Abstract
Some apparently valid arguments crucially rely on context change. To take a kind of example first discussed by Frege, ‘Tomorrow, it’ll be sunny’ taken on a day seems to entail ‘Today, it’s sunny’ taken on the next day, but the first sentence taken on a day sadly does not seem to entail the second sentence taken on the second next day. Mid-argument context change has not been accounted for by the tradition that has extensively studied the distinctive logical properties of context-dependent languages, for that tradition has focussed on arguments whose premises and conclusions are taken at the same context. I first argue for the desiderability of having a logic that accounts for mid-argument context change and I explain how one can informally understand such context change in a standard framework in which the relation of logical consequence holds among sentences. I then propose a family of simple temporal “intercontextual” logics that adequately model the validity of certain arguments in which the context changes. In particular, such logics validate the apparently valid argument in the Fregean example. The logics lack many traditional structural properties (reflexivity, contraction, commutativity etc.) as a consequence of the logical significance acquired by the sequence structure of premises and conclusions. The logics are however strong enough to capture in the form of logical truths all the valid arguments of both classical logic and Kaplan-style “intracontextual” logic. Finally, I extend the framework by introducing new operations into the object language, such as intercontextual conjunction, disjunction and implication, which, contrary to intracontextual conjunction, disjunction and implication, perfectly match the metalinguistic, intercontextual notions of premise combination, conclusion combination and logical consequence by representing their respective two operands as taken at different contexts.
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Notes
Throughout, by ‘infer’ and its relatives I mean the activity of drawing conclusions from premises.
Throughout, by ‘sentence’ I mean a meaningful abstract syntactic structure (of the usual type)—i.e., roughly, an abstract syntactic structure (of the usual type) whose basic elements are taken together with their meaning.
Henceforth, by ‘mid-argument context change’ and its like I mean the kind of mid-argument context change instantiated in the example just presented in the text.
Throughout, I use ‘entail’ and its relatives to denote the relation that is the converse of the relation denoted by ‘logically follow’ and its relatives.
Notoriously, Frege (1918), p. 64 went so far as to claim, in similar cases, that exactly the same thought—in Frege’s technical sense of ‘thought’—is expressed in \(c_{0}^{\mathrm {FE}}\) and \(c_{1}^{\mathrm {FE}}\). Our discussion will be neutral with respect to this stronger claim.
Notice that, strictly speaking, while useful in first introducing the idea, it is actually rather misleading to say that ‘Tomorrow, it’ll be sunny’ at (precisely) \(c_{0}^{\mathrm {FE}}\) entails ‘Today, it’s sunny’ at (precisely) \(c_{1}^{\mathrm {FE}}\), for \(c_{0}^{\mathrm {FE}}\) and \(c_{1}^{\mathrm {FE}}\) contain a lot of information that is utterly irrelevant to the issue whether the Fregean Entailment holds (since, for the purposes of semantic analysis, contexts must be individuated in an extremely fine-grained way)—indeed, they contain so much information as to determine, for every sentence \(\varphi \), whether \(\varphi \) is true or false in them, and so rob of any real modal force the claim in the text that “the meanings themselves of ‘today’ and ‘tomorrow’ guarantee that, if the premise is true in \(c_{0}^{\mathrm {FE}}\), the conclusion is true in \(c_{1}^{\mathrm {FE}}\)” (at least if, in that claim, ‘if’ expresses material implication and if contexts are guaranteed to have the parametre values which they happen to have). The ‘at’-modifications that are crucial for the Fregean Entailment and similar entailments to hold consist not in taking the relevant sentences at maximally specific individual contexts, but in taking them at very abstract types of contexts, in particular at relationally specified types of contexts (for example, in the case of the Fregean Entailment, the crucial ‘at’-modifications are that ‘Tomorrow, it’ll be sunny’ entails ‘Today, it’s sunny’ with the latter taken at any context whose time is located at the day after the day at which the time of the context at which the former is taken is located, and whose values of all the other relevant parametres are identical to those of the context at which the former is taken). A virtue of the families of logics developed in Sects. 3–5 is that they account for the Fregean Entailment and similar entailments at such level of abstraction (see also the second challenge raised in this section for a logic of utterances). Having noted this, in order to avoid unnecessary verbosity in the following ‘context’ will be used in a suitably flexible way, applying on an occasion to those types of contexts that are at the level of abstraction appropriate for that occasion.
Throughout, by ‘utterances’ I mean particular speech acts (i.e. particular events of asserting, asking, commanding etc.).
Gumb (1979) introduces an informal notion of diachronic inconsistency among tensed statements that is sensitive to the times at which the statements are made (without noting that it is merely a particular case of the general phenomenon of mid-argument context change). He then provides a formal framework that in effect takes sentence-context pairs as logical-consequence bearers. Gumb does not consider a language with context-dependent expressions like ‘today’, and actually the most natural extension of his framework to include such expressions does not validate the Fregean Entailment. Gumb also provides an adequate tableaux-style deductive system for the resulting logic (while this paper leaves an analogous task for future work). After presenting an earlier version of the material in this paper at the 3rd Arché Foundations of Logical Consequence Workshop on Propositions, Context and Consequence (University of St Andrews), I was informed that Geoff Georgi and Alexandru Radulescu both have work-in-progress that also in effect takes sentence-context pairs as logical-consequence bearers. Both Georgi and Radulescu do consider a language with context-dependent expressions like ‘today’. All of Gumb’s, Georgi’s and Radulescu’s discussions are largely motivated by the same broad kind of considerations that I’m offering in this section. Given limitations of space (plus, as for Georgi’s and Radulescu’s work, the fact that it is still unpublished), a detailed critical comparison of these works with the present one is however better reserved for a future occasion (although see the challenges raised in this section and in fn 32 for a logic of sentence-context pairs). Thanks to an anonymous referee for alerting me to the existence and relevance of Gumb’s work.
