t ]\!] \ B$| and |$a\not \Vdash [\![ t ]\!] \ C$|⁠. Then there is |$x$| such that |$S_tax$| and |$x\not \Vdash C$|⁠. As |$a\Vdash [\![ t ]\!] \ B$|⁠, |$x\Vdash B$|⁠. For some |$b\in P$|⁠, |$Rbxx$|⁠. By assumption, |$b\Vdash B\mathbin{\rightarrow } C$|⁠, whence |$x\Vdash C$|⁠, contradicting the assumption. Therefore, |$\models _{{\textsf{L.KCR}}}{[\![ t ]\!] \ B\mathbin{\rightarrow }[\![ t ]\!] \ C}$|⁠, as desired. To prove completeness, we need to construct a canonical model. We can reuse much of the extant work on canonical model constructions, e.g. the cited work by Fuhrmann or by Restall, with some alterations to the cases involving the modal accessibility relation. Where |$X$| is a set of formulas and |$t\in{\textsf{Terms}}$|⁠, let |$X^{-t}=\{ A:[\![ t ]\!] \ A\in X\}$|⁠. For each |$t\in{\textsf{Terms}}$|⁠, define the accessibility relation |$S_t$| in the canonical model |$M$| for the logic as follows, adapting the rest of the canonical model from the example in §1.2. |$S_tab$| iff |$a^{-t}\subseteq b$| There are a couple of things to verify in order to show that the canonical model is based on an RMJ-frame and that it is a model, which are proved in turn, where being a frame for the base L is proved as usual. Lemma 3.4 The following are true of the canonical model |$M$|⁠: If |$a\leq b$|⁠, then |$S_tb\subseteq S_ta$|⁠. If |$S_{t\cdot s}(Rab)c$|⁠, then |$R(S_ta)(S_sb)c$|⁠. Proof. For (1), note that |$a\leq b$| iff |$a\subseteq b$|⁠. The conclusion then follows immediately. For (2), assume |$S_{t\cdot s}(Rab)c$| to show |$R(S_ta)(S_sb)c$|⁠. So, for some |$d$|⁠, |$Rabd$| and |$S_{t\cdot s}dc$|⁠. Suppose |$A\mathbin{\rightarrow } B\in a^{-t}$| and |$A\in b^{-s}$|⁠. Then |$[\![ t ]\!] (A\mathbin{\rightarrow } B)\in a$| and |$[\![ s ]\!] \ A\in b$|⁠. As |$[\![ s ]\!] \ A\mathbin{\rightarrow }[\![ t\cdot s ]\!] \ B\in a$|⁠, |$[\![ t\cdot s ]\!] \ B\in d$|⁠, so |$B\in c$|⁠, as desired. We can then use the prime extension lemma on the theories |$a^{-t}$| and |$b^{-s}$| to obtain appropriate prime theories extending each, respectively, to obtain |$R(S_t a)(S_s b)c$|⁠. Next, we need to show that the canonical model is really a model. Lemma 3.5 For all |$A$| and |$a\in K_M$|⁠, |$A\in a$| iff |$a\Vdash A$|⁠. Proof. The proof is by induction on the complexity of |$A$|⁠. Most of the cases are covered in other completeness proofs for relevant logics, so I will just do the new modal case. Let |$A$| be of the form |$[\![ t ]\!] \ B$|⁠. Suppose |$[\![ t ]\!] \ B\in a$| and suppose |$S_tab$|⁠. Since |$B\in a^{-t}$|⁠, |$B\in b$|⁠, so |$a\Vdash [\![ t ]\!] \ B$|⁠. Next, suppose |$a\Vdash [\![ t ]\!] \ B$|⁠. For reductio, suppose |$[\![ t ]\!] \ B\not \in a$|⁠. So, |$B\not \in a^{-t}$|⁠. Using the prime extension lemma, following the arguments of [50, 256ff.], we can obtain a prime theory |$b\supseteq a^{-t}$| such that |$B\not \in b$|⁠. However, then |$S_tab$| and |$b\not \Vdash B$|⁠, so |$a\not \Vdash [\![ t ]\!] \ B$|⁠, contradicting the assumption. We can then finish the argument for completeness by taking a prime theory excluding, and so falsifying, the non-theorem |$A$|⁠. Theorem 3.6 If |$\models _{{\textsf{L.KCR}}}{A}$|⁠, then |$\vdash _{{\textsf{L.KCR}}}{A}$|⁠. The basic logic L.KCR is fairly weak. As noted, even when |${\textsf{L}}$| is a relatively strong relevant logic, L.KCR does not prove |$[\![ t\cdot s ]\!] \ A\mathbin{\rightarrow }[\![ s\cdot t ]\!] \ A$|⁠. Before turning to some strengthenings of the basic relevant justification logics, I will briefly comment on RN. One can add the rule RN to the logic L.KCR. Using the frame condition that for all |$t\in{\textsf{Terms}}$| \begin{equation*} \forall x\forall y(x\in P\,\&\, S_{t}xy\mathbin{\Rightarrow} y\in P) \end{equation*} one can obtain soundness and completeness.32 Note that in the present context, one cannot adopt RN and K in lieu of RM. The argument that the latter follows from the combination of the former breaks down, as K uses at least two distinct justification modals, yielding only |$[\![ t ]\!] \ A\mathbin{\rightarrow }[\![ t\cdot t ]\!] \ B$| from the derivability of |$A\mathbin{\rightarrow } B$|⁠. Adding RN to the logic largely defeats the purpose using justification modals, as then, all terms will justify all theorems. Given that the logics are polythetic, it is hard to see a motivation for requiring that justifications treat theorems uniformly in this way. 4 Strengthenings One can strengthen the justification logic along two dimensions, adding traditional modal principles, such as the T axiom, |$[\![ t ]\!] \ A\mathbin{\rightarrow } A$|⁠, or adding principles relating justifications, such as |$[\![ t\cdot s ]\!] \ A\mathbin{\rightarrow } [\![ s\cdot t ]\!] \ A$|⁠. I will briefly explore the latter option. Here are some additional axioms that one might want to add, for all |${s,t,u\in{\textsf{Terms}}}$|⁠. Com|$[\![ t\cdot s ]\!] \ A\mathbin{\rightarrow } [\![ s\cdot t ]\!] \ A$| Assoc1|$[\![ (t\cdot s)\cdot u ]\!] \ A\mathbin{\rightarrow } [\![ t\cdot (s\cdot u) ]\!] \ A$| Assoc2|$[\![ t\cdot (s\cdot u) ]\!] \ A\mathbin{\rightarrow } [\![ (t\cdot s)\cdot u ]\!] \ A$| W|$[\![ t\cdot t ]\!] \ A\mathbin{\rightarrow } [\![ t ]\!] \ A$| Mi|$[\![ t ]\!] \ A\mathbin{\rightarrow } [\![ t\cdot t ]\!] \ A$|33 In|$[\![ t ]\!] \ A\mathbin{\rightarrow }[\![ t\cdot s ]\!] \ A$| These have more or less straightforward frame conditions to which they correspond. FCom|$\forall x\forall y(S_{s\cdot t}xy\mathbin{\rightarrow } S_{t\cdot s}xy)$| FAssoc1|$\forall x\forall y(S_{t\cdot (s\cdot u)}xy\mathbin{\rightarrow } S_{(t\cdot s)\cdot u}xy)$| FAssoc2|$\forall x\forall y(S_{(t\cdot s)\cdot u}xy\mathbin{\rightarrow } S_{t\cdot (s\cdot u)}xy)$| FW|$\forall x\forall y(S_{t}xy\mathbin{\rightarrow } S_{t\cdot t}xy)$| FMi|$\forall x\forall y(S_{t\cdot t}xy\mathbin{\rightarrow } S_{ t}xy)$| FIn|$\forall x\forall y(S_{t\cdot s}xy\mathbin{\rightarrow } S_{ t}xy)$| The final axiom, In, has the feel of irrelevance to it, although that does not transfer to the conditional of the logic. The axiom says that the dot operation is increasing, or cumulative, on the right, with an obvious analog on the left.34 I will note that something similar can be done with the RME-frames, putting conditions on the evidence functions, rather than the accessibility relation. Theorem 4.1 Each of the axioms