Semantic studies of intuitionistic logic

Previous intuitionistic justification logics included explicit justifications for all admissible rules of intuitionistic logic in order to get completeness with respect to provability semantics. We present the justification logic iJT4, which does not have these additional justification terms. We establish that iJT4 is complete with respect to modular models and that there is a realization of intuitionistic S4 into iJT4. Hence iJT4 can be seen as an explicit version of intuitionistic S4.

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

Beliefs have been formally modelled in the last decades using doxastic logics. The possible worlds model and its associated Kripke semantics provide an intuitive semantics for these logics, but the...

In intuitionistic justification logic, evidence terms represent intuitionistic proofs, thatis a formula r:A means r is an intuitionistic proof of A. A natural principle in thiscontext is the internalized disjunction property (IDP), which is: for each term r thereexists a term s such that r:(A or B) implies s:A or s:B.We introduce a light extension of iJT4, in which IDP is valid. Our proof relies ona model construction that enforces sharp evidence relations and a tight connectionbetween syntax and semantics. This makes it possible to switch between proofs andmodels, which will be the key to proving IDP.

It’s widely accepted that normativity is not subject to truth values. The underlying reasoning is that truth values can only be predicated of descriptive statements; normative statements are prescriptive, not descriptive; thus truth value predicates cannot be assigned to normative statements. Hence, deonticity lacks logical semantics. This semantic monism has been challenged over the last decades from a series of perspectives that open the way for legal logics with imperative semantics. In the present paper I will go back to Kant and review his understanding of practical judgments, presenting it as supported by a pluralistic semantics. From this perspective a norm of Law is a logical expression that includes as content a generic description of a possible behavior by a generality of juridical agents, and assigns to that content the assertion of its obligatory character, accompanied by a disincentive for non-compliance. From this perspective legal norms can be syntactically formalized and assigned appropriate semantic values in such terms that they can be incorporated into valid inferential schemes. The consequence is that we can put together legal logics that handle both the phenomenal and the deontic dimensions of legality.

Invertible map equivalences are approximations of graph isomorphism that refine the well-known Weisfeiler-Leman method. They are parameterized by a number k and a set Q of primes. The intuition is that two equivalent graphs G equiv^IM_{k, Q} H cannot be distinguished by means of partitioning the set of k-tuples in both graphs with respect to any linear-algebraic operator acting on vector spaces over fields of characteristic p, for any p in Q. These equivalences have first appeared in the study of rank logic, but in fact they can be used to delimit the expressive power of any extension of fixed-point logic with linear-algebraic operators. We define {LA^{k}}(Q), an infinitary logic with k variables and all linear-algebraic operators over finite vector spaces of characteristic p in Q and show that equiv^IM_{k, Q} is the natural notion of elementary equivalence for this logic. The logic LA^{omega}(Q) = Cup_{k in omega} LA^{k}(Q) is then a natural upper bound on the expressive power of any extension of fixed-point logics by means of Q-linear-algebraic operators. By means of a new and much deeper algebraic analysis of a generalized variant, for any prime p, of the CFI-structures due to Cai, Furer, and Immerman, we prove that, as long as Q is not the set of all primes, there is no k such that equiv^IM_{k, Q} is the same as isomorphism. It follows that there are polynomial-time properties of graphs which are not definable in LA^{omega}(Q), which implies that no extension of fixed-point logic with linear-algebraic operators can capture PTIME, unless it includes such operators for all prime characteristics. Our analysis requires substantial algebraic machinery, including a homogeneity property of CFI-structures and Maschke's Theorem, an important result from the representation theory of finite groups.

In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion scheme. We show how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors (or priorities) in higher-order model-checking. We show in particular how to reunderstand in this way the type-theoretic approach to higher-order model-checking developed by Kobayashi and Ong. We briefly explain in the end of the paper how this analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order recursion schemes.

Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. In this paper we present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides the basis for an effective system for automating inductive reasoning.

Interacting with the Web of Things

Carl Posy’s Mathematical Intuitionism is an installment in the increasingly impressive Cambridge Elements in the Philosophy of Math series, edited by Penelope Rush and Stewart Shapiro. Three earlier volumes, on the history of mathematics, on structuralism, and on innovation and certainty in mathematics have been previously reviewed in this journal [Carson, 2020; Sereni, 2020; Gillies, forthcoming]. Posy’s brief volume consists of three main chapters addressing intuitionistic mathematics, intuitionistic logic, and the philosophy of intuitionism respectively, with a brief introductory chapter (‘The three faces of intuitionism’) motivating this organization of the material. The discussion, in a sense, focuses squarely on the founder of intuitionism, L.E.J. Brouwer, and presents the material in terms of a development of Brouwer’s philosophical and foundational account of intuitionism, an account that Posy calls the Intuitionistic Standpoint. This is not to say that other later intuitionists such as Arend Heyting and Michael...

