“Anything goes in; anything goes out./Fish, bananas, old pajamas, mutton, beef, and trout.” (Cole Porter, “Anything goes”)
Abstract
What sorts of sequent-calculus rules succeed in specifying a legitimate piece of vocabulary? Following on Arthur Prior’s discussion of the connective tonk, there have been a flurry of criteria offered. Here, I step back a bit, examining the role of structural rules in an inferentialist theory of meaning, and sketch a theory on which any way at all of giving left and right sequent rules for a piece of vocabulary is ok. Tonk, among other things, is a full citizen of coherent-idea-land.
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Notes
In fact, I came to this topic through its potential applications to paradoxes in the first place (see e.g., Cobreros et al. 2012, 2014; Ripley 2013a, b). Others have explored related avenues for addressing these paradoxes as well; see for examples Zardini (2008), Schroeder-Heister (2012), Weir (2005).
Notation, etc: capital Roman letters are individual formulas; capital Greek letters (those that aren’t also capital Roman letters!) are structures, with comma as sole structural connective. For much of the paper, you can think of these structures as multisets; I’ll only look for a moment at any finer distinctions between structures. \(A/B\) can be either \(A\) or \(B\).
I promised I wouldn’t say too much about paradox, but here I can’t resist: note how similar this is to the situation created by your favorite paradox.
There are ways to formulate classical logic without some structural rules, by giving different operational rules that build in the needed structural effects. See for example the system G3c in Troelstra and Schwichtenberg (2000, p. 77). The negation and conjunction rules of G3c can determine classical negation and conjunction somewhat more directly; rules of contraction and weakening are dispensed with. But the ‘weakened reflexivity’ axioms of G3c still need to be justified before we can get full classical negation and conjunction—so the situation is not really very different. Other possible ways to divide up the structural/operational work would also have to be dealt with case by case.
Although see Ripley (2014b) for worries about pulling off the mutation in certain cases.
Compare the ‘Venusian connective’ \( {\mathbb {V}}\) in Humberstone (2012, pp. 592–593). As Humberstone points out, if the Venusians allege that they have a connective \( {\mathbb {V}}\) such that \( {\mathbb {V}}\! B\) doesn’t follow from \(A \wedge {\mathbb {V}}\! B\), the right thing to conclude is not that \( {\mathbb {V}}\) is very strange—it’s that the Venusians don’t understand how our \(\wedge \) works. Similarly, if the Venusians allege that \( {\mathbb {V}}\!B, {\mathbb {V}}\!B \vdash A\) but \( {\mathbb {V}}\!B \not \vdash A\), it can’t just be that \( {\mathbb {V}}\) is very strange—instead, they must not understand how \(\vdash \) works.
You might, of course, offer a different conception of the turnstile, and hopefully that would make it clear what contraction would amount to in that context. For some purposes, contraction is clearly no good (see Restall 2000 for a bunch of examples). Crucially, though, it’s only with some conception of what the turnstile amounts to that we can have any idea of what we’re differing about, if we’re differing about contraction. And just saying ‘consequence’ isn’t enough to get this benefit—there are too many candidates for what that might amount to.
For an example such formulation of classical logic, see the calculus G1c of Troelstra and Schwichtenberg (2000, p. 61). There are plenty of different formulations, though, that vary in ways that don’t matter for my purposes here.
There are a few different versions of it out there, but with both contraction and weakening in play, some of them turn out equivalent. Since we haven’t assumed anything like compactness, some dimensions of variation still remain (see Shoesmith and Smiley 1978; Humberstone 2012); the version of cut I’m using is a particularly weak version, and even it will have to go, so that tells you what I think of stronger versions.
A rule is admissible in a sequent calculus iff whenever the rule’s premises are derivable in the calculus, then so is its conclusion.