Thanks to an anonymous referee for urging me to be more explicit about these issues.
Throughout, I use square brackets to disambiguate constituent structure in natural language.
Throughout, I use ‘properties’ in a general sense, including relations with arbitrarily high arity.
Other suitable entities prominently include (to remain within the human realm) all kinds of broadly social entities, like clubs, teams, countries etc.: for example, Spain may collaborate with Argentina. Also for these other entities we can observe the circumstance that they may collaborate or fail to collaborate in particular capacities: for example, Spain in its capacity as current holder of the Presidency of the EU Council may collaborate with Argentina in its capacity as current holder of the Presidency of the Mercosur Council without Spain in its capacity as current holder of the Presidency of the Directive Council of the Organisation of Ibero-American States collaborating with Argentina in its capacity as current holder of the Presidency Pro Tempore of UNASUR. And, again, we should all agree that, in such cases, collaborating is merely a relation between Spain and Argentina, not among Spain, Argentina and the relevant capacities. Thanks to an anonymous referee for reminding me of such cases.
(Warning: what follows goes into rather subtle issues about collaborating, to an extent most likely to be unprecedented by any previous discussion of logical consequence. However, as far as I can tell, analogous considerations would eventually be required by any other natural example that could be used instead to make the point that I’m making in the text. Thus, plausibly, the structure of the following discussion of collaborating is actually an essential component of the argument about logical consequence that I’m developing in the text.) It might be claimed that, at least in other cases, collaborating does hold between capacities. For example, there is arguably a reading of ‘In this town, the president of the local wine club collaborates with the president of the local cheese club’ meaning, roughly, that the collaboration in question is a matter of tradition going beyond the individual holders of the relevant positions, and it might be conjectured that an adequate analysis of such reading will have to posit that collaborating holds between capacities. It might then be further suggested that, if so, this somehow speaks in favour of the thesis that, in the example considered in the text, collaborating is after all either a relation among people and capacities or a relation between capacities, in either case spoiling the intended analogy. In reply, I’ll make two points. First, granting the conjecture, that would in effect simply amount to recognising capacities as being among the other suitable entities between which collaborating can hold (see fn 12). Now, the fact that collaborating can hold 2arily between capacities does not seem to speak much in favour of the idea that, in the example considered in the text, it holds 4arily among people and capacities (the first disjunct of the suggestion); at best, it could speak in favour of the idea that, in the example considered in the text, it still holds 2arily between capacities (the second disjunct of the suggestion). But the latter claim is extremely problematic given that it is extremely plausible that, in the example considered in the text, people are among the bearers (for example, assuming that both Joan and Juan are tall, the discourse in the text can be glossed as ‘Two tall people collaborate, one in his capacity as president of the local wine club and the other in his capacity as president of the local cheese club’). Second, similar glosses also show the conjecture itself to be extremely problematic (for example, assuming that, as a matter of tradition, both the president of the local wine club and the president of the local cheese club must be tall, the original sentence in this fn can be glossed as ‘In this town, two tall people collaborate, the president of the local wine club and the president of the local cheese club’). In my view, all such sentences are better analysed as involving an implicit modal element, so that, for example, the original sentence in this fn is roughly synonymous with ‘In this town, as a matter of tradition, the president of the local wine club collaborates with the president of the local cheese club’ (whose analysis arguably does not require positing that collaborating holds between capacities, just as the analysis of ‘As a matter of tradition, the bridegroom kisses the bride’ arguably does not require positing that kissing holds between wedding roles). Thanks to an anonymous referee for putting forth to me this kind of example.
The correlation is not perfect: sometimes bearers are not expressed by syntactic arguments (for example, albeit an adjunct, the second prepositional phrase in ‘Mexico sold California to the US for \(\$15,000,000\)’ plausibly expresses a bearer of the relation of selling), sometimes syntactic arguments do not express bearers (for example, albeit a syntactic argument, the expletive pronoun in ‘It is raining’ does not plausibly express a bearer of the property of raining), sometimes modifications are not expressed by adjuncts (for example, albeit a syntactic argument, the adverbial phrase in ‘Juana worded this line beautifully’ plausibly expresses a modification of the relation of wording) and sometimes adjuncts do not express modifications (as per the first example mentioned in this fn). Having noted these points, the correlation, imperfect as it may be, is very suggestive all the same.
