On the Interpretation of Common Nouns: Types Versus Predicates

Abstract

When type theories are used for formal semantics, different approaches to the interpretation of common nouns (CNs) become available w.r.t whether a CN is interpreted as a predicate or a type. In this paper, we shall first summarise and analyse several approaches as found in the literature and then study a particularly interesting and potentially challenging issue in a semantics where some CNs are interpreted as types – how to deal with some of the negated sentences and conditionals. When some CNs are interpreted as types (e.g., $Man :Type$), a sentence like John is a man can be given a judgemental interpretation $j:Man$, rather than the traditional Montagovian interpretation man(j). In such a setting, the question is then how to interpret negated sentences like John is not a man (or more complicated sentences like conditionals). A theory for predicational forms of judgemental interpretations is introduced and is shown to be able to deal with negated sentences and conditionals appropriately. A number of examples are considered to show that the theory provides an adequate treatment in various situations. Furthermore, experiments in the proof assistant Coq are performed in order to provide more supporting evidence for this adequacy. Besides the above, we also briefly study the use of indexed types in order to deal with CNs exhibiting temporal sensitivity and gradability.

S. Chatzikyriakidis—Supported by the Centre of Linguistic Theory and Studies in Probability in Gothenburg.

Z. Luo—Partially supported by the research grants from Royal Academy of Engineering, EU COST Action CA15123, and the CAS/SAFEA International Partnership Program for Creative Research Teams.

Appendices

A. Coq: A Gentle Introduction

Coq is an interactive theorem prover, i.e. a proof-assistant, that implements the Calculus of Inductive Constructions (pCIC, see The Coq Team 2007), in effect an MTT.¹⁴ The basic idea behind Coq is as follows: it helps one to see whether propositions based on statements previously pre-defined or user defined (definitions, parameters, variables) can be proven or not. To give a very short example on how Coq operates, imagine that we want to prove the following propositional tautology:

• (73) $((P \vee Q) \wedge (P \rightarrow R) \wedge (Q \rightarrow R)) \rightarrow R$

Given Coq’s typed nature we have to introduce the variables P, Q, R as being of type Prop (P, Q, R : Prop). Using the command Theorem, we put Coq into proof mode:

• (74) Theorem A :  $((P \vee Q) \wedge (P \rightarrow R) \wedge (Q \rightarrow R)) \rightarrow R$

The idea from this point onwards is to guide the prover into the completion of the proof using proof tactics that are either already predefined or can be defined by the user using the tactical language Ltac. For the case interested, we first introduce the antecedent as an assumption using intro:

We split the hypothesis into individual hypotheses using destruct

Now, we can apply the elimination rule for disjunction (elim) which will basically result in two subgoals:

The two subgoals are already hypotheses. We can use the assumption tactic that matches the goal with an identical hypothesis and the proof is completed:

Now, what we need to do in order to reason about NL is to implement our theoretical accounts we have provided in Coq. For example, one must introduce the universe $\textsc {cn}$ and further declare the types of the universe. For the moment, since universe construction is not an option in Coq, we define $\textsc {cn}$ to be Coq’s predefined Set universe. We show these, plus the introduction of individuals John and Mary and quantifier some. We also introduce the subtyping relations $Man<Human<Animal<Object$:

With the above definitions one can perform reasoning tasks, for example one might want to check whether some man walks follows from John walks. We formulate this as a theorem named EX:

The next step here is to unfold the definition for the quantifier. This is done using the cbv command, which basically performs any reduction possible. Then, we move the antecedent as a hypothesis using intro:

At this point, we use the exists tactic to substitute John for x. Using assumption the theorem is proven.

B. Coq Data and Assumptions

The following gives all of the data and assumptions made in Coq in order to deal with all the examples discussed in this paper.

We further assume that all the types in the universe D are totally ordered and dense; i.e., they respect the following axioms:

$$\begin{aligned} (reflexivity)&\forall A:D.\ \forall d:A.\ d\le d \\ (anti\texttt {-}symmetry)&\forall A:D.\ \forall d, d_1:A.\ d\le d_1\wedge d_1 \le d \rightarrow d = d_1 \\ (transitivity)&\forall A:D.\ \forall d, d_1,d_2:A.\ d\le d_1\wedge d_1 \le d_2 \rightarrow d \le d_2 \\ (density)&\forall A:D.\ \forall d, d_1:A.\ d\le d_1\rightarrow \exists d_2:A. d \le d_2 \le d_1 \end{aligned}$$

Remark

There is a pending question w.r.t the formal properties of grades. One of the common assumptions in the literature is that grades are dense, total, transitive, anti-symmetric and reflexive. The question is whether these assumptions are enough or different weaking might be needed depending on the grade in each case. What is rather difficult to tackle is the formal properties of abstract grades like idiocy.