CET and the Structure of Proof

is, in effect, is a CET-equivalent claim applied to the notions of proof and inference. It amounts to saying that behind the haze of use logical thinkingisinstantiated and it is logic (of course!) that dictates its essential features.

All that needs doing to vindicate (sound) everyday reasoning, then, is to ll out the various gaps that economy of conversation leaves open and then show that the properly expanded pieces of contentandprop- erly expanded pieces of reasoning conform to those principles that we have identi ed at the formal level.

So, just as the surface structure of NL sentences is generallydefec- tiveto the extent that in the absence offurtherspeci cations it leaves its content underdetermined, so is the surface structure of inference (we reason by leaps and bounds, but the gaps, on the CET view, can alwaysbe lled appropriately).

Accordingly, CET does double duty as the essential tool in our ef- forts fully to capture the formal (inferential) properties of thoughts (as the senses of eternal sentences) while also allowing us to capture the equally precise structure of proof and inference (as appropriately disciplined movement between precisely speci ed thoughts).

e CET-equivalent claim with respect to proof (and reasoning in general) is thus that any piece of informal reasoning can be trans- formed into an extensionally equivalent formalisation where no step is omitted and all transitions are shown to be in accordance (or oth- erwise) with the basic laws of logic.

Properly seen, then, CET is not just a technical claim about NL semantics: it also encases an ideal of rationality astheoretical reason (whatever reasoning move we make is translatable into a fully explicit, rigorously assessable form; whatever piece of content we entertain

_{ere are times when it seems as if Frege is saying that only thinking that accords with logical}

lawsisthinking properly speaking (i.e. atruth-directedactivity), see e.g. Frege (1893/1998: xv) and (1897b: 128, 149).

_{A claim of this sort is explicitly mooted in e.g. Priest (2006: 198).}

_{Indeed, right from the start Frege made clear that his great technical achievement, namely,}

the making explicit of the hitherto glossed-over structure of proof, was also a profoundlyepistemic

one of hugenormativeimport, for it traced the ultimate justi cation for our privileged forms of reasoning back toa priorilaws of logic that determine what counts as properly rational thinking, indeed as thinking at all.

Situating Consequence |  can be made fully univocal).

In fact, the two roles of CET cannot be torn asunder: a concep- tion of content as the fully determinate inferential potential of the thoughts we standardly entertain (implemented by the notion of eter- nal, fully de-contextualised sentences assigned absolute truth values by the semantics) goes hand-in-hand with a conception of inference as codi ed by precisely stipulated relations holding between eternal sentences.

Let me assemble these re ections into one principle:

Determinacy esis for Logic(DETL): LSR⇒CET

What the thesis says is that any logic that is Fregean in scope, that is, any genuinelyuniversallogic, requires pieces of determinate content to operate upon. e threat here of course is contraposition—without determinacy for the relata of the consequence relation (CR) (andfor the logical constants embodying its properties), we cannot systema- tise reasoning.

If CET is unimplementable, then the claims of logic becomeat best conditional in character: shouldwe succeed in making content deter- minate, then the CR will have a eld to operate on; otherwise it will be empty.

Let me now recall two claims I made in earlier chapters. I argued that CETisunimplementable and that the logical constants too dis- play indeterminacy features (or rather: that attempts to make their content determinate end up in dead alleys).

If so, DETL fails. ere are no candidate contentful items that we can take our non-logicaland logical vocabulary to range over. And if content has to be fully determinate, then unless underspeci city is

_{Let me anticipate my sympathy for the view of rationality sketched in Ryle (1962) and Hacking}

(1983). On that view, CET is wrong on both counts. More about this in chapter 6.

_{As remarked in Black (1937: 77). Let me wave away the classic objection (raised by Hartry}

Field in conversation) that asks why we should worry about determinacy of content for thesub- stituentsof the non-logical constants. e very notion of topic-neutrality presupposes that nothing but the meaning of the logical constants will determine whether an inference counts as genuinely logical. Indeed, we can useobscurepieces of content toexemplifylogicality—classic case: the move from ‘Every tove is slithy’ and ‘Alice is not slithy’ to ‘Alice is not a tove’ (Bell and Machover 1977: 5). As long as interpretation is kept xed across different tokenings, we are told, everything is all right. Reply: the point I am considering is whether we can so much as make sense of what it meansto keep interpretation xed synchronically, let alone diachronically (for worries aboutthat, see e.g. Williams 2008: §II). Similarly, it is not clear that we can prevent the indeterminacy worries from extending to what we normally class as logical vocabulary. If indeterminacy-as-instability is an essential feature of the sign, why should logicalsignsbe immune to it?

 | Chapter 

banished we can entertain no content at all.

e upshot is that we have to give up on LSR altogether and restrict the scope of logic to mathematics and the study of the properties of logical systems at a purely formal level.

Logic would thus turn out to be merely a highly specialised tool used in meta-reasoning of a restricted kind, rather than the most gen- eral of disciplines.

Note here the connection with the discussion of NS back in chap- ter : to neutralise worries about nonsense, semantics has to assume that meaningfulness questions have been settled before it gets started; similarly, logic will have to assume that underspeci city questions have been settled before it can get going.

Uncertainty on both points makes both semantics and logic con- ditional enterprises—somethingelse does the dirty job for them, and their claim to full generality is thus severely curtailed. Not quite the picture the standard view requires, I’d say.