Distinguish two degrees of modifiability of a property: the first degree of modifiability is such that the presence of one modification and the absence of another modification entails the relevant unmodified claim, the second degree of modifiability is such that the presence of one modification and the absence of another modification entails that the relevant unmodified claim is (in a natural sense) ill-defined. For example, the modifiability by cooking methods of the relation of cooking is of first degree (for instance, the modified ‘Ampar cooked the rice in the pan’ and ‘Ampar did not cook the rice in the oven’ entail the unmodified ‘Ampar cooked the rice’), while the modifiability by capacities of the relation of collaborating is of second degree (for instance, the modified ‘Joan in his capacity as president of the local wine club collaborates with Juan in his capacity as president of the local cheese club’ and ‘Joan in his capacity as president of the local Barcelona club does not collaborate with Juan in his capacity as president of the local Real Madrid club’ entail that the unmodified ‘Joan collaborates with Juan’ is ill-defined). Notice that the second degree of modifiability is in itself compatible with unmodified claims being well-defined (indeed, either true or false), and even with the presence of one modification defeasibly entailing the relevant unmodified claim (for example, [if the presidency of the local wine club and the presidency of the local cheese club are respectively Joan’s and Juan’s only capacities, the unmodified ‘Joan collaborates with Juan’ is well-defined (indeed, in the example in the text, true)], and the modified positive ‘Joan in his capacity as president of the local wine club collaborates with Juan in his capacity as president of the local cheese club’ defeasibly entails the unmodified ‘Joan collaborates with Juan’). Now, the point of using as running example in the text an example with modifiability of second degree is that the modifiability by contexts of the relation of logical consequence is itself arguably of second degree (for example, the modified ‘ ‘Tomorrow, it’ll be sunny’ at \(c_{0}^{\mathrm {FE}}\) entails ‘Today, it’s sunny’ at \(c_{1}^{\mathrm {FE}}\)’ and ‘ ‘Tomorrow, it’ll be sunny’ at \(c_{0}^{\mathrm {FE}}\) does not entail ‘Today, it’s sunny’ at \(c_{0}^{\mathrm {FE}}\)’ arguably entail that the unmodified ‘ ‘Tomorrow, it’ll be sunny’ entails ‘Today, it’s sunny’ ’ is ill-defined).
I thought it interesting to show how some basic features of the framework can already be developed under this relatively weak assumption about the structure of days. In Sect. 4, I’ll strengthen the assumption to the more usual one of a discrete strict linear ordering. In the other direction, I’m still requiring that the structure of days be linear: that property is notoriously controversial in view of the problem of the open future, but relinquishing it would lead to complications that would be quite unnecessary for the purposes of this paper (see fn 35).
Notice that, given the very limited expressive resources of \(\fancyscript{L}\), both contexts and circumstances can be identified with days.
I’ll henceforth adopt standard scope conventions to save on brackets, assuming right-associativity for 2ary connectives.
\(\mathsf {min}(n,m)\) is the non-strictly smaller natural number between \(n\) and \(m\).
As perZardini (2012), I actually think that precisely context dependence shows that there are substantial gaps between logical consequence and truth preservation. However, without going into the nitty-gritty details, I simply note that those gaps are compatible with what this paper assumes about logical consequence and truth preservation.
\(\mathsf {ran}(\Gamma )\) is the range of the function \(\Gamma \).
Humberstone (1988) studies logics whose general abstract feature is that premises and conclusions are thought of as evaluated by two different valuations. Humberstone (1988), p. 403 explains that one particular instantiation of this general abstract feature results if, with a tensed language including ‘yesterday’ and ‘tomorrow’, one in effect thinks of the premises as uttered at the day immediately before the day at which the conclusions are uttered, and notes that, on this scheme, reflexivity fails whereas close relatives of the Fregean Entailment hold (of the other particular instantiations that Humberstone discusses, the only one that seems to lend itself naturally to an account along the broad lines proposed in this paper is the one that results if, with two semantically different but syntactically overlapping languages, one thinks of the premises as being in one language and of the conclusions as being in the other language). While a detailed critical comparison of Humberstone’s work with the present one lies beyond the scope of this paper, it is in order to comment on what, for our purposes, the main difference arguably is between the two. Whereas Humberstone’s logics only account for a single context change which occurs between the premises and the conclusions, the intercontextual logic of this paper accounts for arbitrarily many context changes which occur with each new premise or conclusion (indeed, in the extension of the basic construction to be developed in Sect. 5, with each new relevant component of a premise or conclusion). (It should be mentioned though that both developments are anticipated as possible avenues of further inquiry by Humberstone (1988), p. 432, n. 6 and Humberstone (1988), p. 402 respectively.) It is a straightforward consequence of this difference that, while the substructurality of Humberstone’s logics only involves failures of reflexivity and transitivity, as will become apparent in Theorem 7 the substructurality of the intercontextual logic of this paper also extends to failures of monotonicity, and even allows for failures of properties like contraction and commutativity, with the latter circumstance forcing premises and conclusions to be combined into objects that are finer-grained than sets (as sequences are). Thanks to an anonymous referee for alerting me to the existence and relevance of Humberstone’s work.
\(\mathsf {dom}(\Gamma )\) is the domain of the function \(\Gamma \).
\(\mathsf {max}(X)\) is the maximum of the set \(X\) under the contextually salient ordering (in this case, the standard well-ordering \(\le \) on the ordinals).
\(\mathsf {lub}(X)\) is the least upper bound of the set \(X\) under the contextually salient ordering (in this case, the standard well-ordering \(\le \) on the ordinals).