Here an anonymous referee objects: ‘So what? If multiplicative conjunction is added to the vocabulary, contraction is no longer admissible, and if relevant implication is added, thinning [weakening] is no longer admissible.’ But this is not the case. In Sect. 3.1, I argued that contraction and weakening govern \(\vdash \) (on the particular reading in play here!) come what may. A full theory of \(\vdash \), then, will impose contraction and weakening (along with reflexivity, permutation, associativity).
If multiplicative conjunction is added to such a theory, contraction remains admissible (because derivable!). Similarly for relevant implication and weakening. Only rules that are not independently justified—like swap and cut—are at risk from new vocabulary. See also footnote 9.
To answer an anonymous referee’s query: this thus requires no restriction on the number of times a new piece of vocabulary can occur in its rules, or on how deeply embedded it might occur, or on whether it occurs on both sides of the turnstile in the conclusion-sequent. As long as cut is not present, as long as no old rules eliminate the new vocabulary, and as long as the new vocabulary itself occurs in the conclusion-sequent of its rules, these rules will be conservative. See also Kremer (1988) for discussion.
This referee also asks whether it is true in general that a connective definable by an arbitrary set of conservative rules (in the present setting) has an equivalent definition in terms of left and right sequent rules, adding ‘A positive answer to this general question would be a strong reason to restrict one’s attention to left and right rules’.
On a certain understanding, the answer is indeed positive—but the understanding may well be unsatisfying. Suppose we read ‘equivalent’ as ‘determines the same consequence relation’. Consider a calculus \(C\) determining a consequence relation \(\vdash _C\), conservatively extended with rules for a new piece of vocabulary to yield a calculus \(C'\) determining a consequence relation \(\vdash _{C'}\). Consider the set \(X\) of sequents valid in \(C'\) but not in \(C\); since the extension to \(C'\) was conservative, the new vocabulary occurs in each sequent in \(X\). Now, add to \(C\), as the rule for the new vocabulary, a rule allowing us to conclude any sequent in \(X\) from no premise-sequents—or, what amounts to the same, add a new rule for each sequent in \(X\), allowing us to conclude that sequent from no premise-sequents. Nothing besides sequents already in \(X\) will follow from this addition, since \(C'\) was conservative.
One reason this may be unsatisfying is this: there may be no schematic formulation of the addition. Another is that there may be members of \(X\) involving the new vocabulary on both the left and right, or deeply embedded. Another—and this seems to me the most important—is that this is a very weak notion of equivalence between sequent systems. I don’t know to what extent these features are avoidable.
In Ripley (2013a), I consider the argument as presented in Restall (2005). In all cases, Restall considers so-called ‘additive cut’ rather than the ‘multiplicative’ version I’ve used here, but since I agree with Restall that contraction and weakening are in force for \(\vdash \), this makes no difference.
Enough structural rules are uncontroversial here that this is well-defined even though positions use sets.
To get from cut to RP, assume that \(\langle \Gamma \cup \{A\}, \Delta \rangle \) is out of bounds; that is, \(\Gamma , A \;\vdash \; \Delta \). Now we must show, using cut, that \(\langle \Gamma , \Delta \rangle \) is equivalent to \(\langle \Gamma , \Delta \cup \{A\} \rangle \); that is, that \(\Gamma , \Gamma ' \;\vdash \; \Delta , \Delta '\) iff \(\Gamma , \Gamma ' \;\vdash \; \Delta , \Delta ', A\), for any \(\Gamma ', \Delta '\). LTR: \(\Gamma , \Gamma ' \;\vdash \; \Delta , \Delta ', A\) follows from \(\Gamma , \Gamma ' \;\vdash \; \Delta , \Delta '\) by weakening on the right. RTL: \(\Gamma , \Gamma ' \;\vdash \; \Delta , \Delta '\) follows from \(\Gamma , \Gamma ' \;\vdash \; \Delta , \Delta ', A\) together with \(\Gamma , A \;\vdash \; \Delta \) (which we assumed) by cut and contraction on both sides.
