Situating Consequence
5.3. Content, Validity and Logical Revision
Now, there is an obvious line of resistance to my argument above. I seem to be arguing for the (confused) conclusion that considera- tions to do with content could falsify claims about a discipline that has traditionally prided itself on itsdisdainfor content. Wasn’t logic supposed to transcend all particulars, and haven’t I just quoted that Fregean remark myself?
Well, yes, but things are not that simple.
Let me try to further motivate my insistence on the pivotal role of CET with a few, very basic examples from the textbook and from the history of logical revision.
I have already quoted Frege’s claim that his logic was superior to Boole’s because of its better discriminatory powers with respect to inference.
DETL is most readily associated with Frege’s work of course. I think we should be reluctant to
think it is an easily discarded thesis however. Frege was remarkably forthright in confronting the issue head on. Most writers aer him have for the most part ducked the issue. Or, if discussing it at all, have set it aside with a shrug of the shoulder (see e.g. Lewis 1982: §3). Russell (1923) is one (partial) exception: but even he ducks out at the last minute, appealing to some sort of Humean naturalism. Williamson (1994: ch. 2, 6, 7) tackles the issue head on, and so does his (2000). As far as I can tell, Williamson’s position amounts to a sophisticated version of LIR (see §5.4). To that extent it does suffer from the problems I discuss later in the text.
e requirement is most explicit in Lewis (1982: 107). Strikingly, his discussion in the follow-
ing two pages concedes that there is no guarantee CET can be made to work.
Situating Consequence | For my purposes, it is crucial to note that Frege articulated his claim along three lines: i)that his logic had greater expressive powers; ii) that his logic could account for alargerclass of intuitively valid pat- terns of inference; andiii)that his logic achieved the dual (and simul- taneous) task of codifying inferenceandconcept-formation processes (whereas Boolean and Aristotelian logic had topresuppose concepts were already formedin a particular wayand in good standing).
It is received wisdom that Frege’s work represented the greatest ad- vance in the history of modern logic. But it was an advance trig- gered by (and implemented through) considerations to do withcon- tent(through that advance, a better grasp of content was to be had).
Crucially, content-individuation and codi cation of inference are co-occurring acts; and I am urging that we cannot separate worries about content from claims about logicality.
Let’s see why.
According to semanticism, the logical constants are re-absorbed into the form of the expressions they govern. What counts asthelog- ical form of an expression, however, is clearly a function of our best conception of content in force at a time.
Defenders of modal logic, for instance, have claimed that Frege (: §) missed a trick in dismissing modal content as belonging merely to a theory of force because by doing so he overlooked the effect of modality on inferential potential and hence on content.
ere is therefore a very close connection between content- individuation and the range of inferences we are interested in de ning as logical—quite clearly, the proper description of the dispute here is
See the already quoted Frege (1880/81: 14-5). Frege speaks here of concept-formation but
elsewhere (1897b: 127) stresses that we grasp, rather than generate, thoughts. I take ‘formation’ to refer to the process ofmodellingconcept-formation in our logic.
Well, that it is received seems undoubted. at it is wisdom has been questioned by e.g. Put-
nam (1982) and Boolos (1994).
See e.g. Frege (1882: 75/55; 74/53), where the role of logic is given precisely as that of split-
ting content into its “ultimate components”, thus showing “the manifold logical relations that join thoughts together”. Sullivan (2004: 696) has worried about this aspect in Frege’s conception of logic, since it seems impossible to settle inferential properties independently of (and in advance of) settling questions of identity of content.
I am not claiming that from the undoubted fact that our conception of logicality is indexed to
our conception of content it follows that logicality is anintrinsicallyrelative question. In fact I am not sure we can even make sense of theabsolute/relativecontrast in this (or in any other) case. I am simply arguing for the point that logical revision (and the attendant re-drawing of the logical/non- logical distinction) is a content-driven process. at’s why CET matters to the issue. Burge (1998: 354) is useful in this respect (the determination relation between logical structure and the study of inference runs both ways).
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that Frege thought modal operators do not express a speci c content but merely ag the speaker-relative degree of epistemic entrenchment with respect to a given proposition; advocates of modal logic by con- trast think there is such a thing as modal content and (or rather: be- cause) it does affect inference in ways that rightfully qualify as logi- cal.
is sort of dispute concerns which (possibly contentious) schemata we ought to incorporate in our logic. But consider also a very basic example familiar from the textbook involving perfectly uncontroversial schemata: p ⊢ p∨q, we are informed, is either an instance of the schemaφ⊢ψor ofφ⊢φ∨ψ.
We are then told that only the latter is the appropriate schema under which to subsume sentences of that form and the moral is supposed to be that we need to uncoverjust as muchinternalstructureas is re- quired to give a plausible account of validity (the rst schema would be insensitive to a logical particle in the conclusion which ensures va- lidity: content was carved too coarsely under that schematisation). Quite. But given existing controversies with regard to the valid- ity of certain contentious schemata, I maintain that it is still an open question whether we have cracked this particular nut (the one relating to the appropriate connections that have to obtain between schemata and their instantiations).