Notice that, while Theorem 1 can be strengthened so as to cover every index satisfying (C) in its \(\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}}\supseteq \mathbf {Intra}_{\mathcal {YTM}}\) direction, the theorem cannot be so strengthened in its \(\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}}\subseteq \mathbf {Intra}_{\mathcal {YTM}}\) direction. For consider, for example, an index \(\mathfrak {i}\) satisfying (C) and such that, for every \(\fancyscript{L}\)-structure \(\mathfrak {S}\), for every \(\alpha <\kappa \), it is not the case that both \(\mathsf {pre}_{\mathfrak {S}}(\mathsf {ass}^{\mathfrak {i}}(\alpha , \mathfrak {S}))\) does not exist and \(\mathsf {suc}_{\mathfrak {S}}(\mathsf {ass}^{\mathfrak {i}}(\alpha , \mathfrak {S}))\) exists (some such index exists, for, clearly, there is no \(\fancyscript{L}\)-structure \(\mathfrak {S}\) such that, for every \(d\in D_{\mathfrak {S}}\), both \(\mathsf {pre}_{\mathfrak {S}}(d)\) does not exist and \(\mathsf {suc}_{\mathfrak {S}}(d)\) exists): \(\lnot \mathcal {Y}(\varphi \vee \lnot \varphi )\wedge \mathcal {M}(\varphi \vee \lnot \varphi ) \vdash ^{\mathfrak {i}}_{\mathbf {Inter}_{\mathcal {YTM}}}\varphi \wedge \lnot \varphi \) then holds, whereas \(\lnot \mathcal {Y}(\varphi \vee \lnot \varphi )\wedge \mathcal {M}(\varphi \vee \lnot \varphi )\vdash _{\mathbf {Intra}_{\mathcal {YTM}}}\varphi \wedge \lnot \varphi \) does not hold.
Notice that, given the properties of \(<\) and \(\lhd _{\mathfrak {S}}\), the result of replacing in (ii) the conditional consequent by its converse, but not the result of replacing in (iii) the conditional consequent by its converse, is entailed by (i). Notice also that (iii) could be strengthened by adding an analogous clause for limit ordinals. However, given that such strengthening does not play any role in the development of intercontextual logic in this paper, it has not been officially included in (IM).
Since \(\mathfrak {i}\) satisfies (IM), actually many versions of the kind of possibility contemplated in the text cannot be turned into formal \(\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}}\)-countermodels, for condition (iii) of (IM) is incompatible with the day of \(c_{1}^{\mathrm {FE}}\) having a predecessor. Still, formal \(\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}}\)-countermodels are available in which the day of \(c_{1}^{\mathrm {FE}}\) does not have a predecessor (so that, in such models, the relevant instance of condition (iii) of (IM) is vacuously satisfied).
To complete the picture, \(\Gamma ,\mathcal {M}\varphi , \mathcal {T}\lnot \varphi \vdash ^{\mathfrak {i}}_{\mathbf {Inter}_{\mathcal {YTM}}} \Delta \) holds, but \(\Gamma \vdash ^{\mathfrak {i}}_{\mathbf {Inter}_{\mathcal {YTM}}} \Delta ,\mathcal {T}\varphi , \mathcal {Y}\lnot \varphi \) does not hold, with \(\varphi \) not containing \(\mathcal {Y}\), \(\mathcal {T}\) or \(\mathcal {M}\).
I myself am a fervent believer in the philosophical significance of substructural logics: in earlier works, I’ve developed and defended an approach to vagueness which relies on the adoption of a non-transitive logic (see for example Zardini (2008a)) as well as an approach to truth which relies on the adoption of a non-contractive logic (see for example Zardini (2011)). Neither of those two approaches goes however so far as to abandon properties like reflexivity, monotonicity or commutativity, as the present approach to context change on the contrary does. Relatedly, and with a focus on failure of commutativity, while the non-transitive approach to vagueness is consistent with premises and conclusions being combined into sets, and while the non-contractive approach to truth is consistent with premises and conclusions being combined into multisets, the present approach to context change requires that premises and conclusions be combined into sequences (see (Moruzzi and Zardini 2007, pp. 180–187; Zardini 2014a) for more relevant background on substructurality).
\(\langle .\rangle \) is the empty sequence (so that \(\mathsf {dom}(\langle .\rangle )=\varnothing \)).
The construction naturally suggests a variation whose distinctive feature would be to take utterances rather than sentences to be the logical-consequence bearers and which would roughly consist in requiring truth preservation from the premise utterances to the conclusion utterances taking each at the context in which they are made but allowing for reinterpretation of their non-logical vocabulary (see (Zardini 2008b, 2014c, e) for more relevant background on utterance truth). As per Sect. 2, I should reiterate that the spirit of this paper is by no means opposed to this and other alternative ways of accounting for mid-argument context change and that the objective here is rather to show that, by exploiting the fine structure into which premises and conclusions can be combined, an account of mid-argument context change is still possible even in a standard framework in which sentences are the logical-consequence bearers. I should also note, however, that I do happen to think that the sentence-based account proposed here has certain advantages over the alternative utterance-based account just sketched. In addition to the three more general problems that I’ve already presented in Sect. 2, let me now briefly introduce a more specific problem for the latter account. An utterance of ‘If Dave is here, Dave is here’ made successively pointing at two different places \(p_{0}\) and \(p_{1}\), with Dave being at \(p_{0}\) but not at \(p_{1}\), is false. But such utterance is true in the context in which it is made under any standard reinterpretation of its non-logical vocabulary (since any standard reinterpretation of its non-logical vocabulary assigns the same referent to the two occurrences of ‘here’). Thus, under the utterance-based account just sketched, such utterance would be a logical truth, even though it is false, which does violence to our notion of logical truth. (In Sect. 5, I’ll introduce a conditional connective that makes available a reading of ‘If Dave is here, Dave is here’ such that an utterance of that sentence on that reading may not be a logical truth. Even so, the problem observed in this fn would still persist for an utterance of ‘If Dave is here, Dave is here’ on the \(\supset \)-reading.) Thanks to Peter Fritz for discussion of these issues.