Com, Assoc1, Assoc2, W, Mi and In is valid in a class of RMJ-frames |$\mathcal{F}$| if |$\mathcal{F}$| satisfies the condition FCom, FAssoc1, FAssoc2, FW, FMi and FIn, respectively. Proof. The proofs for each case are similar, so I will present only the case for Com. Suppose that |$\mathcal{F}$| obeys FCom and, for an arbitrary |$F\in \mathcal{F}$| and a model |$M$| on |$F$| such that |$a\Vdash [\![ t\cdot s ]\!] \ A$|⁠. Then, for all |$x$| such that |$S_{t\cdot s}ax$|⁠, |$x\Vdash A$|⁠. As |$S_{s\cdot t}a\subseteq S_{t\cdot s}a$|⁠, |$a\Vdash [\![ s\cdot t ]\!] \ A$|⁠, as desired. Theorem 4.2 Let |${\textsf{X}}\subseteq \{ {\textsf{Com}}, {\textsf{Assoc1}}, {\textsf{Assoc2}}, {\textsf{W}}, {\textsf{Mi}}, {\textsf{In}}\}$|⁠. Then |$\models _{{\textsf{L.KCRX}}}{A}$| only if |$\vdash _{{\textsf{L.KCRX}}}{A}$|⁠. Proof. The canonical model construction for L.KCR can largely be reused. It remains to verify that the canonical frame obeys the frame condition corresponding to each axiom in X. The cases are similar, so I will present the case for Com and FCom. Suppose that |$S_{s\cdot t}ab$|⁠. To show |$S_{t\cdot s}ab$|⁠, suppose that for some |$B$|⁠, |$B\in a^{-(t\cdot s)}$| but |$B\not \in b$|⁠. So, |$[\![ t\cdot s ]\!] \ B\in a$|⁠. As |$\vdash _{{\textsf{L.KCRX}}}{[\![ t\cdot s ]\!] \ B\mathbin{\rightarrow } [\![ s\cdot t ]\!] \ B}$|⁠, |$[\![ s\cdot t ]\!] \ B\in a$|⁠. Therefore, |$B\in a^{-(s\cdot t)}$|⁠. Since |$a^{-(s\cdot t)}\subseteq b$|⁠, |$B\in b$|⁠, contradicting the assumption. Therefore, |$S_{t\cdot s}ab$|⁠. The other dimension along which one can strengthen is by adding modal axioms. In work on justification logic, the 4 and 5 axioms are usually accompanied by an enrichment of the language with new term-forming operators, ‘!’ and ‘?’.35 I will focus only on the T axiom, |$[\![ t ]\!] \ A\mathbin{\rightarrow } A$|⁠, which does not require any change to the language.36T is valid on all RMJ-frames that satisfy reflexivity for all binary accessibility relations, i.e. |$\forall x S_t xx$|⁠. Let L.KCRT be the logic |${\textsf{L.KCR}}$| with the addition of T. Theorem 4.3 |$\vdash _{\textsf{L.KCRT}}{A}$| iff |$\models _{\textsf{L.KCRT}}{A}$|⁠. Proof. The soundness direction is straightforward. The scheme |$[\![ t ]\!] \ A\mathbin{\rightarrow } A$| is valid on frames satisfying |$\forall x S_txx$|⁠.37 The completeness direction piggybacks on the earlier completeness result. We need to show that the canonical frame satisfies the condition |$\forall xS_t xx$|⁠, for each |$t\in{\textsf{Terms}}$|⁠. Let |$a\in K_M$| be arbitrary and suppose that it is not the case that |$S_taa$|⁠. Then there is some formula |$B$| such that |$B\in a^{-t}$| but |$B\not \in a$|⁠. Since |$B\in a^{-t}$|⁠, |$[\![ t ]\!] \ B\in a$|⁠. As |$a$| is a L.KCRT-theory and |$\vdash _{{\textsf{L.KCRT}}}{[\![ t ]\!] \ B\mathbin{\rightarrow } B}$|⁠, |$B\in a$|⁠, contradicting the assumption. There are further strengthenings that would be good to explore, but I will not pursue them here.38 Instead, I will turn to an option for weakening the logic, dropping RM for RE. While this weakens the logic, it arguably fits more neatly with the philosophical ideas motivating the use of justification logics. 5 Neighbourhoods The use of the binary accessibility relations for justification terms, without evidence functions, results in the evidence sets being theories in the given relevant justification logic. This captures the idea that evidence is transmitted along logical entailment. One might take this to be a little too loose with evidence.39 Some epistemological or justificatory scenarios may incline one to think that evidence should not be transmitted freely by logical entailment, even when that entailment relation is a relatively discerning one, as is the case with the relevant logics being considered as base logics. There is a natural way of weakening the evidence sets that has a neat philosophical interpretation and results in a clean logic. That way is via neighbourhood functions.40 In the classical setting, a neighbourhood function |$N$| is a function from a set of worlds |$W$| to |${\mathcal{P}}({\mathcal{P}}(W))$|⁠. The idea is that an arbitrary set of worlds forms a proposition and |$\Box p$| is true at a world just in case the proposition that |$p$| is in the neighbourhood of that world. Some work must be done to carry that idea over to RM-frames, where it is not arbitrary sets of points that form propositions, but rather hereditarily closed sets that do.41 Definition 5.1 For a given |$F$|⁠, a set |$X\subseteq K$| is hereditarily closed in |$F$| iff |$a\in X$| and |$a\leq b$| only if |$b\in X$|⁠. Let |${\mathcal{H}}_{F}=\{ X\subseteq K: X\ \textrm{is hereditarily closed in}\ F\}$|⁠. Define the following operations for sets of points |$X,Y\subseteq K$|⁠: |$-X=\{ a\in K: a^*\not \in X\}$| |$X\Rrightarrow Y=\{ a\in K: \forall b\forall c(Rabc\,\&\, b\in X\mathbin{\Rightarrow } c\in Y ) \}$| With those definitions in place, I can define RM neighbourhood frames.42 Definition 5.2 Let an RM neighbourhood frame, or RMN-frame be an RM-frame |$F$| with a non-empty set |${\textsf{PROP}}\subseteq \mathcal{H}_{F}$| and a set of neighbourhood functions, |$N_t: K\mapsto{\mathcal{P}}({\textsf{PROP}})$|⁠, for each |$t\in{\textsf{Terms}}$|⁠, that obeys the following conditions: |$a\leq b\mathbin{\Rightarrow } N_t(a)\subseteq N_t(b)$|⁠. |$Rabc\mathbin{\Rightarrow } \hat{R}N_t(a)N_s(b)N_{t\cdot s}(c)$|⁠, where for |$X, Y ,Z \in{\mathcal{P}}({\textsf{PROP}})$|⁠, \begin{equation*} \hat{R}XYZ\mathbin{=_{Df}} \forall U,V\in{\textsf{PROP}}( \{ x\in K:\forall u\forall v(Rxuv \,\&\, u\in U\mathbin{\Rightarrow} v\in V)\}\in X\,\&\, U\in Y\mathbin{\Rightarrow} V\in Z). \end{equation*} If |$X,Y\in{\textsf{PROP}}$|⁠, then |$-X\in{\textsf{PROP}}$|⁠, |$X\Rrightarrow Y\in{\textsf{PROP}}$|⁠, |$X\cap Y\in{\textsf{PROP}}$|⁠, and |$X\cup Y\in{\textsf{PROP}}$|⁠. If |$X\in{\textsf{PROP}}$|⁠, then for each |$t\in{\textsf{Terms}}$|⁠, |$\nu _{t}X\in{\textsf{PROP}}$|⁠, where |$\nu _{t}X=\{ a\in K: X\in N_{t}(a)\}$|⁠. An RMN-model |$M$| is a pair |$\langle F,V \rangle$| of an RMN-frame |$F$| and a valuation function |$V:{\textsf{At}}\times K\mapsto \{ 0,1\}$| such that if |$a\leq b$| and |$V(\,p,a)=1$|⁠, then |$V(\,p,b)=1$|⁠, and |$\{ a\in K: V(\,p,a)=1\}\in{\textsf{PROP}}$|⁠, for all |$p\in{\textsf{At}}$|⁠. Models