Consider, that is, live issues such as the relevantist logician’s attack
One might object that this problem affects only Fregean views of content, where content in-
dividuation is given in inferential terms. I deny this is the case at all. Any account of content in- dividuation will suffer from the same conceptual stumbling block, for logical inference will always be de ned in the same terms as those chosen for giving identity criteria for content. Wealways
draw the logical/non-logical demarcation in the very act of choosing identity criteria—as a brief examination of accounts given in terms of concept possession, acceptance conditions or even plain truth-conditions will show.
Oen enough, this sort of example is used to introduce students to the transition between
propositional and predicate logic (i.e. from a coarse to a ner-cutting logic).
Here’s yet another problem: logic’s main claim topracticalutility is its (purported) ability to
makeconfusingNL argumentsclearerby bringing out theirstructure—see e.g. Kalish and Montague (1964: 14) and Forster (2003: 3-4). Since there seems to be no way of isolating structurally impor- tant aspects of content independently of our views of logicality, it seems as if this task cannot be carried out either. is is so because the individuation of the logical components of those aspects
cannotbe neutral with respect to our intuitions regarding whatshouldcount as a valid argument. Logic’s ability to clarify the structure of argumentation, that is, essentially depends on our (pre- existing) judgements regarding validity: onlyaerwe have settled on a view regarding logicality can we then use the tools thus individuated to clarify muddled arguments. e paradox then is that we are supposed to establish what counts as valid reasoning by making manifest what is hidden be- hind the confusing cloak of NL expressions (and the haze of use), but we can’t do that until we have demarcated the logical/non-logical divide, which in turn presupposes we havealreadyclari ed the muddle those tools were supposed to dispel.
Situating Consequence | on the paradoxes of implication and McGee’s purported counterex- amples tomodus ponensreasoning.
In the latter case, much of the controversy is over whether McGee’s cases are genuine instances of the MPP schema. e claim, that is, is that there are hidden bits of content that once made manifest do show that everything is all right (the defender of MPP thinks it in- conceivable that MPP could fail: hence a better account ofcontentis proposed to salvage the logicality of the schema).
In the case of the relevantist critique, the proposal is that the rules for the conditional (and other operators) need to be restricted. But again, the relevantist’s broad requirement of variable-sharing is rst and foremost one to do with the content of expressions (we need to cut them more nely than on the classical view: we need to represent moreof their content in the logical form that regulates that particu- lar inferential step; and this time the ner account of content that is proposed is at the service ofdownsizingthe class of validities).
I could easily give more examples and in greater detail, but I trust that the general point is fairly clear.
Faced with questions about validity thrown up by controversial examples, logicians will either insist that we cut the content of the propositions involved more nely or that we revise our class of va- lidities so as to incorporate fresh patterns determined by a new view about the content of (or rather: about the form/content demarcation for) the propositions involved.
Sometimes the claims about missing content are not intended as a defence of MPP. See e.g. Ly-
can (2001: 64). Sometimes they are (even when ostensibly about context-switches and assertibility), as in Gauker (2005: 86-7). In general, the discussion of McGee conditionals shows thatphilosophers standardly assume that we can pack an awful lot into content. MPP is salvaged, that is, by showing that acceptance of a certain “bare” sentence always involves acceptance of a complex set of beliefs somehow attached to the content of the sentence—see for instance the discussion in Bennett (2003: §61).
For the requirement see e.g. Brady (2006: 4ff.). For an unusual take on relevance see Lewis
(1988). e discussion in Burgess (2009: ch. 5) is instructive. Burgess rst notes that relevantistic logic (as he calls it) insists there betopic overlapacross an inferential step for it to be valid. Using Craig Interpolation eorem, he easily shows that even the classical logician can ensure (trivial) topic-overlapin all but thedegeneratecases(the ones involving the paradoxes of material implica- tion). Burgess then concludes that because of the indispensability to mathematical practice of those degenerate cases there is no reason to insist on the topic-overlap requirement in full generality. So, we have here apragmaticexplanation (indispensability) for the insistence that our logic need not discriminate content as nely as the relevantist logician wants.
One more example, vagueness. ere have been recent attempts (e.g. Cobreroset al.2010 and
Zardini 2008) to impose local restrictions on some structural features of the CR, e.g. transitivity, to deal with vague terms. Yet, these attempts face the problem that unless wecandiscriminate vague terms from their precise counterparts, if any, there is no way to implement the proposal: before we can disambiguate the CR we need to have settled whether or not the pieces of content concerned
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To sum up: logic may indeed be de ned as a topic-neutral disci- pline. at claim, however, is one we can legitimately makeonly when taking logic asnormalscience.
Whenever pressure is applied on the precise shape of its laws, logic must (and typically does) pay heed to content. In short, logical paradigm-shis are content-driven, they are shis in the ways we take ourselves to be grasping content, in the ways we model our grasp of content (indeed, in the way we read UaGS).