In some cases, I find that the natural extended understanding [of the structural properties mentioned in Theorem 7] required by indices satisfying (GIM) but not (IM) is particularly illuminating. For example, the natural extended understanding of contraction required for Theorem 7 to hold for indices satisfying (GIM) but not (IM) is such as to encompass contractions “across the turnstile”, since it needs contraction to be counterexampled, for instance, by the fact that [\(\mathcal {M}P,\lnot \mathcal {Y}P \vdash ^{\mathfrak {i}}_{\mathbf {Inter}_{\mathcal {YTM}^{\mathbb {Z}}}} \lnot \mathcal {M}P\) holds, but \(\mathcal {M}P\vdash ^{\mathfrak {i}}_{\mathbf {Inter}_{\mathcal {YTM}^{\mathbb {Z}}}} \bot , \mathcal {Y}P\) does not hold (if \(\mathsf {per}^{\mathfrak {i}}(0)=0\) and \(\mathsf {per}^{\mathfrak {i}}(1)=2\))]. Thus, the natural extended understanding of contraction required for Theorem 7 to hold for indices satisfying (GIM) but not (IM) is such as to imply that the implication from \(\varphi , \lnot \bot \vdash \lnot \varphi \) holding to \(\varphi \vdash \bot ,\bot \) holding is an instance of contraction, thereby revealing even the intuitionistically acceptable version of reductio ad absurdum (i.e. the implication from \(\varphi \vdash \lnot \varphi \) holding to \(\langle .\rangle \vdash \lnot \varphi \) holding) to be essentially just an instance of contraction (at least in those situations in which \(\varphi \vdash \lnot \varphi \) can be treated as equivalent with \(\varphi , \lnot \bot \vdash \lnot \varphi \) and \(\varphi \vdash \bot ,\bot \) can be treated as equivalent with \(\langle .\rangle \vdash \lnot \varphi \)). Zardini (2014b) has more discussion of the relation between contraction and reductio ad absurdum.
Analogous situations are known to occur in many areas in which informal notions of order are represented using the formal notion of a sequence. Thus, to take a simple example, the relation of loving is sensitive to an abstract order on people (the one who loves must be distinguished from the one who is loved), but is not sensitive to any of the particular ways in which such abstract order is concretely represented by the sets of pairs \(\{\langle x,y\rangle :\) \(x\) loves \(y\}\), \(\{\langle x,y\rangle :\) \(y\) loves \(x\}\), \(\{\langle x,y\rangle :\) if \(x\) and \(y\) have the same height, \(x\) loves \(y\), otherwise \(y\) loves \(x\}\) etc. All these sets can be seen as equally good alternative ways of representing the relation of loving, just as, restricting to \(\mathfrak {i}\)s satisfying (GIM), all the \(\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}^{\mathbb {Z}}}\)s can be seen as equally good alternative ways of representing \(\mathbf {Inter}_{\mathcal {YTM}^{\mathbb {Z}}}\). Thanks to Sebastiano Moruzzi and Eugenio Orlandelli for discussion of \(\mathbf {Inter}_{\mathcal {YTM}^{\mathbb {Z}}}\).
If we generalise the framework developed in this paper so as to cover other fragments of intercontextual logic beyond the one concerning ‘yesterday’, ‘today’ and ‘tomorrow’, the main idea should presumably be that each index represents with a concrete ordering the way in which context changes mid-argument. However, once a generalisation to other context-dependent expressions is made, there will typically be many incompatible ways in which context can change mid-argument: for example, considering ‘you’, ‘I’ and ‘she’, the public may become the agent, or, instead, may become a demonstratum. This contrasts with the particular case of the fragment concerning ‘yesterday’, ‘today’ and ‘tomorrow’, where the only way in which context can change mid-argument is determined by the passage of time. Since Theorems 10 and 11 can be seen as relying in effect both on the property of any index \(\mathfrak {i}\) satisfying (GIM) of representing with \(\sqsubseteq ^{\mathfrak {i}}\) the passage of time and on the fact that such passage determines the same way in which context changes mid-argument, keeping fixed that property analogues of these theorems cannot be expected to hold once a generalisation to other context-dependent expressions is made (for then that fact is no longer guaranteed to obtain). An analogous comment applies if we generalise the framework developed in this paper so as to cover the case in which the structure of days is not linear (see fn 16). It is important to note that both these comments presuppose that premises and conclusions remain linearly ordered (so that, given the property mentioned in the second last sentence, each index satisfying (IM), and hence also each index satisfying (GIM), is in effect forced to track only a particular chain in each relevant partially ordered structure). Such presupposition is far from being unquestionable, and further philosophical reflection on and technical investigation of intercontextual logic may well lead to its motivated rejection. Thanks to an anonymous referee for comments that led to substantial changes in this fn.
