According to the logicality of language hypothesis (hereinafter, LLH), logical considerations are relevant for syntactic formation to the point that they are needed to explain some ungrammaticalities (Gajewski 2002, 2009; Fox & Hackl 2006; Chierchia 2013, 2021; Abrusán 2014; Del Pinal 2019, 2021). In other words, the syntactic unacceptability of some linguistic constructions is traced back to their logical status, suggesting that speakers judge a word-sequence as ungrammatical or ill-formed when it is always false or always true. The aim of this work is to examine the recent literature that attempts to offer alternative explanations and test the solidity of the objections it proposes. In particular, we will look at cases concerning identity and co-binding.

Chief evidence in support of LLH comes from word-sequences, such as (1)–(4), judged ungrammatical qua contradictory.

(1) ^*Some students but John passed the exam

(2) ^*There are any cookies left

(3) ^*How fast didn’t you drive?

(4) ^*How did John regret that he behaved at the party?

Other word-sequences, such as (5)–(7), are judged ungrammatical qua tautological.

(5) ^*There is every fly in my soup

(6) ^*Mary is taller than no student is

(7) ^*At least zero students smoked

However, in general, contradictions and tautologies are not ungrammatical.

(8) This table is red and it is not red

(9) It is raining or it is not raining

Within LLH, the syntactic acceptability of (8)–(9) is not questioned. As a result, the asymmetry between the ungrammaticality of (1)–(7), due to their logical status, and the grammaticality of (8)–(9), notwithstanding their logical status, has to be accounted for. This is the so-called «analyticity puzzle».

According to an early articulation of LLH (Gajewski 2002, 2009; Fox & Hackl 2006; Chierchia 2013), one must assume a purely linguistic logic and associate linguistic constructions with particularly austere representations, which reveal truly contradictory or tautological contents only in the case of constructions later judged ungrammatical. According to an alternative articulation, on the other hand, contradictions and tautologies are ungrammatical insofar as it is not possible to make them informative, by modulating their lexical content (Del Pinal 2019, 2021; Pistoia-Reda & Sauerland 2021; Pistoia-Reda & San Mauro 2021).

There have been attempts to explain the rejection of (1)–(7) typologically, i.e., in terms of a failure of composition (Abrusán, Asher & Van de Cruys 2021). To that extent, (1)–(7) would be similar to examples of semantic anomaly, such as (10)–(11).

(10) #Tigers are Zermelo-Fraenkel sets

(11) #My toothbrush is pregnant

According to LLH, (1)–(7) are only superficially uninterpretable: when analysed, they receive an interpretation, which is that of being either contradictory or tautological. According to Abrusán, Asher & Van de Cruys (2021), (1)–(7) are instead uninterpretable for, in building up their semantic representation, an insuperable semantic problem is encountered.

The critical objective of Abrusán, Asher & Van de Cruys (2021) is primarily the early articulation proposed by Gajewski (2002, 2009). However, according to them, the alternative articulation proposed by Del Pinal (2019, 2021) also faces problems. In particular, it could not explain the syntactic acceptability of (12) where, they say, we insist on making the two predicates in (8) be identical, hence leaving no space for meaning modulation.

(12) This table is red₁ and not red₂ and the property red₁ is identical to the property red₂

However, sentences concerning identity are not always interpreted as strict identities. In other words, they do not always entail that the pre-copular and the post-copular DPs are co-extensional (Fiorin & Delfitto 2024). Indeed, in a pragmatically supportive context, even possessive binding becomes acceptable:

(13) Anne_i is identical to her_i mother

In (13) we do not take Anne and her mother to be co-extensional, but to share certain typical characteristics (e.g.: they behave in the exact same way, or they have a striking physical resemblance). We will argue that, similarly, (12) does not necessarily entail that red₁ and red₂ are co-extensional, therefore pragmatic considerations can still make the word-sequence non-contradictory.

Furthermore, we will argue that moving from logical identity to logical identification is the key to account for other examples, involving co-binding, considered problematic for language logicality, such as (14) and (15).

(14) John_i is smarter than himself_i

(15) John_i is himself_i

To conclude, non-typological explanations of (1)–(7) have not yet been decisively ruled out, and the typological explanation offered by Abrusán, Asher & Van de Cruys (2021) does not represent a defeat for language logicality but a redefinition of it: given that the peculiarity of (1)–(7) is due to types denoting a context-invariant logical meaning, logical considerations are relevant for syntactic formation after all.

References

Abrusán M., 2014, Weak Island Semantics, Oxford, Oxford University Press.

Abrusán M., N. Asher & T. Van de Cruys, 2021, «Grammaticality and Meaning Shift» in G. Sagi, J. Woods (eds.), The Semantic Conception of Logic, Cambridge (MA), Cambridge University Press, pp. 249–276.

Chierchia G., 2013, Logic in Grammar: Polarity, Free Choice, and Intervention, Oxford, Oxford University Press.

Chierchia G., 2021, «On Being Trivial: Grammar vs. Logic» in G. Sagi, J. Woods (eds.), The Semantic Conception of Logic, Cambridge (MA), Cambridge University Press, pp. 227–248.

Del Pinal G., 2019, «The Logicality of Language: A New Take on Triviality, “Ungrammaticality”, and Logical Form», Noûs, 53, 4, pp. 785–818.