are well defined, as ensured by the following lemma. Lemma 5.3 Let |$F$| be an RMN-frame with |$X\subseteq K$| and |$Y\subseteq K$|⁠. If |$X$| and |$Y$| are hereditarily closed, then so are |$-X, X\cap Y, X\cup Y, X\Rrightarrow Y,$| and |$\nu _{t}X$|⁠. Proof. Suppose that |$F$| is an RMN-frame with |$X$| and |$Y$| hereditarily closed sets. The cases for |$X\cap Y$| and |$X\cup Y$| are straightforward. I will do the remaining cases. Suppose |$a\leq b$| and |$a\in -X$|⁠. Then |$a^*\not \in X$|⁠. From the definition of RM-frame, |$b^*\leq \ a^*$|⁠. As |$X$| is hereditarily closed, |$b^*\not \in X$|⁠, so |$b\in -X$|⁠. Suppose |$a\leq b$| and |$a\in X\Rrightarrow Y$|⁠, so |$\forall x\forall y(Raxy\,\&\, x\in X\mathbin{\Rightarrow } y\in Y)$|⁠. Suppose that |$b\not \in X\Rrightarrow Y$|⁠. Then it follows that there are points |$c$| and |$d$| such that |$Rbcd$|⁠, |$c\in X$| and |$d\not \in Y$|⁠. From the definition of RM-frame it follows that |$Racd$|⁠. By assumption, |$Racd\,\&\, c\in X\mathbin{\Rightarrow } d\in Y$|⁠, so |$d\in Y$|⁠, which contradicts the supposition. So, |$b\in X\Rrightarrow Y$|⁠. Suppose |$a\leq b$| and |$a\in \nu _{t}X$|⁠. Then |$X\in N_{t}(a)$|⁠. As |$a\leq b$|⁠, |$N_{t}(a)\subseteq N_{t}(b)$|⁠. So |$X\in N_{t}(b)$|⁠, so |$b\in \nu _{t}X$|⁠. For a given model |$M$|⁠, define the truth set for a formula |$A$| as |$[A]_M=\{ a\in K: a\Vdash A\}$|⁠.43 The truth condition for the justification modal in this context is |$a\Vdash [\![ t ]\!] \ A$| iff |$[A]\in N_t(a)$|⁠. The other truth conditions remain the same. The new condition on models that the set of points at which an atom is true is a member of PROP is not a great restriction, as such sets are hereditarily closed anyway. The condition ensures that for a given model |$M$|⁠, the truth set of each formula is proposition according to the model, i.e. for every formula |$A$|⁠, |$[A]\in{\textsf{PROP}}$|⁠. Lemma 5.4 For a given model |$M$|⁠, the following hold for all formulas |$A$| and |$B$|⁠: |$[A\,\&\, B]=[A]\cap [B]$| |$[A\lor B]=[A]\cup [B]$| |$[A\mathbin{\rightarrow } B]=[A]\Rrightarrow [B]$| |$[\mathord{\sim } A]=-[A]$| |$[[\![ t ]\!] \ A]=\nu _{t}[A]$| Proof. The proof is by induction on formula complexity. The base case is given by the definition of model and the inductive cases are straightforward. Corollary 5.5 For a model |$M$|⁠, for all formulas |$A$|⁠, |$[A]\in{\textsf{PROP}}$|⁠. The definitions for holding and validity are adapted as one would expect. We can adapt the heredity lemma to the present context. Lemma 5.6 If |$a\leq b$| and |$a\Vdash A$|⁠, then |$b\Vdash A$|⁠. Proof. The case for the justification modal is the only one I will present here. Suppose that |$A$| is of the form |$[\![ t ]\!] \ B$| and that |$a\leq b$| and |$a\Vdash [\![ t ]\!] \ B$|⁠. Since |$a\Vdash [\![ t ]\!] \ B$|⁠, |$[B]\in N_t(a)$|⁠. Since |$a\leq b$|⁠, |$N_t(a)\subseteq N_t(b)$|⁠, |$[B]\in N_t(b)$|⁠, whence |$b\Vdash [\![ t ]\!] \ B$|⁠. Given a base logic L, we can define the basic neighbourhood justification logic based on L to be L.EK, which is L with the addition of RE and K. One can prove soundness and completeness results, adapting standard methods. I will begin by proving a small lemma that will clarify the definition of |$\hat{R}$| in the definition of RMN-frames before proceeding to soundness. Lemma 5.7 Let |$M$| be an RMN-model and |$B$| and |$C$| formulas. Then \begin{equation*} \{ x\in K:\forall y\forall z(Rxyz\,\&\, y\in [B]\mathbin{\Rightarrow} z\in [C])\}=[B\mathbin{\rightarrow} C]. \end{equation*} Proof. \begin{equation*} \begin{array}{ccc} a\in[B\mathbin{\rightarrow} C] & \textrm{iff} & a\Vdash B\mathbin{\rightarrow} C \\ &\textrm{iff} & \forall y,z(Rayz\,\&\, y\Vdash B\mathbin{\Rightarrow}z\Vdash C) \\ &\textrm{iff} & \forall y,z(Rayz\,\&\, y\in[B]\mathbin{\Rightarrow} z\in[C]) \\ &\textrm{iff} & a\in\{ x\in K:\forall y,z(Rxyz\,\&\, y\in[B]\mathbin{\Rightarrow} z\in[C])\} \\ \end{array} \end{equation*} Theorem 5.8 |$\vdash _{{\textsf{L.EK}}}{A}$| only if |$\models _{{\textsf{L.EK}}}{A}$|⁠. Proof. As most of the proof is standard, I will only demonstrate the soundness of K and RE. For K, let |$M$| be a model and let |$a$| be a point such that |$a\Vdash [\![ t ]\!] (B\mathbin{\rightarrow } C)$|⁠. To show that |$a\Vdash [\![ s ]\!] \ B\mathbin{\rightarrow }[\![ t\cdot s ]\!] \ C$|⁠, let |$b,c$| be points such that |$Rabc$| and |$b\Vdash [\![ s ]\!] \ B$|⁠. From the definition of RMN-frame, |$\hat{R}N_t(a)N_s(b)N_{t\cdot s}(c)$|⁠, which yields \begin{equation*} \{ x\in K: \forall y\forall z(Rxyz\,\&\, y\in[B]\mathbin{\Rightarrow} z\in[C])\}\in N_{t}{(a)}\,\&\, [B]\in N_{s}(b)\mathbin{\Rightarrow} [C]\in N_{t\cdot s}(c). \end{equation*} By Lemma 5.7, this is |$[B\mathbin{\rightarrow } C]\in N_{t}{(a)}\,\&\, [B]\in N_{s}(b)\mathbin{\Rightarrow } [C]\in N_{t\cdot s}(c)$|⁠. Since |$b\Vdash [\![ s ]\!] \ B$|⁠, |$[B]\in N_{s}(b)$|⁠. By assumption, |$a\Vdash [\![ t ]\!] (B\mathbin{\rightarrow } C)$|⁠, which implies |$[B\mathbin{\rightarrow } C]\in N_{t}(a)$|⁠. Therefore, |$[C]\in N_{t\cdot s}(c)$|⁠, whence |$c\Vdash [\![ t\cdot s ]\!] \ C$|⁠. So, |$a\Vdash [\![ s ]\!] \ B\mathbin{\rightarrow }[\![ t\cdot s ]\!] \ C$|⁠, as desired. Suppose that |$\models _{\textsf{L.EK}}{A\mathbin{\leftrightarrow } B}$|⁠. Let M be an arbitrary |$\textsf{L.EK}$| model. Then, for all |$a\in K$|⁠, |$a\in [A]$| iff |$a\in [B]$|⁠, so |$[A]=[B]$|⁠. Suppose |$a\Vdash [\![ t ]\!] \ A$|⁠. This is the case iff |$[A]\in N_t(a)$|⁠, iff |$[B]\in N_t(a)$|⁠, as |$[A]=[B]$|⁠. However, |$[B]\in N_t(a)$| iff |$a\Vdash [\![ t ]\!] \ A$|⁠. We will carry over much of the definition of a canonical model |$M$| from before. Define |$||A||=\{ a\in K: A\in a\}$|⁠. I will use the following definition for the set PROP and the canonical neighbourhood functions for |$M$|⁠: |${\textsf{PROP}}_{M}=\{ ||A||: A\ \textrm{is a formula}\}$|⁠. |$N^M_{t}(a)=\{ X\in{\textsf{PROP}}: \exists A([\![ t ]\!] \ A\in a\,\&\, X=||A||)\}$|⁠. As usual, |$a\leq b$| iff |$a\subseteq b$|⁠. We can then show that the canonical frame for |${\textsf{L.EK}}$| satisfies the conditions for being an RMN-frame for |${\textsf{L.EK}}$|⁠, in addition to satisfying the conditions for being an RM-frame for |${\textsf{L}}$|⁠. Lemma 5.9 The canonical frame |$F$| underlying the canonical model |$M$| for the logic |${\textsf{L.EK}}$| satisfies the