The deduction theorem fails not only in the sense that implication fails to connect premises with conclusions, but also in the sense that conjunction fails to connect premises with premises (even if \(\mathfrak {i}\) satisfies (IM) and \(\varphi \) does not contain \(\mathcal {Y}\), \(\mathcal {T}\) or \(\mathcal {M}\), \(\Gamma , \mathcal {T}\varphi \wedge \mathcal {Y}\lnot \varphi \vdash _{\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}^{\mathbb {Z}}}}\Delta \) does not hold, while, by Theorem 3, \(\Gamma ,\mathcal {T}\varphi , \mathcal {Y}\lnot \varphi \vdash _{\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}^{\mathbb {Z}}}}\Delta \) holds) and in the sense that disjunction fails to connect conclusions with conclusions (even if \(\mathfrak {i}\) satisfies (IM) and \(\varphi \) does not contain \(\mathcal {Y}\), \(\mathcal {T}\) or \(\mathcal {M}\), \(\Gamma \vdash _{\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}^{\mathbb {Z}}}}\Delta , \mathcal {M}\varphi \vee \mathcal {T}\lnot \varphi \) does not hold, while, by Theorem 5, \(\Gamma \vdash _{\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}^{\mathbb {Z}}}}\Delta ,\mathcal {M}\varphi , \mathcal {T}\lnot \varphi \) holds). It may be worth noting that, even so, the deduction theorem does not fail in the very weak sense that negation still moves premises to conclusions and conclusions to premises (if \(\mathfrak {i}\) satisfies (IM), [\(\Gamma , \varphi \vdash _{\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}^{\mathbb {Z}}}}\Delta \) holds only if \(\Gamma \vdash _{\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}^{\mathbb {Z}}}}\lnot \varphi , \Delta \) holds, and \(\Gamma \vdash _{\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}^{\mathbb {Z}}}}\varphi ,\Delta \) holds only if \(\Gamma ,\lnot \varphi \vdash _{\mathbf {Inter}^{\mathfrak {i}}_{\mathcal {YTM}^{\mathbb {Z}}}} \Delta \) holds]).
If we understand context so as to include, for example, discourse referents and the like, the broad tradition of dynamic logics and semantics provides many examples of logics that allow for different premises and conclusions to be at different contexts and that also allow for occurrences of sentences that are all components of the same premise or conclusion to be at different contexts (with the ensuing failure of [the structural properties mentioned in Theorem 7]). While a detailed critical comparison of the relevant work done in that tradition with the present one lies beyond the scope of this paper, it is in order to comment on what, for our purposes, the main difference arguably is between the two. On the one hand, in the dynamic tradition it is the occurrences of specific expressions that trigger context change, doing so in virtue of and to the extent determined by certain semantic properties of such expressions; hence, given a particular language, only specific kinds of context changes are allowed (to take an influential example, in the dynamic predicate logic of Groenendijk and Stokhof (1991) it is the occurrences of the existential quantifier that trigger context change, doing so in virtue of and to the extent determined by the existential quantifier’s semantic property of selecting assignments of discourse referents that can successfully be processed by the embedded expression; hence, given the language of dynamic predicate logic, only change of discourse referents is allowed). On the other hand, in intercontextual logic it is the sheer numerical identities and differences of premises and conclusions (and, as we’ll see in this section, the sheer numerical identities and differences of the relevant components of the same premise or conclusion) that trigger context change, doing so independently of and to an extent unconstrained by the semantic properties of any expressions in the language; hence, even given a particular language, any kind of context change is in principle allowed (although in this paper I’ve focussed on the change in value of the time parametre). Concisely put, while in the dynamic tradition it is the interpretation of the language that governs context change, in intercontextual logic it is context change that governs the interpretation of the language. Thanks to an anonymous referee for discussion of the relation between the dynamic tradition and intercontextual logic.
The precise syntax of the language is of some importance for the purposes of this section. Throughout, I assume a standard syntactic framework with the two primitive relations of precedence and dominance.
The traditional notion of context is arguably associated with two potentially different roles: that of being the kind of entity such that every utterance can eventually be interpreted with respect to one single such entity, and that of being the kind of entity that fixes the interpretation of certain expressions. While these two roles are typically indiscriminable in formal semantics, they will come apart in the extension of intercontextual logic developed in this section: the first role will typically be played by pairs (of pairs of pairs...) of days, whereas the second role will be played by days. After some deliberation, I’ve chosen to use ‘context’ and its relatives ambiguously to refer to either kind of entities, leaving it to, hum, context to disambiguate. The rationale for this apparently perverse choice is that I think that doing so will actually facilitate seeing the very substantial connections between the extension of intercontextual logic developed in this section and the previous work done in this paper and, more generally, in formal semantics. Thus, for example, I mean the kind of entities playing the first role when I say that “we need [...] to change our conception of contexts”, whereas I mean the kind of entities playing the second role when I say that “\(\otimes \) [...] allows for the passage of time (and so for context change)”. (Notice that, in some situations, some pairs (of pairs of pairs...) of days will not play the first role and will rather behave similarly to days.) An analogous comment applies for ‘circumstance’ and its relatives.
An alternative, tolerant rather than strict clause to the effect that \([\![\varphi ]\!]_{\pi _{0},\pi _{1},\mathfrak {M}}=\mathsf {max}([\![\varphi ]\!]_{\pi _{0},\pi _{2},\mathfrak {M}},[\![\varphi ]\!]_{\pi _{0},\pi _{3},\mathfrak {M}})\) would also have been possible. Given the other properties of the construction, the resulting system wouldn’t have diverged greatly from the present one.
