A Logic for Easy Linking Semantics

Most semantic frameworks assume that the denotations of verbs expect their arguments in a certain specific order. In fixed word order languages, hence, we could say that order codes case marking. Moreover, all syntax-semantic mappings have to provide a solution for the fact that DPs can denote individual concepts of (extensional) type e as well as generalized quantifiers (〈〈e, t〉, t〉). The paper presents a new variant of type logic which offers a lean syntax-semantics interface for semantic representation in a Montagovian format. Specifically, the syntax-semantics mapping does not require obligatory quantifier raising (as Heim+Kratzer, 1998) and does not force the semanticist to make claims about a fixed underlying order of arguments of the verb. The latter feature will facilitate semantic research on free word order languages and semantic research on languages where no syntactic analysis in a Minimalist framework is as yet available. 1 Linking: Troubles and a Vision Which syntax feeds semantics? In the present paper, I want to address the syntax-semantics interface from the back end, so to speak, and propose a new logical backbone for semantics, one that is better suited to host syntax. I should stress that this is a service article. I will not criticize, defend or propose any linguistic analysis but want to present a linking formalism that is easy to handle and can be adapted for a wide range of potential semantic analyses. Nevertheless, my work was inspired by linguistic questions which I will briefly review. Type mismatch problem: It is a common assumption that verbs denote relations between entities. We can use names, indexicals or definite NPs to refer to entities. Moreover, we can use DPs that denote quantifiers over entities. In that case, a type mismatch between verb argument and DP denotation has to be resolved. While some theories endorse the assumption that verbs denote relations between generalized quantifiers, most people prefer to retain the original logical type of verbs. For these, Heim + Kratzer (1998) develop the by now standard way to resolve the type mismatch between verb and quantifiers. They propose an analysis where quantifier raising, coindexing and the interpretation of traces as variables serves, not only to settle matters of scope, but also as the standard way to enable semantic composition of verb projection and quantificational DP. Hence, the type mismatch problem is considered as solved by many semanticists. However, the semantic composition of even a simple sentence like John likes most Fellini movies requires quantifier raising, interpreted traces, coindexing, and lambda abstraction. Order codes argument structure: Standard semantic treatments of English and other languages assume a fixed (underlying) order of arguments of the verb. Word order, rather than case marking, is the factor that ensures that each DP or PP instantiates the correct argument place of the verb. According to this standard analysis, free word order languages where argument structure is exclusively determined by case marking should not exist. If a language is suspected to be of that type (see Haug, 2009 on Ancient Greek), or if a language is not as yet sufficiently well understood to make claims about word order, semantic analysis requires to stipulate a basic order of verbal arguments. This common feature of truth conditional semantics in the Montagovian format can even lead scholars to adopt other semantic frameworks which allow for a more direct impact of case marking in semantic interpretation. Hence, Montagovian semantics with interpreted case marking should be an attractive generalization of the standard framework. The tacit argument problem: Many analyses propose that the verb has arguments that are not instantiated by overt phrases in the sentence. One example is provided in recent papers on tense by von Stechow (von Stechow et al., 2009). His analysis rests on a tense argument of the verb. In order to instantiate this argument in matrix clauses, he has to assume that there is a tacit temporal PRO, used as a dummy syntactic object that figures in quantifier raising. PRO leaves a trace which is interpreted as a time variable and instantiates the temporal argument of verbs. PRO is not a generalized quantifier, so it can not initiate lambda abstraction. In non-embedded sentences, von Stechow has to assume that PRO passes its index to an independent lambda operator and gets deleted afterwards. While Minimalist syntax allows to delete non-interpretable material, the entire process looks like an artifact of a specific kind of theory rather than an insight about the logical structure of language. The event problem: In a standard Davidsonian analysis, event modifiers can apply to the event argument of the verb at many levels in syntax. In the standard fixed word order paradigm, we have to make a claim whether the event argument should be the first, or the second, etc. or the last argument of the verb. There is no agreed answer to this question and authors tend to avoid any principled position. I will discuss two possible options here. Solution 1: We could claim that the event is an early argument of the verb such that, for instance, love denotes λeλyλxLOVE(x, y, e). λe gets instantiated by the trace xe of an uninterpretable dummy E-PRO. E-PRO is co-indexed with xe and has to be raised to all positions immediately below an event modifier MOD. In that position, it has to pass its index to an independent lambda operator that makes xe accessible. After combination of MOD and verb projection, another trace of PRO instantiates the event argument of the verb, thereby making the argument inert until needed the next time. (Note: if there is more than one event modifier in a sentence, we will need a chain of traces of PRO). Solution 2: We could alternatively claim that the event is a late argument of the verb, and our example verb love denotes λyλxλeLOVE(x, y, e). If an event modifier wants to combine with the verb before the verb has met all its DP arguments, the modifier has to use some standard procedure to instantiate the innermost argument of an n-place relation and to reopen all other arguments after modification. Such modes of combination can certainly be defined. Still, the resulting analysis again carries the flavor of repairing theory-internal problems rather than offering insights about the logical structure of language. It should be pointed out that Kratzer (2002/unpublished) might offer a solution: She assumes that each quantificational DP binds the (currently open) event argument with an existential quantifier, and at the same time introduces a new, plural event argument that remains accessible and consists of the sum of all smaller events. Following this proposal, a sentence like Sally fed all chicken in one hour then means ∃E∀x(Chicken(x) → ∃eFeed(Sally, x, e) ∧ e ⊂ E) ∧ τ(E) = 1hour) (ignoring further minimality requirements on events). Her analysis is motivated by the observation that different event modifiers can take scope below and above nominal quantifiers in one and the same sentence. Yet, the event problem originally is not a scope problem. If we want to generalize Kratzer’s solution to a mechanism where the event parameter is accessible at each syntactic level, we’d have to claim that any DP (including definite noun phrases, proper names and other non-scope-taking DPs) existentially binds the event argument of the verb, combines with the verb, and afterwards introduces a new plural event that has the existentially bound first event as its part. Hence, a sentence like Sally fed Prillan will receive the following interpretation (again, leaving minimality conditions on E aside): ∃E(∃e(Feed(Sally, Prillan, e) ∧ e ⊂ E)) Even though this may not be wrong in a strictly logical sense, it is at least redundant. Event semantics would loose much of its original appeal: Events should make semantic representations elegant and perspicuous, and not redundant and unperspicuous. In this paper, I will define Linking Logic, a type logic on finite variable assignments, and Easy Linking Logic which endorses variables that are indexed with abstract case labels. This will allow us to design Easy Linking Semantics, a format for semantic analysis and composition that is independent of any specific grammatical framework and yet draws on earlier Montagovian semantics in a maximally conservative manner.