following conditions: |${\textsf{PROP}}\subseteq \mathcal{H}$|⁠. |$a\leq b\mathbin{\Rightarrow } N_t(a)\subseteq N_t(b)$|⁠. |$Rabc\mathbin{\Rightarrow } \hat{R}N_t(a)N_s(b)N_{t\cdot s}(c)$|⁠. If |$X,Y\in{\textsf{PROP}}$|⁠, then |$-X\in{\textsf{PROP}}$|⁠, |$X\Rrightarrow Y\in{\textsf{PROP}}$|⁠, |$X\cap Y\in{\textsf{PROP}}$|⁠, and |$X\cup Y\in{\textsf{PROP}}$|⁠. If |$X\in{\textsf{PROP}}$|⁠, then for each |$t\in{\textsf{Terms}}$|⁠, |$\nu _{t}X\in{\textsf{PROP}}$|⁠. Proof. Suppose |$X\in{\textsf{PROP}}$|⁠, |$a\leq b$|⁠, and |$a\in X$|⁠. From the definition, |$X=||B||$|⁠, for some |$B$|⁠, so |$B\in a$|⁠. As |$a\leq b$| only if |$a\subseteq b$|⁠, |$B\in b$|⁠, so |$b\in X$|⁠. Thus, |$X\in \mathcal{H}$|⁠. Suppose |$a\leq b$| and |$X\in N_t(a)$|⁠. Then for some |$B$|⁠, |$X=||B||$| and |$[\![ t ]\!] \ B\in a$|⁠. Since |$a\leq b$|⁠, for some |$d\in P$|⁠, |$Rdab$|⁠. As |$\vdash _{{\textsf{L.EK}}}{[\![ t ]\!] \ B\mathbin{\rightarrow } [\![ t ]\!] \ B}$|⁠, |$[\![ t ]\!] \ B\mathbin{\rightarrow } [\![ t ]\!] \ B\in d$| so |$[\![ t ]\!] \ B\in b$|⁠, which is sufficient to conclude |$X\in N_t(b)$|⁠. Suppose |$Rabc$|⁠. Suppose that not |$\hat{R}N_t(a)N_s(b)N_{t\cdot s}(c)$|⁠, so there are some |$Y,Z\in{\textsf{PROP}}$| such that |$\{ x\in K: \forall y\forall z(Rxyz \,\&\, y\in Y\mathbin{\Rightarrow } z\in Z)\}$||$\in N_t(a)$|⁠, |$Y\in N_{s}(b)$| but |$Z\not \in N_{t\cdot s}(c)$|⁠. From the definition of |${\textsf{PROP}}$|⁠, for some |$B$| and |$C$|⁠, |$Y=||B||$| and |$Z=||C||$|⁠. It then follows that |$\{ x\in K: \forall y\forall z(Rxyz \,\&\, y\in Y\mathbin{\Rightarrow } z\in Z)\}=||B\mathbin{\rightarrow } C||$|⁠. From the definitions and the next lemma, as |$||B||\in N_{s}(b)$|⁠, |$[\![ s ]\!] \ B\in b$|⁠. Similarly, |$[\![ t ]\!] (B\mathbin{\rightarrow } C)\in a$| and |$[\![ t\cdot s ]\!] \ C\not \in c$|⁠. As |$\vdash _{{\textsf{L.EK}}}{[\![ t ]\!] (B\mathbin{\rightarrow } C)\mathbin{\rightarrow }([\![ s ]\!] \ B\mathbin{\rightarrow }[\![ t\cdot s ]\!] \ C)}$|⁠, |$([\![ s ]\!] \ B\mathbin{\rightarrow }[\![ t\cdot s ]\!] \ C)\in a$|⁠. However, by assumption, |$Rabc$|⁠, from which it follows that |$[\![ t\cdot s ]\!] \ C\in c$|⁠, contradicting an assumption. Therefore, |$\hat{R}N_t(a)N_s(b)N_{t\cdot s}(c)$|⁠, as desired. Straightforward from the definitions. Straightforward from the definitions and the next lemma. There is an additional lemma that will be appealed to in showing that the canonical model is a model. Lemma 5.10 For a canonical frame |$F$| for L.EK, if |$||A||\subseteq ||B||$|⁠, then |$\vdash_\textsf{L.EK}{A\mathbin{\rightarrow } B}$| Proof. Suppose that |$||A||\subseteq ||B||$| but that |$A\mathbin{\rightarrow } B$| is not a theorem. So |$\langle{\{A\}},{\{B\}} \rangle$| is an L.EK-consistent pair. Using the prime extension lemma, we can construct a prime theory |$b$| with |$A\in b$| and |$B\not \in b$|⁠. Since |$||A||\subseteq ||B||$|⁠, |$B\in b$|⁠, which is a contradiction. The neighbourhood functions for the canonical model |$M$| are then well defined. If |$||A||=||B||$|⁠, then |$A\mathbin{\leftrightarrow } B$| is a theorem, so |$[\![ t ]\!] \ A\mathbin{\leftrightarrow } [\![ t ]\!] \ B$| is a theorem. Then |$[\![ t ]\!] \ A\in a$| iff |$[\![ t ]\!] \ B\in a$|⁠. So, if |$||A||\in N_t(a)$|⁠, there is a C such that |$[\![ t ]\!] \ C\in a$| and |$||C||=||A||$|⁠, whence |$[\![ t ]\!] \ A\ {\leftrightarrow }\ [\![ t ]\!] \ C$| is a theorem. Then, |$[\![ t ]\!] \ A\in a$|⁠, so |$[\![ t ]\!] \ B\in a$|⁠, and then |$||B||\in N_{t}(a)$|⁠, as desired. Lemma 5.11 For the canonical model |$M$| for L.EK defined above, |$A\in a$| iff |$a\Vdash A$|⁠. Proof. The only new case to deal with is the justification modal case. Let |$A$| be of the form |$[\![ t ]\!] \ B$|⁠. Suppose |$[\![ t ]\!] \ B\in a$|⁠. Since |$||B||$| is a set such that for the formula |$B$|⁠, |$[\![ t ]\!] \ B\in a$| and |$||B||=||B||$|⁠, |$||B||\in N_t(a)$|⁠. By the inductive hypothesis, |$||B||=[B]$|⁠, so |$[B]\in N_{t}(a)$|⁠. Therefore, |$a\Vdash [\![ t ]\!] \ B$|⁠, as desired. Suppose |$a\Vdash [\![ t ]\!] \ B$| and |$[\![ t ]\!] \ B\not \in a$|⁠. Then |$[B]\in N_t(a)$|⁠. By the inductive hypothesis, |$[B]=||B||$|⁠, so |$||B||\in N_t(a)$|⁠. Then there is some formula |$C$| such that |$[\![ t ]\!] \ C\in a$| and |$||C||=||B||$|⁠. However, then, |$C\mathbin{\leftrightarrow } B$| is a theorem, so by RE, |$[\![ t ]\!] \ C\mathbin{\leftrightarrow }[\![ t ]\!] \ B$| is too. Therefore, |$[\![ t ]\!] \ B\in a$|⁠, contradicting the assumption. That gives us the resources for completeness. Theorem 5.12 |$\models _{{\textsf{L.EK}}}{A}$| only if |$\vdash _{{\textsf{L.EK}}}{A}$|⁠. We can strengthen L.EK with C, RM and RN by requiring, respectively, the following. C: For |$X,Y\in{\textsf{PROP}}$|⁠, if |$X\in N_t(a)$| and |$Y\in N_t(a)$|⁠, then |$X\cap Y\in N_t(a)$|⁠. RM: For |$X,Y\in{\textsf{PROP}}$|⁠, if |$X\in N_t(a)$| and |$X\subseteq Y$|⁠, then |$Y\in N_t(a)$|⁠. RN: For |$X\in{\textsf{PROP}}$|⁠, if |$P\subseteq X$|⁠, then |$\forall x\in P(X\in N_{t}(x))$|⁠.44 Soundness for C, RM and RN with respect to frames satisfying these conditions is straightforward. Lemma 5.13 If the logic L.EKX extends L.EK with |${\textsf{X}}\subseteq \{ {\textsf{C,RM,RN}}\}$|⁠, then |$\vdash _{{\textsf{L.EKX}}}{A}$| only if |$\models _{{\textsf{L.EKX}}}{A}$|⁠. Proof. I will present the cases for the new axiom and rules. Case: C. Suppose |$a\Vdash [\![ t ]\!] \ B\,\&\,[\![ t ]\!] \ C$|⁠. So |$[B]\in N_{t}(a)$| and |$[C]\in N_{t}(a)$|⁠. By the frame condition for C, |$[B]\cap [C]\in N_{t}(a)$|⁠. Therefore, |$a\Vdash [\![ t ]\!] (B\,\&\, C)$|⁠, as desired. Case: RM. By the inductive hypothesis, |$\models _{{\textsf{L.EKX}}}{B\mathbin{\rightarrow } C}$|⁠. Suppose |$a\Vdash [\![ t ]\!] \ B$|⁠, so |$[B]\in N_{t}(a)$|⁠. Let |$x\in [B]$|⁠. By definition, there is some |$y\in P$|⁠, such that |$Ryxx$|⁠. By assumption, |$y\Vdash B\mathbin{\rightarrow } C$|⁠, so |$x\Vdash C$|⁠, whence |$x\in [C]$|⁠. By the frame condition for RM, as |$[B]\subseteq [C]$|⁠, |$[C]\in N_{t}(a)$|⁠. So, |$a\Vdash [\![ t ]\!] \ C$|⁠, as desired. Case: RN. By the inductive hypothesis, |$\models _{{\textsf{L.EKX}}}{B}$|⁠. Therefore, |$P\subseteq [B]$|⁠. By the frame condition for RN, for all |$x\in P$|⁠, |$[B]\in N_{t}(x)$|⁠. Therefore, |$\models _{{\textsf{L.EKX}}}{[\![ t ]\!] \ B}$|⁠, as desired. I will briefly present the arguments