On one natural understanding, a connective is a monster iff, roughly, it non-trivially selects contexts in which to evaluate the immediate components of the compound whose main connective it is. Under such understanding, a context-change-dependent connective like \(\otimes \) is a monster. On one natural understanding, a connective is intensional iff, roughly, it non-trivially selects circumstances at which to evaluate the immediate components of the compound whose main connective it is. Under such understanding, a context-change-dependent connective like \(\otimes \) is intensional. On one natural, weaker understanding, a connective is context dependent iff, roughly, it non-trivially selects contexts in or circumstances at which to evaluate the immediate components on the basis of the context in which the compound whose main connective it is is evaluated. Under such understanding, a context-change-dependent connective like \(\otimes \) is context dependent. On another natural, stronger understanding, a connective is context dependent iff, roughly, it non-trivially selects circumstances at which to evaluate the immediate components on the basis of the context in which the compound whose main connective it is is evaluated. Under such understanding, a context-change-dependent connective like \(\otimes \) is not context dependent.
Focussing on occurrences of sentences, I stress that such introduction of entities that are finer-grained than sentences is in no tension with the philosophical picture sketched in Sect. 2. First, in what follows occurrences will only play a very limited role that is fully compatible with the idea of Sect. 2 that it is sentences (at contexts) that entail sentences (at contexts), since occurrences will only be appealed to in the syntactic part of the theory, in their (usual) role of bearers of the dominance relation; in particular, neither logical nor semantic properties will be attributed to occurrences (thus, in the latter respect, the theory remains firmly within the bounds of “expression-based semantics” in the sense of Salmon (2006)). Second, even restricted to this very limited role, occurrences are introduced as a technical ersatz construct in the expressively poor background mathematical theory, with the only purpose of modelling in the theory the notion of [an expression as it occurs at a particular point in an atomisation], and that notion, on the construal proposed in the text, merely involves commitment to expressions, points and atomisations. Third, sentence occurrences are in any event not to be identified with utterances (or sentence-context pairs): taking for example the argument \(\mathcal {M}P \vdash \mathcal {T}P\), there is only one occurrence of \(\mathcal {M}P\) in that argument, even if there are indefinitely many utterances of \(\mathcal {M}P\) (and indefinitely many sentence-context pairs figuring \(\mathcal {M}P\) in their first coordinate) that are associated with that argument (relatedly, as I’ve indicated in fn 6, points do not represent any individual context nor any non-relationally specified type of context). (It is unfortunate that Kaplan (1989a), p. 522 deviated from the traditional usage of ‘occurrence’—which I’m following—and called sentence-context pairs with the same word.)
An alternative, forwards-looking rather than backwards-looking clause to the effect that \(\mathsf {ass}^{\mathfrak {i}}_{\Gamma }(\alpha ,\mathfrak {S})= \mathsf {ass}^{\mathfrak {i}}_{\Gamma }(\gamma ,\mathfrak {S})\) would also have been possible. Given the other properties of the construction, the resulting system wouldn’t have diverged greatly from the present one.
A well-known conjunctive connective broadly similar to \(\otimes \) is the ‘and next’-connective \(\boxtimes \) introduced by von Wright (1963), pp. 28–34 and further studied for example in von Wright (1965). Von Wright’s favoured understanding of \(\boxtimes \) (which is officially defined simply in terms of an axiomatic system) treats it as an intensional 2ary connective that is neither context-change dependent (as for example \(\otimes \) is) nor context dependent (as for example \(\mathcal {T}\) is), thus resembling in these respects for example the Priorean intensional 1ary ‘it will be the case that’-connective (see for example Prior (1955)). More specifically, adapted to the framework of this paper, \(\boxtimes \) enjoys the kind of basic semantics developed in Sect. 3 and can be taken to be governed by the following clause for an \(\fancyscript{L}\)-\(\mathbb {Z}\)-model \(\mathfrak {M}\) on an \( \fancyscript{L}\)-\(\mathbb {Z}\)-structure \({\mathfrak {S}}\):
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\([\![\varphi \boxtimes \psi ]\!]_{d_{0},d_{1},\mathfrak {M}}= \mathsf {min}([\![\varphi ]\!]_{d_{0},d_{1},\mathfrak {M}},[\![\psi ]\!]_{d_{0},\mathsf {suc}_{\mathfrak {S}}(d_{1}),\mathfrak {M}})\).
