Translating a Fragment of Natural Deduction System for Natural Language into Modern Type Theory

In this paper, we investigate the possibility of translating a fragment of natural deduction system (NDS) for natural language semantics into modern type theory (MTT), originally suggested by Luo (2014). Our main goal will be to examine and translate the basic rules of NDS (namely, meta-rules, structural rules, identity rules, noun rules and rules for intersective and subsective adjectives) to MTT. Additionally, we will also consider some of their general features.


Introduction
In this paper, we will examine two proof-theoretic approaches to natural language semantics. Specifically, we will explore the possibility of embedding natural deduction systems (NDS, Francez 2015) into modern type theory (MTT, Chatzikyriakidis and Luo 2017b), originally hinted at by Luo (2014).
Our main goal will be to examine and try to translate the basic rules of NDS (namely, meta-rules, structural rules, identity rules, noun rules and rules for intersective and subsective adjectives) to MTT. Additionally, we will also consider some of their general features.

NDS and MTT: A Preliminary Overview
MTT is closely related to Martin-Löf's constructive type theory (Martin-Löf 1984) and it fully utilizes its rich type structures (dependent types, inductive types, . . . ). NDS is more similar to the standard logical approach based on Gentzen's natural deduction. In practice, this means that with NDS we are devising a purely proof-theoretic framework, but with MTT we are allowed more liberties due to its type-theoretic nature. This earned MTT some criticism from Francez, who regards MTT as 'a modeltheoretic semantics, but one constrained by proof-theoretic constraints' (see Francez and Dyckhoff 2010, pp. 474-475). This point was not contested but rather embraced by Luo (see Luo 2014), who views his MTT as having both proof-theoretic and model-theoretic features (see Luo 2014, pp. 177-178).
The choice of a base system also dictates what will be the main vehicles for content: in MTT we work with judgements of the form a : A, where a is a so-called proof object (proof term, witness, justification, . . . ) and A a proposition/type, while in NDS we work with formulas (or (pseudo-)sentence in the case of natural language fragment). From a technical standpoint, probably the most important difference between judgements and formulas is that judgements have effective procedural content, i.e., they are decidable. More specifically, given a judgement a : A we should be always able to compute whether a is an object of the type A. Consider e.g., the judgement 1 : N at, i.e., the judgement that 1 is a natural number. The number 1 is in constructive type theories usually defined simply as s(0) : N at, i.e., as the successor of 0. This form alone tells us that 1 is indeed a natural number, since in MTT all natural numbers are either 0 or have the form s(a) where a is a natural number. 1 This is not the case with formulas, which are generally undecidable. For example, suppose that a ∈ P is a formula of predicate logic capturing the fact that a has some property P , more specifically that a is in the set P . Whether some element is a member of a set or not is, however, generally not decidable. Consequently, a ∈ P is undecidable as well. 2 Other points of discord could be found as well. For example, if our translation of NDS into MTT succeeds, we lose some of the nice 'philosophical' properties of NDS (e.g., fewer ontological commitments). 3 On the other hand, if we are solely interested in formal semantics, this does not need to concern us. So far we have discussed only the differences between NDS and MTT, but we can identify important similarities as well. The key intuition of proof-theoretic semantics that meanings are constituted via canonical proofs 4 is, of course, present in both systems. As Francez states: For compound sentences, sentential meanings are defined as the (contextualised) collection of canonical derivations [. . . ]. This is very much in the spirit of the modern approach 'propositions as types ' (for example, Martin-Löf 1984), the inhabitants of a type being the the proofs [. . . ]. (Francez 2015, p. 46) As expected, both systems rely on the standard scheme of introduction rules mirrored by the corresponding elimination rules. MTT adds to this mix, however, also formation rules and computation rules (also called equality rules), which can be understood as rules for assembling well-formed terms and for term reductions, respectively. Both frameworks also avoid the Fregean function-argument form of predication (F (a)) and move towards the more classical subject-predicate predication (S is P ) reminiscent of Aristotelian logic. More specifically, NDS utilizes pseudo-sentences of the general form a isa A, while MTT operates with judgements of the general form a : A. NDS also relies on the so-called reification of meaning, which puts it apart from most PTS approaches, but closer to MTT: The approach I am proposing in this book is rather to conceive of PTS as providing an explicit definition of meanings by meaning-conferring rules. Thus, if ξ is some meaning bearing expression, PTS should provide some proof-theoretic semantic value of the form [ [ξ] ] = df. · · · as the meaning of ξ. I refer to this semantic value as a reified meaning. (Francez 2015, p. 7) This reification is similar to MTT and its underlying conception of propositions as sets of proof objects. Assume that we have two proof objects λx.x and λy.y for the proposition A ⊃ A. These two proof objects differ only in the names of bound variables, i.e., they are α-equivalent. In MTT, we can express all this as λx.x : A ⊃ A, λy.y : A ⊃ A, and finally λx.x = α λy.y : A ⊃ A. Compare this with NDS and its expression [[A ⊃ A]] Ic denoting the set of all I-canonical proofs of A ⊃ A. Since λx.x and λy.y are essentially understood as reified proofs or codes for proofs, we can see that both λx.x = α λy.y : A ⊃ A and [[A ⊃ A]] Ic capture a similar intuition.