that the canonical models for L.EK with the addition of any of C, RM and RN, obey the appropriate frame conditions. Lemma 5.14 If the logic L.EKX extends L.EK with |${\textsf{X}}\subseteq \{ {\textsf{C,RM,RN}}\}$|⁠, then the canonical model for L.EKX satisfies the appropriate frame conditions. Proof. I will present the cases for the axiom and rules. Case: C. Suppose |$X,Y\in \textsf{PROP}$|⁠, |$X\in N_{t}(a)$|⁠, and |$Y\in N_{t}(a)$|⁠. For some formulas |$B$| and |$C$|⁠, |$X=||B||$| and |$Y=||C||$|⁠. Then |$[\![ t ]\!] \ B\in a$| and |$[\![ t ]\!] \ C\in a$|⁠. As |$a$| is a theory, |$[\![ t ]\!] \ B\,\&\,[\![ t ]\!] \ C\in a$|⁠. Since |$\vdash _{{\textsf{L.EKX}}}{[\![ t ]\!] \ B\,\&\,[\![ t ]\!] \ C\mathbin{\rightarrow }[\![ t ]\!] (B\,\&\, C)}$|⁠, |$[\![ t ]\!] (B\,\&\, C)\in a$|⁠. It follows that |$||B\,\&\, C||\in N_{t}(a)$|⁠, so |$X\cap Y\in N_{t}(a)$|⁠. Case: RM. Suppose |$X,Y\in{\textsf{PROP}}$|⁠, |$X\in N_{t}(a)$|⁠, and |$X\subseteq Y$|⁠. For some formulas |$B$| and |$C$|⁠, |$X=||B||$| and |$Y=||C||$|⁠. Since |$||B||\subseteq ||C||$|⁠, |$\vdash _{{\textsf{L.EKX}}}{B\mathbin{\rightarrow } C}$|⁠. By RM, |$\vdash _{{\textsf{L.EKX}}}{[\![ t ]\!] \ B\mathbin{\rightarrow }[\![ t ]\!] \ C}$|⁠. Since |$||B||\in N_{t}(a)$|⁠, |$[\![ t ]\!] \ B\in a$|⁠, so |$[\![ t ]\!] \ C\in a$|⁠. It follows that |$||C||\in N_{t}(a)$|⁠, which is |$Y\in N_{t}(a)$|⁠, as desired. Case: RN. Suppose |$X\in{\textsf{PROP}}$| and |$P\subseteq X$|⁠. For some |$B$|⁠, |$X=||B||$|⁠. Since |$P\subseteq ||B||$|⁠, |$\vdash _{{\textsf{L.EKX}}}{B}$|⁠. To see this, suppose |$B$| is not a theorem. Then by the prime extension lemma, there is a regular, prime theory excluding |$B$|⁠, but then it is not the case that |$P\subseteq ||B||$|⁠, contradicting the assumption. So, |$\vdash _{{\textsf{L.EKX}}}{B}$|⁠. By RN, |$\vdash _{{\textsf{L.EKX}}}{[\![ t ]\!] \ B}$|⁠. Suppose |$a\in P$| is arbitrary. Then |$[\![ t ]\!] \ B\in a$|⁠, so |$||B||\in N_{t}(a)$|⁠, so |$X\in N_{t}(a)$|⁠. Therefore, |$\forall x\in P(X\in N_{t}(x))$|⁠, as desired. The logic can be strengthened with additional conditions on the neighbourhood functions, but I will not pursue that direction here. Instead, I will turn to the philosophical interpretation of the logic L.EK. A problem that I highlighted above with L.KCR is that the evidence sets are theories. A consequence is that justification is transmitted along the implication of the logic. One might doubt that justification should be transmitted in that fashion, even when the implication is as strict as one finds in relevant logics. Evidence sets for a neighbourhood model are obtained via the following definition, |${\mathcal{E}}(a,t)\mathbin{=_{Df}}\{ A\in \mathscr{L}: [A]\in N_t(a)\}$|⁠. The logic L.EK then represents a natural weakening according to which evidence sets need not be theories. This represents a happy intermediate position between the evidence sets in the style of Fitting, where evidence sets can be almost arbitrary sets of formulas, and evidence sets in RMJ-models, where they are theories. An issue with the Fitting-style evidence sets is that they are sensitive to syntactic details. Being justification for something is, plausibly, hyperintensional, but it is a further step to require that justifications distinguish |$p$| and |$\mathord{\sim }\mathord{\sim } p$| or between |$p\lor p$| and |$p$|⁠. That sort of syntactic sensitivity does not fit nicely with the idea of being justification for a certain proposition, since that suggests some level of abstraction away from concrete formulas. The logic L.EK delivers an appropriate level of abstraction. It does not distinguish justification for |$p$| from justification for |$\mathord{\sim }\mathord{\sim } p$| or for |$p\lor p$|⁠. Evidence sets in models for L.EK are not just syntactic filters. Rather, they abstract from a concrete formula to the set of all logically equivalent formulas. The fact that we are taking a relevant logic as the base logic L means that we can still distinguish justifications for theorems of L.EK, as not all theorems are equivalent. I will now turn to the concluding discussion. 6 Conclusions In this paper, I have presented three kinds of frames for logics that add justification-logical machinery to base relevant logics. The RME-frames place almost no closure conditions on the evidence sets, and yield correspondingly weak justification logics. The RMJ-frames and RMN-frames have sound and complete axiomatizations and use natural closure conditions on evidence sets. They provide somewhat stronger logics than those of the RME-frames. One issue I raised in the introduction was seeing the dot of the justification terms as reminiscent of theory fusion in relevant logics. I will close with some further consideration of this issue. As shown in §3, the dot does not inherit the properties of theory fusion from the conditional (or fusion, if it is in the language). This was shown by the failure of |$[\![ t\cdot s ]\!] \ A\mathbin{\rightarrow }[\![ s\cdot t ]\!] \ A$| in R.KCR. The basic axioms of justification logic, K and whichever of RE, RM, M and C are adopted, do not provide any way to manipulate dots in justification modals apart from introducing them. Nonetheless, we can strengthen the dot to make it more like theory fusion. This is done by adopting additional axioms governing the dot, as appropriate to the relevant logic under consideration. In addition to the axioms suggested in §4, the rules MonR|$[\![ t ]\!] \ A\mathbin{\rightarrow } [\![ t ]\!] \ B\mathbin{\Rightarrow } [\![ t\cdot u ]\!] \ A\mathbin{\rightarrow }[\![ t\cdot u ]\!] \ B$|⁠, and MonL|$[\![ t ]\!] \ A\mathbin{\rightarrow } [\![ t ]\!] \ B\mathbin{\Rightarrow } [\![ u\cdot t ]\!] \ A\mathbin{\rightarrow }[\![ u\cdot t ]\!] \ B$|⁠, could be plausible additions, capturing the monotonicity of fusion. There are other potential axioms to add, as the axioms in §4 are mostly keyed to the logic R, with Mi appropriate to the stronger R-Mingle. As apparent from the proofs of completeness, prime theories are important for the meta-theory of relevant logics. In the basic justification-logical language used above, one scheme that plausibly captures primeness is |$[\![ t ]\!] (A\lor B)\mathbin{\rightarrow } [\![ t ]\!] \ A\lor [\![ t ]\!] \ B$|⁠. Whether this is a plausible rending of primeness will depend on