Logical revision, then, is rst and foremost revision of the tools we use to individuate content (each time we move bits of our vocabulary across the non-logical/logical divide, each time we impose or li re- strictions on existing logical particles, we are thereby changing not just the ne-grainedness of our logical tools but also our conception of the content they operate upon).
If CET cannot be implemented, however, I do not think we can be as con dent as we seem to be that the standard account is telling the whole story about logic (how can we legitimately draw the logical/non-logical divide in the absence of an assurance that we have individuated the content of the propositions involved precisely enough?).
Before I can move on, there is an immediate objection to the argu- ment of this section which I need to address.
One could argue that all that logic is required to do is identify, for each piece of content,initial segmentsof their CET expansion.
It will then be a matter of deriving general logical principles from the appropriately logical patterns that emerge from examination of those segments. As long as we can agree on the character and struc- ture of the initial segments it doesn’t much matter whether or not we
are vague. A further problem concerns thead hocnature of the restrictions on transitive chains (3 elements for Zardini, 2 for Cobreroset al.). Why not 4 or 3? Or 6 or 243? And how can we tell if we can get away with chaining a hundred pieces of reasoning (for by their own admission transitivity is notalwaysbad)? ese, it seems to me, are insuperable problems. No formal approach will ever give a theory-independent answer to these questions.
In Kuhn’s (1962) sense.
If LSR is to hold, the content it oversees must be made up of the subjects of our attitudes. We
may argue about the needed level of transparency, but reasoning is moving between evidence-based beliefs, and hencesomedegree of transparency is needed. ).
Logicality is oen de ned as invariance under permutation. Wouldn’t that escape my CET-
based argument? Well, as is familiar, this proposal helps give necessary conditions for logicality but it over-generates. Casanovas (2007) is a good recent discussion (with references to the relevant literature). As far as I can tell, at present there is no de nitive result in this eld that could secure a precise and uncontroversial demarcation of the logical/non-logical divide.
Situating Consequence | cancompletethe expansion process,
e objection has initial plausibility, but if one goes back to my discussion of ‘and’ in the preceding chapter, it should be clear that we have reason to doubt that we could individuate even the initial segments with the sort of precision assumed by the logician (for the worries we raised there concerned precisely the individuation of the initialfragment; the questions before us were: is theresuch a thingas a temporal/instrumental/causal conjunction operator? isitproperly classi ed as logical?—and those questions arose well before doubts about CET-completionwould ensue).
e fact is, we test validity by appeal to examples stated in NL. Counterexamples are formulated in NL. Relevant facts about con- textual embedment and its impact on assertibility are spelled out by clauses given in NL.
If CET is a chimera, so is the idea of precision even with respect to initial segments of the expansion process.
In case you are still unconvinced, let me say a little more. 5.3.1. Disambiguation and Inference
I have been arguing that the act of marking out the division between the logical and the non-logical iseo ipsothe act of individuating cer- tain pieces of content as the pieces theyareby virtue of their inferen- tial properties (or acceptance/possession conditions, or contribution to truth-conditional content). We do both things at the same time. e two acts are conceptually inseparable.
e difficulty was: how can we establish that the logical vocabulary is appropriately insensitive to theparticularitiesof content if we can- not individuate contentsemanticallyin the rst place (especially: not in a logic-independent manner)?
Here’s a strategy to deal with the problem, suggested in Lewis (: §): suppose we abandon the notion of truth-preservationsimpliciter and adopt a notion oftruth-preservation on some disambiguation of the sentences involved. We assume (or rather: pretend) that CET can be implemented, and implemented uniformly. We then check for truth-values distributions under (potential) disambiguation. Prob- lem sorted.
Indeed, those doubtsspeci callyconcerned whether anyfurthersteps in the expansion process
were needed (e.g. from ‘and then’ to ‘and then as a result’, and so forth).
I’m aware of no-one who has given a convincing proof that CETcanbe implemented—and
the close of Lewis (1982) is remarkably (and disarmingly) indecisive in that respect: no such proof
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Well, I don’t think so. is response, I contend, would be far too quick and in fact question-begging.
Firstly, Lewis (: ) simply stipulates that wecanreach a stage in the disambiguating process where we have settled on a range of truth-values assignments under disambiguation. But that is precisely what the anti-CET objection challenges and it cannot be assumed without argument (and neither Lewis nor the original Quinian strat- egy of eternalisation have provided one).
More importantly, the Lewis view assumes that we can make sense of sameness of disambiguation, a notion that we require in order to enforce the principle of uniform substitution.
But the anti-CET proponent will challenge that assumption too, for whether there can benon-syntacticcriteria for what can count as the samesentence is precisely the point in dispute.
Given the CET worries, why shouldagiven stage in the expansion process qualify as expressing thesamedisambiguation across an in- ference if what supposedly does the disambiguation is just more ex- pressions?
If we need tostart the expansion process in the rst place, we are conceding that signs are unstable (that syntax alone cannot disam- biguate); but then it’s obscure why we should take any stage in the completion process to be anylessunstable.