From NDS to MTT
In our translation, we start with meta-rules (3.1), then we consider identity rules (3.2), rules for proper names (3.3), and finally we will examine rules for intersective and subsective adjectives (3.4). As we shall see, all the discussed NDS rules can be embedded into MTT semantics and justified either as admissible rules or derivable rules.
The translation method we utilize is based on the suggestion made by Luo (2014). Generally speaking, the translation method has two steps: 1) identifying the suitable expressions of NDS for the application of translation function [[ ]] (a syntactic step), and 2) finding the appropriate translations in MTT (a semantic step). As an simple example, suppose we have an expression Alice isa student, which is a proper sentence of NDS, hence we can apply the translation function [[Alice isa student]]. As the corresponding translation in MTT, we get the judgement Alice : Student. Although the translation is not always as straightforward as Luo's quote might suggest, we will show that in general it can be successfully deployed for all the basic rules (meta-rules, identity rules, noun rules, adjective rules).

Meta-Rules
The meta-rules for NDS (see below) are intended to confer meaning of sentences from the natural language fragment containing only in/transitive verbs, determiner phrases with a singular noun, determiners 'every' and 'some' and a copula 'is' (see Francez 2015). The rules for determiners come in pairs of introduction rules (I-rules) and (generalized) elimination rules (E-rules) and they behave in accordance with the standard intuitionistic explanations of the corresponding quantifiers.
First, some additional explanations are in order: j, X, and S are meta-variables for individual parameters (determiner phrases, . . . ), nouns (including compound nouns), and (affirmative) pseudo-sentences, 6 respectively, while isa serves as a copula. Furthermore, expression of the form S[j] means that j occupies a determiner phrase position in the sentence S. every and some are determiners (for more, see Francez 2015).
First, we present all the translated variants for meta-rules, then we add comments and examples. Comments. The rule (Ax) is justified by the structural rule assump from MTT. The rules (eI) and (sI) are justified by the rules Π-intro and Σ-intro (or more precisely, via ∀-intro and ∃-intro that are based upon them), respectively. Analogously for the rules (eE) and (sE) . The context Γ from NDS, i.e., a finite list of formulas, is translated into a list of judgements. More specifically, in MTT, Γ a : A is a hypothetical judgement properly unpacked as a 1 : A, . . . , a n : A a : A where n is the number of assumptions in the context. The copula 'isa' is used for predication in NDS. In MTT, predication is achieved with the use of colon ':', so translating j isa X as j : [[X]] seems as a good fit. This decision dictates the rest of the translation: if we replace isa with :, then the left-hand side has to be some object and the right-hand side has to be its type. The most straightforward way to treat the determiner every 5 Since we will not be interested here in the issue of quantifier scope ambiguity, we omit the corresponding explicit scope indicators from the rules. For example, the rule (sI) in its fully disclosed variant looks like . 6 A pseudo-sentence (of the object language) is a schematic sentences with occurrences of at least one parameter, e.g., j isa X. Example of a (pseudo-)sentence might be e.g., j isa student. seems to be simply to take it as the universal quantifier ∀, which is defined in MTT via the Π type. 7 In other words, we will capture S[(every X)] as sentential function over individual parameters, i.e., as an indexed family of types over the objects of type X. Analogously for the determiner some that can be treated via the Σ type. 8 Examples. The following derivation from NDS: Γ, j isa girl j smiles (eI) Γ every girl smiles gets as its MTT variant the following derivation: Γ, j : Girl s(j) : Smiles(j) (eI) Γ λj.s(j) : (∀j : Girl)Smiles(j) Note that in MTT, the noun girl is captured as the type Girl and the predicate smiles as the dependent type Smiles(j). Furthermore, note that the relationship between j and S in NDS, i.e., S[j], is captured in MTT by interpreting S[j] as a type of sentence (proposition) depending on the assumption j : Girl. We can also see that this formalization is in accord with Francez's own approach: A proof of S[(every X)] is a function mapping each proof of j isa X (for an arbitrary fresh parameter j) into a proof of S[j]. (Francez 2015, p. 247) On our approach, the proof of (∀j : Girl)Smiles(j) is the proof object λj.s(j) which is a function (or rather a function name) that takes a proof object j and returns a proof object s(j). 9 As a more complicated example with a transitive verb, the NDS derivation: k isa boy Γ, j isa girl j loves k (eI) Γ every girl loves k (sI) Γ every girl loves some boy becomes: k : Boy