how the justification modal is understood. If |$[\![ t ]\!] \ A$| is meant to be understood as saying that |$A$| is in the theory |$t$|⁠, this is plausible. If |$[\![ t ]\!] \ A$| is meant to understood as saying that for any theory |$u$| extending |$t$|⁠, |$u$| contains |$A$|⁠, then this formula will be inappropriate for expressing primeness. In many of the RMJ-models, the interpretation of the justification modal will be closer to the second of the above options, as there will often be more than one point accessible via |$S_t$| and each such point will be a prime extension of a theory containing |$A\lor B$|⁠. The basic justification language may be adequate, in a sufficiently strong logic, to capture important facts about theories in the negation-free fragment of the language, but it appears that accommodating negation will require further expressive resources. For any prime theory |$a$|⁠, its star theory, |$a^{*}$| is the theory |$\{ A\in \mathscr{L}:\mathord{\sim } A\not \in a\}$|⁠. An operation on justification terms yielding star theories seems ill at ease with some of the intuitions behind interpreting the terms as justifications or reasons for claims. Reasons provide some positive support for claims, and there seems to be an asymmetry in the idea of a reason between positive support and absence of evidence against. However, the star operation would yield justification for (the negations of) claims not supported.45 Instead of using a star operation on justifications, a different approach would be to use an incompatibility relation on justification terms, adapting an idea from work on compatibility models for negation in non-classical logics.46 With a binary relation ‘|$\bot$|’ in the language, we could add the axiom \begin{equation*} [\![ s ]\!]\ A\mathbin{\rightarrow}([\![ t ]\!]\mathord{\sim} A\mathbin{\rightarrow} s\bot t), \end{equation*} which captures a sense in which two bits of evidence are incompatible, when one justifies the negation of something justified by the other.47 There is, then, a question about what |$\bot$|-elimination axioms would be appropriate. In the present language, good options seem scarce. With propositional quantification, a natural idea would be \begin{equation*} s\bot t\mathbin{\rightarrow} \exists p([\![ s ]\!]\ p\,\&\,[\![ t ]\!]\mathord{\sim} p). \end{equation*} Rather than involve the full resources of propositional quantification, we might instead add another operator on justification terms, |$\downarrow$|⁠, with the interpretation that |$\downarrow \! t$| is the conjunction of all claims justified by |$t$|⁠. With this addition, the axiom \begin{equation*} s\bot t\mathbin{\rightarrow} [\![ s ]\!]\mathord{\sim}\!\downarrow\! t, \end{equation*} suggests itself.48 There is more to say about incompatibility and justifications, but I will leave further investigation to future work. While the initial intuition that the dot of justification logic is similar to theory fusion in relevant logics is not borne out by the basic justification logics, additional axioms can be adopted to bring the dot closer to representing theory fusion. Developing this idea thoroughly appears to require enriching the language with additional operators or relations, which raises questions about other plausible axioms and frame conditions. Acknowledgements I would like to thank Greg Restall, Igor Sedlár, Rohan French, Ed Mares, Ted Shear, Dave Ripley, Lloyd Humberstone, and the audience at the Melbourne-Glasgow Formal Philosophy Workshop for discussion and feedback. I would also like to thank the two anonymous referees for their feedback that greatly improved the paper. This research was supported by the Australian Research Council, Discovery Grant DP150103801. Footnotes 1 See [21] and [13] for accessible overviews of relevant logics. In this paper, I will not propose a general characterization of relevant logics. Instead, I will focus on a few of the standard logics in the family. 2 See [40]. 3 See [7] for an overview of the development, with contributions such as [8], [5] and [24]. 4 There has been work on paraconsistent justification logic. Su [63] investigates justification logics in which the modalities are paraconsistent, using many-valued worlds. Fitting [25] considers justification logics over a paraconsistent base logic. 5 Some central topics of the justification logic literature will be omitted in this paper, including constants, internalization and realization. Constants are being omitted as the variables are enough for the main points of this paper, and internalization is not being explored here. The use of additional rules raises some questions for internalization in weaker logics and in cases in which the language contains conjunction. The rules adopted in later sections would seem to pose some additional difficulties for internalization. The use of additional rules beyond modus ponens would appear to need new operators on justification terms to record them, which would be relevant for realizations. Both internalization and realization will be left for future work. I would like to thank an anonymous referee for clarifying the importance of these topics for the justification logic research programme to me. 6 Urquhart [64] 7 Fine [22] 8 Such an approach would have to take the justification terms to be formulas. 9 Unpacking the definition, ‘|$[\![ \,p ]\!] \ p\mathbin{\rightarrow } p$|’ would be ‘|$(\,p\mathbin{\rightarrow } p)\mathbin{\rightarrow } p$|’, which would be undesirable. 10 I thank Dave Ripley for discussion of the points in this paragraph. 11 For an overview of the latter, see [14] and [32], among others. In a sense, the study of relevant modal logics goes back to the early work of Anderson, Belnap, and Meyer. Anderson and Belnap [1] spend much time investigating the logic E, which has a modal component, but there is also discussion of the addition of a primitive necessity modal introduced to the logic R, coming from [44]. Whereas the focus there was proof theoretic, here my focus will be model theoretic. 12 Dean and Kurokawa [19] develop a response to the knowability paradox using intuitionistic justification logic. 13 Buchili et al. [16] study a justification logic with common knowledge. 14 This follows on the work of [27] and [50]. 