Today, I’d like to discuss the **very** general question of what logical theories are theories *of*, if anything at all, and to inquire into the appropriate terminology to be used in such contexts. In a forthcoming JPL paper (co-authored with Edgar Andrade-Lotero), we start with the following remark:

Let us start with a fairly uncontroversial observation. Generally speaking, a logical system can be viewed from (at least) two equally important but fundamentally different angles: i) it can be viewed as a pair formed by a syntax, i.e. a deductive system, and a semantics, i.e. a class of mathematical structures onto which the underlying language is interpreted; or ii) it can be viewed as a triad consisting of a syntax, a semantics and the target phenomenon that the logic is intended to capture. In the first case, both syntax and semantics are viewed as autonomous mathematical structures, not owing anything to external elements. In the second case, both syntax and semantics are accountable towards the primitive target phenomenon, which may be an informally formulated concept, or even phenomena in the ‘real world’ (e.g. logics of action, logics of social interaction, quantum logic etc.). Indeed, in the second case, both syntax and semantics seek to be a ‘model’ in some sense or another of the target phenomenon.

In Chap. 12 of *Doubt Truth to be a Liar* Graham Priest draws a similar distinction between pure vs. applied logics. As I read him, the distinction is not really intended to differentiate logic systems as such, but rather to outline different attitudes one can have towards a logical system. He then goes on to argue that, from the applied logic point of view, the canonical application of logic is *correct reasoning*. In a similar vein, Paoli (JPL, 2005) borrows Quine’s distinction between immanent and transcendent predicates, and remarks:

According to Quine, in fact, logical connectives are immanent, not transcendent. There is no pretheoretical fact of the matter a theory of logical constants must account for; rather, the vicissitudes of a connective are wholly internal to a speciﬁed formal language, to a given calculus. There is nothing, in sum, that precedes or transcends formalization, no external data to “get right”.

The issue is very general and does not concern logical connectives in particular. The key opposition is between the *internal* features of a logical theory, and its potential relation with *external* phenomena, which the logical theory purports to be a model*of*. Although Quine (in *Philosophy of Logic*) seemed to mean something slightly different with his notions of immanent and transcendent predicates, I find Paoli’s appropriation of the terminology quite fitting.

My question now is: whenever there is something transcendent that a logical system is intended to capture, what is the appropriate terminology to refer to these external phenomena? In practice, the adjectives ‘pre-theoretical’ and ‘intuitive’ are often appended to whatever target phenomenon that a given logical system is ‘about’ (the ‘being about’ part is also what needs to be explained). Presumably, the idea is that the phenomenon is conceptually prior to its systematization within the theory, and this seems right according to the 'transcendent' approach, but there are problems.

Elsewhere, I have objected to the qualification of ‘pre-theoretical’ as attributed to the notion(s) of logical consequence which is (are) presumably the target of the familiar technical accounts of logical consequence (proof-theoretic, model-theoretic). The trouble with the terminology is that it suggests a theoretically neutral target phenomenon, emerging from ‘common, everyday’ practices (terms used in Tarski’s seminal 1936 paper). In truth, the notions in question are inherently couched in robust theoretical frameworks –-T. Smiley (1988) and P. Smith (2010) make similar points; the latter specifically criticizes Field’s misconception of what is ‘squeezed’ in a squeezing argument. My general worry is that, by describing these notions as ‘pre-theoretical’ and ‘intuitive’, we seem to be suggesting that they are transparent and unproblematic, whereas what is often required to make philosophical progress in these discussions is precisely a deeper understanding of the target phenomenon as such. (Shapiro’s and Prawitz’s papers in the 2005 *Handbook *are good attempts in this direction.)

So ‘pre-theoretical’ and ‘intuitive’ are problematic; what could possibly be used instead? I’ve been contemplating using ‘extra-systematic’ or ‘extra-theoretic’, but they don’t sound all the way right either. In a sense, perhaps there is no terminology to be used across the board, precisely because the target phenomena of different logical systems may be widely dissimilar kinds of phenomena. Some of them may come closer to what one could describe as ‘intuitive’ (e.g. the truth predicate as used in everyday language), while others will be grounded in a considerable amount of theorizing (e.g. the validity predicate, extensively discussed in blog posts recently – my general position is that it is a conceptual mistake to treat the truth predicate and the validity predicate on a par, even though there are interesting technical connections). So for the time being, I continue to use the vague and uninformative phrase ‘target phenomenon’, which is more of a place-holder, but this may well be what is required here.

(Alternatively, one may simply maintain that there are no target phenomena that logical systems seek to capture in any interesting way, i.e. that *everything *in a logical system is an immanent matter. Although frustrating for a variety of reasons, this remains an available move for the theorist.)

## Recent Comments