Abstract
In formal semantics based on modern type theories, some sentences may be interpreted as judgements and some as logical propositions. When interpreting composite sentences, one may want to turn a judgemental interpretation or an ill-typed semantic interpretation into a proposition in order to obtain an intended semantics. For instance, an incorrect judgement \(a:A\) may be turned into its propositional form \(\textsc {is}(A,a)\) and an ill-typed application p(a) into \(\textsc {do}(p,a)\), so that the propositional forms can take part in logical compositions that interpret composite sentences, especially those that involve negations and conditionals.In this paper, we propose an operator not that facilitates such a transformation. Introducing not axiomatically, with five axiomatic laws to govern its behaviour, we shall use it to define \(\textsc {is}\) and \(\textsc {do}\) and give examples to illustrate its use in semantic interpretation. The introduction of not into type theories is logically consistent – this is justified by showing that not can be defined by means of the heterogeneous equality \(\textrm{JMeq}\) so that all of the axiomatic laws for not become provable. Therefore, since the extension with \(\textrm{JMeq}\) preserves logical consistency, so does the extension with not. We shall also study conditions under which \(\textsc {is}\) and \(\textsc {do}\) operators can be used safely without the risk of over-generation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In the literature on linguistic semantics, the formula in (3) is usually written as \(\forall x.\ teacher(x)\wedge talk_1(x)\) without the domain \({\text{ e }}\) because, in most of the cases in the Montagovian setting, quantifications are always over \({\text{ e }}\) of all entities.
Note that this is different from Montague semantics: the Montagovian interpretations (7) and (8) are well-formed formulas of type \({\textbf{t}}\) because table, \(talk_1\) and \(eat_1\) are all predicates with domain \({\text{ e }}\) and \(t_1\) and \(e_1\) are entities of type \({\text{ e }}\). Put in another way, they are legal formulas, although their intended interpretations should usually be false.
-
(7)
\(\forall x:{\text{ e }}.\ table(x)\wedge talk_1(x)\)
-
(8)
\(eat_1(t_1,e_1)\)
-
(7)
By Modern Type Theories (MTTs), we refer to the family of formal systems such as Martin-Löf’s intensional type theory (MLTT) (Martin-Löf, 1975; Nordström et al., 1990), the Calculus of Inductive Constructions (pCIC) (Coq, 2010) and the Unifying Theory of dependent Types (UTT) (Luo, 1994). Martin-Löf’s type theory MLTT can be employed as an adequate foundational language for MTT-semantics when it is extended with h-logic – see (Luo, 2019b) for more details.
Another important property of MTTs is decidability, which is necessary for an internal logical system based on the propositions-as-types principle (Curry & Feys, 1958; Howard, 1980). Note that, given a proof candidate a and a proposition A, it should be decidable whether \(a:A\) is correct (i.e., type checking should be decidable), because we are dealing with a finite proof system and, in particular, both a and A are finite. Please also note that this is different from provability of a proposition, which is undecidable because, without a proof candidate, one has to try to find out whether a proof exists.
Formally, (26) is derivable if, for example, \(j:\varSigma x:Student.male(x)\), which is a subtype of both Student and Man with \(Student\le Human\) and \(Man = \varSigma x:Human.male(x)\). Under many other environments, the judgement \(j:Student\) is non-derivable.
The second author is grateful to several researchers who have discussed with him about how to interpret negative sentences in MTT-semantics, including Glyn Morrill (during ESSLLI 2011), Nicholas Asher (in an email communication about a paper in LACL 2014) and Koji Mineshima (during ESSLLI 2014 and subsequent communications with the second author when he was writing (Chatzikyriakidis & Luo, 2017b), where a preliminary NOT-operator was studied – see Footnote 8 in §3.2.)
A preliminary NOT-operator was studied by the second and third authors in Chatzikyriakidis and Luo (2017b), which is a special case for the \(\textsc {not}\)-operator proposed above. The study of the preliminary NOT-operator was limited in its scope – it assumed that there be a top type for all types in cn, its axiomatic laws were not general enough, their justification was not given, and the issue of over-generation was not studied (for the study of these for the \(\textsc {not}\)-operator proposed here, see §3.3, §4 and §5).