Identity Rules
In the natural language fragment, Francez works with a set of rules determining the behaviour of the copula is, which is treated as 'a disguised identity' (Francez 2015, p. 250). Naturally, it behaves in the same way. The collection of rules is as follows: 10 Before we approach the translation of these rules, we have to address that in MTT there are two kinds of identity: propositional (or extensional) and judgemental (intensional, definitional). Probably the most important difference is that the judgemental identity, represented as a = b : A, is decidable, while the propositional identity, usually written as Id (A, a, b), is not. 11 So what type of identity describe the rules above? If their were describing judgemental identity, then the translation of the reflexivity, symmetry and transitivity would be straightforward. For example, the MTT variant of (is − sym) would be: a = b : A symm b = a : A , etc. However, given the fact that the identity rules of NDS have dedicated Iand E-rules, they seem to be describing propositional identity. In MTT, we cannot introduce judgemental identity in a way we would introduce e.g., some logical operator. 12 The translated variants would be: where A is the type of the objects j and k that we would use to represent the individual parameters. For example, if we have j isa girl in NDS, then we assume that j : Girl in MTT.
Comments. Validity of the (isI) rule follows from the fact that, in general, from a = b : A we can deduce ref l(A, a) : Id(A, a, b), which can be derived as a rule from Id-intro using substitution and set equality rules: which in turn justify the rules (is/s) and (is/t) , respectively.

Proper Names Rules
The Iand E-rules for proper names, which are ranged over by meta-variables N, M , are specified as follows (see Francez 2015, p. 251 As a concrete example of this distinction, consider e.g., the difference between numbers 5 + 7 and 12. While their are both extensionally equal (they both denote the number 12), they are not intensionally equal (they do not denote it in the same way). Alternatively, we can view this distinction as a difference between abstract objects and linguistic terms. 12 Recall that a = b : A is one of the basic kinds of MTT judgements. 13 For proofs and definitions of the constants symm and trans, see e.g., Nordström et al. (1990).
The translated variants will become:
Informally, it states that whenever we have two identical names, we can freely swap them in any sentence they appear. For its proof using the Id-elim rule, see e.g., Martin-Löf (1984). The (nE) rule is also justifiable via Id-elim. Examples. As an example, we construct the derivation (