15 In the justification logic literature, there is frequently another set of atomic terms, the constants and another term-forming operator, +. I will not pursue their contributions to the justification logics, so I will omit them here. 16 The notation ‘|$t:A$|’ is commonly used in the justification literature, rather than ‘|$[\![ t ]\!] \ A$|’. 17 Humberstone [31, 103ff.] 18 This presentation is based on that of [15], although Brady’s presentation is non-redundant, whereas this one includes redundancies. 19 See [52] for a detailed survey of optional axioms. 20 I am calling these logics justification logics because the explicit modals of justification logic are the primary focus. Since there are no internalization or realization theorems offered, there is much yet to do for them to be considered full-fledged justification logics. 21 There are alternative models, developed by [46], as well. See [6] for discussion. 22 Although in the classical context, the modal accessibility relation is usually represented by ‘|$R$|’, I use ‘|$S$|’ here to harmonize with the notation in the RME-frames and because in the context of relevant logics, ‘|$R$|’ is used for a ternary relation. 23 The additions are based on the completeness proof from [23]. 24 To be clear, this difficulty does not, by itself, show that L.K is not complete for the class of L.K RME-frames. An alternative method of proving completeness may yet deliver the desired theorem. 25 The ‘single conclusion’ view of proofs is adopted in [33–35] and [2]; the latter of which is not included in the main justification logic tradition. Artemov [4] adopts a more general, ‘multiple conclusion’ view of proofs. Thanks to an anonymous referee for references and clarification. 26 See [66] for discussion of the notion of hyperintensionality. 27 Relevant logics are not monothetic, in the sense of [31, 221]. Roughly, a logic is monothetic if all its theorems are equivalent by its lights. 28 An anonymous referee points out that from the point of view of the proponent of justification logic, the sort of closure conditions considered here are also very bad. Closing the justifications under provable implications, e.g. prevents one from drawing certain distinctions, but, in the context of relevant logics, many distinctions are still preserved. As will be discussed below, the addition of RN prevents the drawing of those distinctions. 29 The behaviour of the conditional is connected to the features of the models as well. In Fine’s models, properties of theory fusion affect the logical behaviour of the conditional, and in Urquhart’smodels, properties of the dot operation affect the logical behaviour of the conditional. 30 See [1, 470]. 31 While the behaviour of the dot is largely independent of the choice of base relevant logic, it is not entirely independent. If one adds the axiom |$A\mathbin{\rightarrow } (A\mathbin{\rightarrow } A)$| to R, obtaining R-Mingle, then one can prove |$[\![ t ]\!] \ A\mathbin{\rightarrow }[\![ t\cdot t ]\!] \ A$| using K and RM. As |$A\mathbin{\rightarrow }(A\mathbin{\rightarrow } A)$| is a theorem, |$[\![ t ]\!] \ A\mathbin{\rightarrow }[\![ t ]\!] (A\mathbin{\rightarrow } A)$| is too, by RM. Using the instance of K|$[\![ t ]\!] (A\mathbin{\rightarrow } A)\mathbin{\rightarrow }([\![ t ]\!] \ A\mathbin{\rightarrow } [\![ t\cdot t ]\!] \ A)$| with C3, yields |$[\![ t ]\!] \ A\mathbin{\rightarrow }([\![ t ]\!] \ A\mathbin{\rightarrow } [\![ t\cdot t ]\!] \ A)$|⁠, which with C4 results in |$[\![ t ]\!] \ A\mathbin{\rightarrow } [\![ t\cdot t ]\!] \ A$| as a theorem. 32 See [27, 507]. 33 The axiom is called Mi in reference to R-Mingle. 34 I thank an anonymous referee for the suggestion to include this axiom. 35 See [47] for soundness and completeness results and [62] for decidability results. 36 Adding new justification-term-forming operators to the language would require additional accessibility relations, appropriate frame conditions for which may be difficult to obtain. To avoid further proliferation of binary relations, it may be worth exploring adaptations of RME-frames that place closure conditions on the evidence functions, in particular closure under conjunction introduction and provable implication. That is to say that the closure conditions would makes the evidence sets theories. I thank Rohan French for suggesting this idea. 37 See [27] or [50]. 38 In this connection, requiring that |$S$| be functional would be particularly natural. I would like to thank Igor Sedlár for suggesting this idea. 39 See [9] or [30], among others, for discussion. 40 For more on neighbourhood models for modal logic, see [17] or [48]. 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Google Scholar Crossref Search ADS WorldCat [66] T. Williamson . Indicative versus subjunctive conditionals, congruential versus non-hyperintensional contexts . Philosophical Issues , 16 , 310 – 333 , 2006 . doi: https://doi.org/10.1111/j.1533-6077.2006.00116.x . Google Scholar Crossref Search ADS WorldCat Appendix In this appendix, I will prove that the canonical frame for T.K does not obey the frame condition |$S(Rab)c\mathbin{\Rightarrow } R(Sa)(Sb)c$|⁠, one of the two conditions for the K axiom. First, I will prove some lemmas. Let the matrix |$\mathcal{M}$| be defined as follows, adjusting the notation from [1, 243]. Let the set of values |$\mathcal{V}=\{ 0,1,2,3,4\}$|⁠, ordered numerically, so |$0< 1< 2< 3< 4$|⁠. Let the designated values |$\mathcal{D}=\{ 1,2,3,4\}$|⁠. Conjunction and disjunction are interpreted as maximum and minimum, respectively, on the order. Negation and the conditional will be interpreted according to the Table A1. Table A1. Five-valued matrix |$\mathcal{M}$| |$\mathbin{\rightarrow }$| 4 3 2 1 0 |$\mathord{\sim }$| 4 1 0 0 0 0 0 3 1 1 0 0 0 1 2 1 1 1 0 0 2 1 1 1 1 1 0 3 0 1 1 1 1 1 4 |$\mathbin{\rightarrow }$| 4 3 2 1 0 |$\mathord{\sim }$| 4 1 0 0 0 0 0 3 1 1 0 0 0 1 2 1 1 1 0 0 2 1 1 1 1 1 0 3 0 1 1 1 1 1 4 View Large Table A1. Five-valued matrix |$\mathcal{M}$| |$\mathbin{\rightarrow }$| 4 3 2 1 0 |$\mathord{\sim }$| 4 1 0 0 0 0 0 3 1 1 0 0 0 1 2 1 1 1 0 0 2 1 1 1 1 1 0 3 0 1 1 1 1 1 4 |$\mathbin{\rightarrow }$| 4 3 2 1 0 |$\mathord{\sim }$| 4 1 0 0 0 0 0 3 1 1 0 0 0 1 2 1 1 1 0 0 2 1 1 1 1 1 0 3 0 1 1 1 1 1 4 View Large To define valuations |$v$| on |$\mathcal{M}$| from the language to |$\mathcal{V}$|⁠, it remains to define the valuations on the justification modals, which will be as follows, where |$f$| is some function from |$\mathscr{L}\times{\textsf{Terms}}$| to |$\mathcal{V}$|⁠: \begin{equation*} v([\![ t ]\!]