In Xue et al. (2018), we have defined the proposition \(P_{A,B}:B\rightarrow Prop\). In the current notation, we have \(P_{A,B}(t)=\textsc {is}_B(A,t)\).
A function \(c:A\rightarrow B\) is injective if, for all \(x_1, x_2:A\), \(c(x_1) = c(x_2)\) implies that \(x_1=x_2\). For instance, the identity function that maps any object to itself is injective. In a coercive subtyping relation \(A\le _c B\), if c from A to B is an injection function, then c is called an injective coercion.
In Coq ‘,’ is used as a ‘separator’ instead of ‘.’
\(\textrm{JMeq}\) is proposed and named ‘John Major equality’ by McBride (2002).
Please note that the notion of ‘negative occurrence’ here is different from that of the term usually used in the literature.
References
Agda. (2008). The Agda proof assistant (version 2). Available from the web page: http://appserv.cs.chalmers.se/users/ulfn/wiki/agda.php
Asher, N. (2012). Lexical meaning in context: a Web of Words. Cambridge University Press.
Bekki, D. (2014). Representing anaphora with dependent types. LACL 2014, LNCS 8535.
Bernardy, J.P. & Chatzikyriakidis, S. (2017). A type-theoretical system for the fracas test suite: Grammatical framework meets coq. In: IWCS 2017-12th International Conference on Computational Semantics-Long papers.
Boldini, P. (2000). Formalizing context in intuitionistic type theory. Fundamenta Informaticae, 42(2), 1–23.
Callaghan, P., & Luo, Z. (2001). An implementation of LF with coercive subtyping and universes. Journal of Automated Reasoning, 27(1), 3–27.
Chatzikyriakidis, S. & Luo, Z. (2013). Adjectives in a modern type-theoretical setting. In: Morrill G, Nederhof J (eds) Proceedings of Formal Grammar 2013. LNCS 8036, pp 159–174.
Chatzikyriakidis, S., & Luo, Z. (2014). Natural language reasoning in Coq. Journal of Logic, Language and Information, 23, 441–480.
Chatzikyriakidis, S., & Luo, Z. (2014). Natural language reasoning using proof-assistant technology: Rich typing and beyond. In EACL Workshop on Type Theory and Natural Language Semantics.
Chatzikyriakidis, S. & Luo, Z. (2016). Proof assistants for natural language semantics. In Logical Aspects of Computational Linguistics 2016, Nancy LNCS 10054
Chatzikyriakidis, S., & Luo, Z. (2017). Adjectival and adverbial modification: The view from modern type theories. Journal of Logic, Language and Information, 26(1), 45–88.
Chatzikyriakidis, S., & Luo, Z. (2017). On the interpretation of common nouns: Types v.s. predicates. Modern perspectives in type-theoretical semantics. Springer.
Chatzikyriakidis, S., & Luo, Z. (2020). Formal semantics in modern type theories. Wiley.
Church, A. (1940). A formulation of the simple theory of types. Journal of Symbolic Logic, 5(1), 56–68.
Cooper, R. (2005). Records and record types in semantic theory. Journal of Logic and Compututation, 15(2), 99–112.
Coq. (2010). The Coq Proof Assistant Reference Manual (Version 8.3), INRIA. The Coq Development Team
Curry, H., & Feys, R. (1958). Combinatory logic (Vol. 1). North Holland Publishing Company.
Dapoigny, R., & Barlatier, P. (2009). Modeling contexts with dependent types. Fundamenta Informaticae, 104(4), 293–327.
Gallin, D. (1975). Intensional and higher-order modal logic: with applications to Montague semantics. North-Holland.
Groenendijk, J. & Stokhof, M. (1990). Dynamic Montague grammar. In Proceedings of the second symposium on logic and language.
Groenendijk, J., & Stokhof, M. (1991). Dynamic predicate logic. Linguistics and Philosophy, 14(1), 39.
Henkin, L. (1950). Completeness in the theory of types. Journal of Symbolic Logic, 15, 81–91.
Howard, W. (1980). The formulae-as-types notion of construction. In: Hindley J, Seldin J (eds) To H. B. Curry: Essays on Combinatory Logic, Academic Press, (Notes written and distributed in 1969.)
Kahle, R. & Schroeder-Heister, P. (eds) (2006). Proof-theoretic semantics. special issue of synthese.