Intersective Adjectives
In this section, we examine and compare NDS and MTT approaches to intersective adjectives. We will take intersective adjectives as specified by adhering to the following two kinds of rules: a is Adj N oun intA 1 a is Adj a is Adj N oun intA 2 a is N oun For example, a is black car intA 1 a is black a is black car intA 2 a is car Hence, intersective adjectives are those adjectives that allow inferring from 'a is Adj N oun' that the underlying object of predication a possesses both its constituents separately: the noun N oun, i.e., 'a is N oun', as well as the intersective adjective Adj, i.e., 'a is Adj'. Observe that the compound Adj N oun of intersective adjective and noun behaves similarly to the logical connective conjunction in standard natural deduction. More specifically, in natural deduction, conjunction has associated two elimination rules: . These two rules correspond in their behaviour to rules intA 1 and intA 1 . Elimination rules for conjunction allow deducing both its conjuncts A and B separately, and, analogously, rules for intersective adjectives allow deducing both its parts Adj and N oun. Hence, we could say that intersective adjectives preserve inferential content. We will utilize this fact later.
In NDS, intersective adjectives appear within ground pseudo-sentences of the form 14 j is A where A is a meta-variable for intersective adjective (see above). The corresponding rules are: From the rule (adjE) we can obtain the following derived rules (see Francez 2015, p. 252): It is easy to check that these two rules correspond to our general rules intA 1 , intA 2 for intersective adjective and, consequently, to conjunction elimination rules. Furthermore, it now becomes clear that the original rule (adjE) corresponds to the generalized conjunction elimination rule (see e.g., Negri et al. 2001) Before we get to the translation of the above rules, we will first discuss how adjectives are treated in MTT. Intersective adjectives are generally analyzed with Σ type, i.e., the same type that is also used for defining conjunction. 15 For example, the expression 'black car' would be captured as the type: (Σx : Car)Black(x), i.e., the type of cars that are black (Black(x) is considered as a property/propositional function). The corresponding proof object is a pair (x, y) such that x : Car and y : Black(x), i.e., y is a justification (proof object) that x is black. Hence, common nouns are interpreted as distinct types, so we will get types of cars, animals, humans, etc. (so-called many-sorted type theory). 16 The corresponding MTT introduction for (adjI) would then be: 17 Γ n : N oun Γ a : intAdj(x) Γ (n, a) : (Σx : N oun)intAdj(x) We mentioned above that intersective adjectives should be conservative with respect to their inferential content. We can test this with projection functions f st and snd. Intuitively, f st and snd are operations that return the first and the second element of the proof object of the pair type (Σx : N oun)intAdj(x), respectively. For example, assume that we have the proof object p such that p = (x, y) of (Σx : Car)Black(x), then f st(p) = x : Car and snd(p) = y : Black(f st(p)). 18 The corresponding elimination rules will then be as follows: The projections f st and snd can be defined using the non-canonical constant E (brought by Σ-elim rule) in the following manner: f st(c) = E(c, (x, y)x) and snd(c) = E(c, (x, y)y), respectively. Now, we can finally get to the translation of the rules themselves (we skip the generalized elimination variant): Comments. As discussed above, the rules for intersective adjectives are justified by the corresponding Σ type rules.

Subsective Adjectives
We specify subsective adjectives by the following two rules: a is Adj N oun subA 1 a is N oun a is Adj N oun subA 2 a is Adj N For example, a is large mouse intA 1 a is mouse a is large mouse intA 2 a is large m Hence, subsective adjectives allow us to infer from 'a is Adj N oun' that the underlying object of predication a possesses both its constituents separately: the noun N oun, i.e., 'a is N oun', as well as the adjective Adj with the proviso it was relativized w.r.t N oun., i.e., 'a is Adj N '. Thus, e.g., 'a is large m ' from the example above can be read as 'a is something large assuming mouse-largeness scale'. Analogously to intersective adjectives, we can see that the compound containing a subsective adjective and a noun behaves similarly to the logical connective conjunction. In NDS, the rules for subsective adjectives are as follows (we omit the generalized elimination rules; Francez 2017): The crucial part for the translaton is the premise Γ j isa A X in (subAI) rule, which captures the fact that j is A only under the assumption that j isa X. Specifically, A X is a family of adjectives over the set of nouns X. Hence, in effect, it makes the meaning of A dependent on the meaning of X. In other words, the meaning of A is restricted only to a certain class of nouns X. So we will have a different types of largeness: e.g., large human, large insect, large mammal, etc.
As Francez describes it: The unfolding of the adjectives can be seen as a purely formal devise to parameterize a subsective adjective: A X is just a family of adjectives originating from A and parameterized by nouns X. The entailments in (4.14) are the basis for the revised I/E-rules for subsective adjectives. [. . . ] The explicit parameterization replaces the dependency in the original rules. (Francez 2017, pp. 12-13) In MTT, we can capture this dependency by making the whole type of subsective adjectives range over the X, i.e., (common) nouns. Thus its type will be ∀α : X.(α → P rop) (see Luo 2013, Chatzikyriakidis andLuo 2017a). After the translation, we get: where Small E denotes the fact that we use 'elephant-smallness' of type: Elephant → P rop. Note that we cannot derive that there is something small (corresponding to snd(l) : Small(f st(l))), only that we have something small w.r.t. an elephant scale.