\ A)= \left\{ \begin{array}{@{}ll} 0 & \textrm{if }A\textrm{ is of the form }B\mathbin{\rightarrow}\textrm{ C}\\ f(A,t) & \textrm{otherwise} \end{array} \right. \end{equation*} The important clause for soundness is the first, as will be demonstrated below. Lemma A.1 For all valuations |$v$| on |$\mathcal{M}$|⁠, and all formulas |$A, B$|⁠, |$v(A\mathbin{\rightarrow } B)\in \{ 0,1\}$|⁠. Proof. By inspection of Table A1. Lemma A.2 For all valuations |$v$|⁠, the following two claims hold: For every axiom |$A$| of T.K, |$v(A)$| takes a designated value. The rules of T.K preserve the property of being designated on |$v$|⁠. Proof. That axioms of T take a designated value on |$v$| is straightforward, albeit tedious to check. For K, note that |$v([\![ t ]\!] (B\mathbin{\rightarrow } C))=0$|⁠, so \begin{equation*} v([\![ t ]\!](B\mathbin{\rightarrow} C)\mathbin{\rightarrow}([\![ s ]\!]\ B\mathbin{\rightarrow}[\![ t\cdot s ]\!]\ C))=1. \end{equation*} As an example of one of the other axioms, I will present the case for C4. If |$v(B\mathbin{\rightarrow } C)=1$|⁠, then the whole axiom gets |$1$|⁠, and we are done. Suppose, then, |$v(B\mathbin{\rightarrow } C)=0$|⁠, so |$v(B)>v(C)$|⁠. Therefore, |$v(B)\neq 0$|⁠. However, then |$v(B\mathbin{\rightarrow } (B\mathbin{\rightarrow } C))=0$|⁠, whence |$v((B\mathbin{\rightarrow } (B\mathbin{\rightarrow } C))\mathbin{\rightarrow } (B\mathbin{\rightarrow } C))=1$|⁠. So, the axiom is designated in all valuations. That the rules preserve being designated on |$v$| is also straightforward but tedious. As an example, take R1. Suppose |$v(B)\in \mathcal{D}$| and |$v(B\mathbin{\rightarrow } C)=1$|⁠. Then, |$v(B)\leq v(C)$|⁠. Since |$v(B)\in \mathcal{D}$|⁠, |$v(C)\in \mathcal{D}$| as well. Therefore, the rule preserves being designated. Let |$v_{0}$| be the valuation on |$\mathcal{M}$| such that |$v_{0}(\,p)=2$|⁠, for all |$p\in{\textsf{At}}$| and the evaluation on the modals is the following: \begin{equation*} v_{0}([\![ t ]\!]\ A)= \left\{ \begin{array}{@{}ll} 2 & \textrm{if }A\textrm{ is }(\,p\mathbin{\rightarrow} q)\,\&\,\textrm{ r and }t=x\\ 0 & \textrm{if }A\textrm{ is of the form }B\mathbin{\rightarrow}\textrm{ C}\\ v_{0}(A) & \textrm{otherwise} \end{array} \right. \end{equation*} Next, I will note a feature of derivability in T.K revealed by |$v_{0}$|⁠. Lemma A.3 Let |$B$| be a disjunction of formulas of the form |$C_{i}\mathbin{\rightarrow } D_{i}$|⁠, for |$0\leq i\leq n$|⁠, and let |$E$| be a disjunction of formulas of the form |$[\![ t_{j} ]\!] (F_{j}\mathbin{\rightarrow } G_{j})$|⁠, for |$0\leq j\leq m$|⁠, where at least one of |$m$| and |$n$| is not |$0$|⁠. It is not the case that \begin{equation*} \vdash_{{\textsf{T.K}}}{[\![ x ]\!]((\,p\mathbin{\rightarrow} q)\,\&\, r)\mathbin{\rightarrow} (B\lor E)}. \end{equation*} Proof. The antecedent of the displayed formula takes the value 2 on |$v_{0}$|⁠, by definition. In the consequent, |$v_{0}([\![ t_{i} ]\!] (F_{j}\mathbin{\rightarrow } G_{j}))=0$|⁠, for each |$j\leq m$|⁠, and |$v_{0}(C_{i}\mathbin{\rightarrow } D_{i})\in \{ 0,1\}$|⁠, for |$i\leq n$|⁠. Then, |$v_{0}(B\lor E)\in \{ 0,1\}$|⁠. However, |$2\mathbin{\rightarrow } 1=2\mathbin{\rightarrow } 0=0$|⁠, so the displayed formula is not provable in T.K. That completes the preliminaries. Now, I will define some theories. There is some prime theory |$c$| such that |$q\not \in c$|⁠. Let |$s$| be an atom distinct from |$p,q$| and |$r$|⁠, and consider the pair |$\langle \{ s\}, \{ [\![ t ]\!]{B}\in \mathscr{L}: t\in{\textsf{Terms}}\,\&\, B\in \mathscr{L}\} \rangle$|⁠. This pair is T.K-consistent. Consider the valuation |$v$| such that |$v(s)=4$| and |$f(A,t)=0$|⁠. Any disjunction |$D$| of formulas from the second member of the pair will take a value strictly less than 4 on |$v$|⁠, so |$v(s\mathbin{\rightarrow } D)=0$|⁠. So, we can extend |$\{ s\}$| to a prime theory disjoint from the second member of the pair. Call this theory |$z$|⁠. Then, |$\{ B\in \mathscr{L}: \exists t\in{\textsf{Terms}}([\![ t ]\!] \ B\in z)\}=\emptyset \subseteq c$|⁠, so |$Szc$|⁠. Next, consider the pair \begin{equation*} \langle \{ [\![ x ]\!]((\,p\mathbin{\rightarrow} q)\,\&\, r)\}, \{ B\mathbin{\rightarrow} C\in\mathscr{L}: B,C\in\mathscr{L}\}\cup\{ [\![ t ]\!](B\mathbin{\rightarrow} C)\in\mathscr{L}: t\in{\textsf{Terms}}\,\&\, B,C\in\mathscr{L}\} \rangle. \end{equation*} This pair is T.K-consistent, as verified by Lemma A.3. So, there is a prime theory |$a\supseteq{\{ [\![ x ]\!] ((\,p\mathbin{\rightarrow } q)\,\&\, r)\}}$| disjoint from the second member of the pair. The theory |$a$| contains no formulas of the form |$B\mathbin{\rightarrow } C$|⁠, by construction. So, |$Radz$|⁠, for any prime theory |$d$|⁠, in particular, a prime theory |$b\supseteq \{ [\![ y ]\!] \ p\}$|⁠. Any prime theory |$u$| such that |$Sau$| will be such that |$((\,p\mathbin{\rightarrow } q)\,\&\, r)\in u$|⁠, so |$(\,p\mathbin{\rightarrow } q)\in u$|⁠. For a prime theory |$w$| such that |$Sbw$|⁠, |$p\in w$|⁠. As |$q\not \in c$|⁠, it will not be the case that |$Ruwc$|⁠. Thus, the condition that |$S(Rab)c\mathbin{\Rightarrow } R(Sa)(Sb)c$| is not satisfied on the canonical frame for T.K There are two things about this argument that are worth noting. First, the valuations used in the argument were not, generally, monotone, so a logic closed under the rule RM would not be sound for them. Second, a key step, obtaining a prime theory that excludes all conditionals, works for neither RW.K nor R.K. In those logics, every formula implies a conditional, as |$\vdash _{{\textsf{RW}}}{A\mathbin{\rightarrow }((A\mathbin{\rightarrow } B)\mathbin{\rightarrow } B)}$|⁠. An argument that the canonical frame for RW.K, or for R.K, fails to satisfy the frame condition would need to proceed via a different route. © The Author(s) 2019. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - Tracking reasons with extensions of relevant logics
JF - Logic Journal of the IGPL
DO - 10.1093/jigpal/jzz018
DA - 2019-07-25
UR - https://www.deepdyve.com/lp/oxford-university-press/tracking-reasons-with-extensions-of-relevant-logics-EG1WICPHGM
SP - 543
VL - 27
IS - 4
DP - DeepDyve
ER -