Kamp, H., & Reyle, U. (1993). From discourse to logic. Kluwer.
Luo, Z. (1994). Computation and reasoning: A type theory for computer science. Oxford University Press.
Luo, Z. (1999). Coercive subtyping. Journal of Logic and Computation, 9(1), 105–130.
Luo, Z. (2009). Type-theoretical semantics with coercive subtyping. Semantics and Linguistic Theory 20 (SALT20), Vancouver
Luo, Z. (2011). Contextual analysis of word meanings in type-theoretical semantics. Logical Aspects of Computational Linguistics (LACL’2011) LNAI 6736
Luo, Z. (2012). Common nouns as types. In: Bechet D, Dikovsky A (eds) Logical Aspects of Computational Linguistics (LACL’2012). LNCS 7351, pp 173–185
Luo, Z. (2012). Formal semantics in modern type theories with coercive subtyping. Linguistics and Philosophy, 35(6), 491–513.
Luo, Z. (2014). Formal semantics in modern type theories: Is it model-theoretic, proof-theoretic, or both? In: Invited talk at logical aspects of computational linguistics 2014 (LACL 2014). Toulouse LNCS, 8535, 177–188.
Luo, Z. (2018). Formal semantics in modern type theories (and event semantics in mtt-framework). In: Invited talk at LACompLing18
Luo, Z. (2019). Formal Semantics in Modern Type Theories (MTT-semantics is both model/proof-theoretic). Proof-Theoretic Semantics: Assessment and Future Perspectives, In: Proceedings of the Third Tuebingen Conference on Proof-Theoretic Semantics, Tuebingen
Luo, Z. (2019). Proof irrelevance in type-theoretical semantics. Logic and Algorithms in Computational Linguistics 2018 (LACompLing2018), Studies in Computational Intelligence (SCI) Springer
Luo, Z. & Callaghan, P. (1998). Coercive subtyping and lexical semantics (extended abstract). In: Logical Aspects of Computational Linguistics (LACL’98).
Luo, Z. & Pollack, R. (1992). LEGO Proof Development System: User’s Manual. LFCS Report ECS-LFCS-92-211, Deptartment of Computer Science, University of Edinburgh
Luo, Z. & Xue, T. (2020). Disjointness of types and negative occurrences. Notes, February 2020.
Luo, Z., Soloviev, S., & Xue, T. (2013). Coercive subtyping: theory and implementation. Information and Computation, 223, 18–42.
Martin-Löf, P. (1975). An intuitionistic theory of types: predicative part. In: HRose, JCShepherdson (eds) Logic Colloquium’73.
McBride, C. (2002). Elimination with a motive. In: Callaghan P, Luo Z, McKinna J, Pollack R (eds) Types for Proofs and Programs, In: Proceedings of TYPES 2000, LNCS 2277, Springer
McBride, C., & McKinna, J. (2004). The view from the left. Journal of Functional Programming, 14(1), 69–111.
Mineshima, K., Martínez-Gómez, P., Miyao, Y. & Bekki, D. (2015). Higher-order logical inference with compositional semantics. In: Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pp 2055–2061
Mönnich, U. (1985). Untersuchungen zu einer konstruktiven Semantik fur ein Fragment des Englischen. University of Tübingen.
Montague, R. (1974). Formal philosophy: Selected papers of Richard Montague. Yale University Press.
Muskens, R. (1996). Combining Montague semantics and discourse representation. Linguistics and Philosophy, 19(2), 143–186.
Nordström, B., Petersson, K., & Smith, J. (1990). Programming in martin-Löf’s type theory: An introduction. Oxford University Press.
Ranta, A. (1994). Type-theoretical grammar. Oxford University Press.
Retoré, C. (2014). The Montagovian Generative Lexicon Ty\(_n\): a type theoretical framework for natural language semantics. In: proofs and programs (TYPES 2013), LIPIcs(26).
Sundholm, G. (1986). Proof theory and meaning. D Gabbay and F Guenthner (eds) In: Handbook of Philosophical Logic, Vol III.
Sundholm, G. (1989). Constructive generalized quantifiers. Synthese, 79(1), 1–12.
Xue, T. (2013). Theory and implementation of coercive subtyping. PhD thesis, Royal Holloway, University of London
Xue, T. & Luo, Z. (2012). Dot-types and their implementation. In: Logical aspects of computational linguistics (LACL 2012) LNCS 7351.