To get from RP to cut, assume the premises of a cut: \(\Gamma \;\vdash \; A, \Delta \) and \(\Gamma ', A \;\vdash \; \Delta '\). By Restall’s principle applied to the second premise, \(\langle \Gamma ', \Delta ' \rangle \) is equivalent to \(\langle \Gamma ', \Delta ' \cup \{A\} \rangle \). Thus, position \(X = \langle \Gamma \cup \Gamma ', \Delta \cup \Delta ' \rangle \) is out of bounds iff position \(Y = \langle \Gamma \cup \Gamma ', \Delta \cup \Delta ' \cup \{A\} \rangle \) is out of bounds. But we have assumed as our first premise that \(\langle \Gamma , \Delta \cup \{A\} \rangle \) is out of bounds; by weakening, it follows that position \(Y\) is out of bounds; and so position \(X\) is out of bounds too. That is, \(\Gamma , \Gamma ' \;\vdash \; \Delta , \Delta '\); this is the conclusion of the cut.
I ignore in this discussion Dummett’s (1991, p. 250) notion of ‘total harmony’. This is conservativeness, which I’ve already discussed. His ‘instrinsic harmony’ is closer to the stream of literature I’m pointing to.
If it is a requirement at all. Consider for example Read (2008, p. 294): ‘[P]erfectly decent connectives can be governed by inharmonious rules. Such cases are unhelpful, in obscuring the meaning of the connective, but the lack of harmony is not itself a source of incoherence, nor does it mean that the rules do not define the meaning of the connectives as logical.’
There are other theories too about what harmony is required for. For example, there is the claim in Dummett (1991, p. 247) that rules must be harmonious in order to be self-justifying. This claim too does not intersect my topic here, since I make no claims about self-justification.
An anonymous referee wonders just what \( {\texttt{tonk }}\)’s meaning is. I reckon the question is misplaced, just as the corresponding question about, say, conjunction would be. We have a full description of the way \( {\texttt{tonk }}\) interacts with norms governing assertions and denials; this is enough—just as it is for conjunction. If we want to find some thing to be the meaning (whether of \( {\texttt{tonk }}\) or of \(\wedge \)), there are multiple candidates: the rules themselves, the constraints the rules put on us, some abstract object determined by the rules, the patterns of conversational norms the rules encode, etc. I doubt it is productive to fix on a single answer.
An anonymous referee gives the following rules for a unary connective \( {\star }\), suggesting that it shows a problem with the present approach.
$$\begin{aligned} \star \hbox {L}:\qquad \frac{\Lambda ,A\;\;\vdash \;\;\Delta }{\Lambda ,\star A\;\;\vdash \;\;\Delta }\qquad \star \hbox {R}:\qquad \frac{}{\Lambda \;\;\vdash \;\;\star A, \Delta } \end{aligned}$$But this is no more of a problem than \( {\texttt{tonk }}\) was; it’s just a new connective, perfectly well-specified. It blocks admissibility of cut in any sensible system, but that’s fine.
The referee points out, by way of objecting, that \( {\star }A \;\vdash \; {\star }B\) is derivable given these rules (in fact, given just \( {\star }\)R), for any \(A, B\). That, though, just can’t be a problem: that much holds as well of uncontroversially legitimate bits of vocabulary, for example \(\square \) in the modal logic Ver (see Hughes and Cresswell 1996, p. 66), or classically where \( {\star }A\) is interpreted as \(A \supset A\), or as \(A \wedge \lnot A\).
For helpful discussion and comments, many thanks to the audience at the MCMP 2012 conference on Paradox and Logical Revision, Suzy Killmister, and Greg Restall. Particular thanks to Julien Murzi and four anonymous referees, whose detailed comments on previous versions of the paper were invaluable.
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Ripley, D. Anything Goes. Topoi 34, 25–36 (2015). https://doi.org/10.1007/s11245-014-9261-8
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DOI: https://doi.org/10.1007/s11245-014-9261-8