Del Pinal G., 2021, «The Logicality of Language: Contextualism vs. Semantic Minimalism», Manuscript, University of Illinois.

Fiorin G. & D. Delfitto, 2024, «Binding in copular sentences and the logic of identification», Presentation delivered at IGG49.

Fox D. & M. Hackl, 2006, «The Universal Density of Measurement», Linguistics and Philosophy, 29, 5, pp. 537–586.

Gajewski J., 2002, «L-analyticity and Natural Language», Manuscript, MIT.

Gajewski J., 2009, «L-triviality and Grammar», Handout, UConn Logic Group.

Pistoia-Reda S. & L. San Mauro, 2021, «On Logicality and Natural Logic», Natural Language Semantics, 29, 3, pp. 501–506.

Pistoia-Reda S. & U. Sauerland, 2021, «Analyticity and Modulation: Broadening the Rescale Perspective on Language Logicality», International Review of Pragmatics, 13, 1, pp. 1–13.

Following recent usage, I’ll use ‘coordination’ as a term for the relation that Fregeans have called ‘sameness of sense’. For the Fregean, sameness of sense is a relation that can hold between representations of the same object. It has a special rational status: when coreferential representations share a sense, their coreference is rationally-relevant, in that it is a determinant of relations of entailment and inconsistency. For example, the Fregean holds that the difference between the subject who (rationally) believes that Dylan is a musician and that Zimmerman is not a musician, and the subject who (irrationally) believes that Dylan is a musician and that Dylan is not a musician, is that in the latter case the two representations of the relevant man are presented via the same sense, and thus their coreference is transparent to the subject.

To systematically characterize the way that the distribution of senses determines rational relations between representations, Fregeans (at least implicitly, and often explicitly) turn to standard first-order logic. They take the formal significance of recurrence of constants in a first-order language to model the rational significance of recurrence of senses in thought or language.

One benefit of using ‘coordination’ over ‘sameness of sense’ is that it allows us to explicitly raise the question of whether the relevant relation is transitive (whatever senses are, sameness of sense is a transitive relation). It is a substantive question whether, when the coreference of a and b is rationally-relevant, and the coreference of b and c is rationally-relevant, the coreference of a and c must also be rationally-relevant. Several recent discussions of coordination have discussed the possibility that coordination might be intransitive (see [1], [2], [3], [4], [5], [6]). But very little attention has been paid to the question of how we should understand the rational relations generated by intransitive coordination. This is a pressing question. If there is no coherent way to answer it, that would be decisive evidence against any theory that posits intransitive coordination. And there is reason to worry. As Fine (2007, pg. 119) writes:

"[...] the failure of transitivity [of coordination] has some radical implications for the conception of logical validity [. . . ] the very idea of validity in virtue of logical form, as this is normally conceived, may also break down."

Without a response to this worry, we should not be confident that intransitive coordination is so much as intelligible.

In this paper, developing ideas by Fine, Pinillos, and Recanati, I show how the proponent of intransitive coordination should model the rational relations induced by intransitive coordination. That is to say, I develop a non-standard logical system that models the rational relations posited by the proponent of intransitive coordination.

I do this in three steps. First, I characterize the Fregean account of coordination in a more-than-usually-abstract way. In brief, we can think of the distribution of senses, for the Fregean, as implementing a more abstract idea: when two representations are coordinated there is a semantic requirement that they corefer (cf Fine 2007).

Second, I show that with only a small tweak, this can be altered to generate an intransitive picture of coordination: when two representations are coordinated there is a semantic requirement that they do not have disjoint reference (cf. Pinillos 2011, Recanati 2016).

Finally, I develop a logical system that models this kind of coordination. It turns out that any logic for intransitive coordination that satisfies certain very weak constraints has a surprising feature: if the logic is conservative with respect to standard first-order logic—in the sense that it gives the same verdict on arguments that involve only a transitive coordination-structure—it will have to employ an intransitive consequence relation. I describe a logical system, incorporating elements of free logic and three-valued logic, that satisfies this constraint. The logic deploys a consequence relation sometimes called “Strawson Entailment”. A set of premises Strawson-entails a conclusion if and only if the truth of the premises guarantees the non-falsity of the conclusion. As is required to model intransitive coordination, Strawson entailment is an intransitive consequence relation. So the system allows that individual valid arguments do not always combine to make new valid arguments.

I close by outlining what this investigation reveals about the choice between the traditional, transitive, picture of coordination and this intransitive picture. There is a sense in which the traditional theorist loses nothing by accepting this new picture: where coordination is, in fact, transitive, the two accounts make the same claims about rational relations. The difference between them is that the intransitive account allows us to characterize the coordination structure in certain cases where classical rational relations break down.

References

[1] Contim, Filipe Drapeau Vieira. 2016. Mental Files and Non-Transitive De Jure Coreference. Review of Philosophy and Psychology, 7(2), 365–388.

[2] Fine, Kit. 2007. Semantic Relationism. Wiley-Blackwell.

[3] Goodsell, Thea. Is de jure coreference non-transitive?. Philosophical Studies 167 (2014): 291-312.

[4] Pinillos, N Ángel. 2011. Coreference and meaning. Philosophical Studies, 154(2), 301–324.

[5] Recanati, François. 2012. Mental files. Oxford University Press.

[6] Recanati, François. 2016. Mental files in flux. Oxford University Press