Xue, T., Luo, Z. & Chatzikyriakidis, S. (2018). Propositional forms of judgemental interpretations. In: Proceedings of workshop on natural language and computer science oxford
Acknowledgements
This work is partially supported by the Nature Science Foundation of Hubei Province (2019CFB742) and the EU research network EUTypes (COST Action CA15123). Stergios Chatzikyriakidis is supported by grant 2014-39 from the Swedish Research Council, which funds the Centre for Linguistic Theory and Studies in Probability (CLASP) in the Department of Philosophy, Linguistics, and Theory of Science at the University of Gothenburg. Many thanks to the anonymous reviewers for their useful comments that have helped to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
A Short Introduction to Coq
The rationale behind Coq, and proof-assistants in general, can be roughly summarized as follows: you use Coq in order to check whether propositions based on statements previously pre-defined or user-defined (definitions, parameters, variables) can be proven or not. Coq is a dependently typed proof assistant and implements the Calculus of Inductive Constructions (pCIC) (Coq, 2010), which is a modern type theory, very similar to the type theory UTT (Luo, 1994); thus, Coq ‘speaks’ so to say the language we use to interpret linguistic semantics in this paper. Coq is a reasoning engine and there are at least ways that can be used in studying linguistic semantics, to an extent overlapping with each other: a) as a formal checker for the semantic validity of proposed accounts in NL semantics and b) Natural Language Inference (NLI), i.e. reasoning with NL.
We proceed by formulating a simple example of NLI in Coq. In effect, NLI is seen as a theorem proving task, where a valid semantic entailment will very simply mean the implication relation between the two semantic structures is a valid theorem. A very simple case of semantic entailment, that of example (115), will be formulated as the following theorem (named ex) in Coq (116):
-
(115)
John walks \(\Rightarrow \) some man walks
-
(116)
Theorem ex: John walks \(\rightarrow \) some man walks
Then, depending on the semantics of the individual lexical items one may or may not be able to prove the theorem in question. Inferences like the one shown in (115) are easy cases to prove. Assuming a semantics of some specifying that given any A of type CN and a predicate of type \(A \rightarrow Prop\), there exists an \(x :A\) such that \(P(x):Prop\), such cases are straightforwardly proven.
A few notes about the lexical entries. We use Coq’s Prop type, corresponding roughly to the type of truth-values (t) in Montague Semantics. We define CN to be Coq’s universe Set and interpret common nouns like man, human as being of type CN (thus we have for example \(Man, Human:CN\)). Verbs are defined as predicates requiring arguments of type \(A:CN\) (the choice of A depends on the verb itself.) Coercive subtyping is supported in Coq. Adjectives are defined as predicates, and adjectival modification as \(\varSigma \) types. Quantifiers and VP adverbs are defined as types ranging over the universe CN. For the example we are interested in, the following are declared:
We have introduced CN as Coq’s universe Set, declared Man, Human and Animal to be of type CN, further introduced the relevant subtyping relations and lastly walk. With walk as being of type \(Human \rightarrow Prop\) and John as being of type Man with Man < Human, we can prove the theorem in (116) easily. We first use the proof tactic intro to move the premises to be hypotheses. Then, we apply the tactic unfold to some (unfold some). Unfold does exactly what it promises: it unfolds the definition associated with a lexical entry (if there is one):
The intro tactic is then used, moving the antecedent to the list of premises. Now, one can existentially instantiate \(x:Man\) with \(John:Man\):
Finally, the tactic assumption finishes the proof.
Coq Code for a Proof of Injectivity
Assuming proof irrelevance, one can show the injectivity of the first projection for a \(\varSigma \)-type whose second component is a proposition.
Justification of NOT by JMeq: Coq Proofs
Here are the Coq proofs of the laws (\(A_1^d\text {-}A_5^d\)), where not is defined by means of the heterogeneous equality JMeq and do by means of not.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Xue, T., Luo, Z. & Chatzikyriakidis, S. Propositional Forms of Judgemental Interpretations. J of Log Lang and Inf 32, 733–758 (2023). https://doi.org/10.1007/s10849-023-09397-y
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10849-023-09397-y