\(\boxtimes \) thus differs fundamentally from \(\otimes \) in that it does not really allow for the passage of time from its lhs to its rhs, but it only partially mimics it by shifting the circumstance at which its rhs is evaluated to the circumstance immediately following the circumstance at which its lhs is evaluated, with the consequence that such shift is only temporary in the sense that it only concerns the evaluation of the rhs of the relevant occurrence of \(\boxtimes \) rather than the evaluation of the whole argumentative material following the occurrence (so that in particular the circumstance at which what immediately follows the rhs is evaluated is in effect reset to be the circumstance at which the lhs is evaluated). For example, while \((\varphi \otimes \psi )\otimes \lnot \psi \) is consistent (since, when testing for consistency, \(\lnot \psi \) is evaluated in the context and at the circumstance immediately following the context in and the circumstance at which \(\psi \) is evaluated), \((\varphi \boxtimes \psi )\boxtimes \lnot \psi \) is not (since \(\lnot \psi \) is evaluated (in the same context and) at the same circumstance as \(\psi \)). An interesting consequence of this fundamental difference is that, while, as is indicated among other things by Theorem 13 and the properties of sequences, \(\otimes \) is “basically associative” in the sense that, as a premise or conclusion, \(\varphi _{0}\otimes \varphi _{1}\otimes \varphi _{2}\) is intersubstitutable with \((\varphi _{0}\otimes \varphi _{1})\otimes \varphi _{2}\) salva validitate (since, when taken as a premise or conclusion, \(\varphi _{0}\otimes \varphi _{1}\otimes \varphi _{2}\) evaluates \(\varphi _{1}\) in the context and at the circumstance immediately following the context in and the circumstance at which \(\varphi _{0}\) is evaluated and evaluates \(\varphi _{2}\) in the context and at the circumstance immediately following the context in and the circumstance at which \(\varphi _{1}\) is evaluated, just as \((\varphi _{0}\otimes \varphi _{1})\otimes \varphi _{2}\) does), as von Wright (1965), p. 297 himself observes \(\boxtimes \) is not (while \(\varphi _{0}\boxtimes \varphi _{1}\boxtimes \varphi _{2}\) evaluates \(\varphi _{1}\) at the circumstance immediately following the circumstance at which \(\varphi _{0}\) is evaluated and evaluates \(\varphi _{2}\) at the circumstance immediately following the circumstance at which \(\varphi _{1}\) is evaluated, \((\varphi _{0}\boxtimes \varphi _{1})\boxtimes \varphi _{2}\) does evaluate \(\varphi _{1}\) at the circumstance immediately following the circumstance at which \(\varphi _{0}\) is evaluated but evaluates \(\varphi _{2}\) at the same circumstance as \(\varphi _{1}\)). (For completeness, I note that, while basically associative in the sense explained above, \(\otimes \) is not unrestrictedly associative in the sense that, as a component of a premise or conclusion, \(\varphi _{0}\otimes \varphi _{1}\otimes \varphi _{2}\) is not intersubstitutable with \((\varphi _{0}\otimes \varphi _{1})\otimes \varphi _{2}\) salva validitate: for example, while \((P\otimes Q)\wedge (R\otimes \lnot R \otimes S)\) is consistent, \((P\otimes Q)\wedge ((R\otimes \lnot R) \otimes S)\) is not.) A related difference emerges if we restrict to \(\otimes \)’s and \(\boxtimes \)’s semantics in the absence of context change (and so to situations in which the only relevant contexts are days): letting both the context and the circumstance be \(d_{0}\), while \(\varphi \otimes \psi \) becomes indiscriminable from \(\varphi \wedge \psi \) in that its truth in \(d_{0}\) at \(d_{0}\) requires \(\psi \) to be true in \(d_{0}\) at \(d_{0}\), \(\varphi \boxtimes \psi \) retains its temporal connotation in that its truth in \(d_{0}\) at \(d_{0}\) requires \(\psi \) to be true in \(d_{0}\) at \(\mathsf {suc}_{\mathfrak {S}}(d_{0})\). Summing up in somewhat pictorial but hopefully helpful terms, while, in the presence of context change, both \(\otimes \) and \(\boxtimes \) function in such a way as to have their rhs evaluated at the circumstance immediately following the circumstance at which their lhs is evaluated, \(\otimes \) achieves this effect by immerging itself into the passage of time and having its two sides “look inward” at the days on which they are respectively uttered, whereas \(\boxtimes \) achieves the same effect by subtracting itself from the passage of time and, although having its lhs “look inward” at the day on which both it and the rhs are uttered, also having its rhs “look forward” at the day immediately following the day on which both it and the lhs are uttered. Thanks to an anonymous referee for recommending a comparison between \(\otimes \) and \(\boxtimes \).
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I let the core idea of this paper already surface in a (fittingly enough) fleeting fashion in Moruzzi and Zardini (2007), p. 180. Earlier versions of the material in the paper have then been presented in 2010 at the 3rd Arché Foundations of Logical Consequence Workshop on Propositions, Context and Consequence (University of St Andrews); in 2011, at the COGITO Philosophy of Language Seminar (University of Bologna) and at the PETAF Mid-Term Conference (University of Aberdeen); in 2012, at the GAP 8 on What may we believe? What ought we to do? (University of Konstanz). I would like to thank all these audiences for very stimulating comments and discussions. Special thanks go to Fabrice Correia, Peter Fritz, Dan López de Sa, Sebastiano Moruzzi, Julien Murzi, Eugenio Orlandelli, Peter Pagin, Jim Pryor, Steve Read, François Recanati, Sven Rosenkranz, Isidora Stojanović and several anonymous referees. At different stages during the writing of the paper, I have benefitted from an AHRC Postdoctoral Research Fellowship, from a UNAM Postdoctoral Research Fellowship and from the FP7 Marie Curie Intra-European Research Fellowship 301493 on A Non-Contractive Theory of Naive Semantic Properties: Logical Developments and Metaphysical Foundations (NTNSP), as well as from partial funds from the project CONSOLIDER-INGENIO 2010 CSD2009-00056 of the Spanish Ministry of Science and Innovation on Philosophy of Perspectival Thoughts and Facts (PERSP), from the FP7 Marie Curie Initial Training Network 238128 on Perspectival Thoughts and Facts (PETAF), from the project FFI2011-25626 of the Spanish Ministry of Science and Innovation on Reference, Self-Reference and Empirical Data and from the project FFI2012-35026 of the Spanish Ministry of Economy and Competition on The Makings of Truth: Nature, Extent, and Applications of Truthmaking
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Zardini, E. Context and consequence. An intercontextual substructural logic. Synthese 191, 3473–3500 (2014). https://doi.org/10.1007/s11229-014-0490-6
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DOI: https://doi.org/10.1007/s11229-014-0490-6