We propose to represent the syntax and semantics of natural languages with labelled proof nets in the implicative fragment of intuitionistic linear logic. Resource-sensitivity of linear logic is used to express all dependencies between the syntactic constituents of a sentence in the form of a proof net. Phonological and semantic labelling of the proof net from its inputs to the unique output are used to produce the well-formed phonological form and the semantic representation of a sentence from entries of a lexicon. In this way we obtain a linguistic model of great flexibility because labelling is not completely determined logically: it is used to introduce linguistic constraints which go beyond the underlying logic. 1

The study explores the practice, meaning and significance of invective and insults in political discourse among Ghanaians. It examines the use of such language forms against people of higher social status in online discourse in Ghana, a country where it is traditionally unacceptable to openly criticize the elderly and authority figures, let alone insult them. Data was sourced from GhanaWeb [www.ghanaweb.com], the most patronized online news website in Ghana that ensures anonymous citizen participation in issues of national interest. The data involved invective and insult-filled online reader comments on news reports about the presidential candidates and the electoral commissioner involved in the 2016 general election in Ghana. Metapragmatic comments (reader reactions) which portray Ghanaian sociocultural perspectives on the use of abusive language against these individuals were also gathered. These two sets of comments were used to build the Corpus of GhanaWeb Comments on Ghana’s 2016 Election (CGCGE16). In addition to CGCGE16, the data for the study included comments from social media platforms, native speakers’ intuitions and the views of Ghanaian anthropological linguists and cultural specialists obtained from interviews, and evidence from scholarly works. The qualitative meaning analysis method is used along with the quantitative discourse analysis method. The study adopts an ethnopragmatic approach, combining corpus-based searches with semantic explications and cultural scripts, following the principles of the Natural Semantic Metalanguage (NSM) approach (Wierzbicka, 1992, 1996; Goddard & Wierzbicka, 2014). The findings indicate that most of the comments in CGCGE16 are personal attacks that violate the Ghanaian communicative norms towards the elderly and people in authority and, furthermore, do not conform to the expected displays of verbal respect. Instead of expressing their views on on-going socio-political issues, the commenters often draw on past socio-political occurrences, their public and sociocultural knowledge to engage in negative emotional characterizations and evaluations of the presidential candidates and the electoral commissioner, thereby, showing disapproval of them. By means of name-calling expressions, derogatory adjectives, and derogatory interjections, these commenters attack the intellectual abilities, moral behaviour, social competence, and physical characteristics of their targets. This negative communicative behaviour is attributed to the fact that on GhanaWeb, like on many online communities, interactants are not obliged to uphold communicative norms. The anonymity of GhanaWeb shields commenters from the repercussions such speech behaviours would otherwise necessitate. It is also attributed to the right to freedom of speech, the repeal of the criminal libel law, and the fact that traditionally, there are some specific settings that allow the use of insults against authority. The study demonstrates that although the practice of using insults against political leaders in online commentary appears to be “normal” in the online context, it is a transgression when one considers the Ghanaian sociocultural norms of communication. Consistent with this, although the interaction on GhanaWeb is asynchronous, some commenters play the role of regulatory figures and traditional gatekeepers of appropriate communicative behaviour by posting disapproving metapragmatic comments. The study identifies many ways in which interactions on GhanaWeb are culturally shaped. These include first, commenters do not only rely on English slang terms and swearwords, they also employ some local language insulting expressions (e.g. kwasea ‘fool’, yaka gbɔmɔ ‘worthless person’) due to the ‘weight’ such expressions carry for Ghanaians. Secondly, the metapragmatic comments do not employ certain key terms such as “offensiveness” that can be found in the literature on online studies related to the speakers of Inner Circle Englishes. Instead, they highlight the clear position of the Ghanaian culture on respect for elders and people in authority. These metapragmatic comments also stress the fact that the exchange of invective and insults in politics can result in violence which may have adverse effects on the development of the nation. Overall, the study is inclined towards the view that instead of generalizing the results of studies on abusive language use in online interactions, there is a need to draw out what distinguishes Ghanaian online interactants from, let’s say, Western countries, Asian countries, African countries, or from interactants elsewhere. The key claim is that understanding the sociocultural backdrop for the speech practices of online interactants is crucial to understanding their language use, as interactants draw on existing communicative repertoires which influence their language choices. Also, the study draws attention to news websites in Ghana as a platform where the use of unmitigated invective and insults against people of higher social status is increasingly becoming the norm, despite the fact it violates Ghanaian communicative values. Thus, this study serves as a springboard for future research into abusive language use in online interactions among Ghanaians.

: With better natural language semantic representations, computers can do more applications more efficiently as a result of better understanding of natural text. However, no single semantic representation at this time fulfills all requirements needed for a satisfactory representation. Logic-based representations like first-order logic capture many of the linguistic phenomena using logical constructs, and they come with standardized inference mechanisms, but standard first-order logic fails to capture the graded aspect of meaning in languages. Distributional models use contextual similarity to predict the graded semantic similarity of words and phrases but they do not adequately capture logical structure. In addition there are a few recent attempts to combine both representations either on the logic side (still, not a graded representation), or in the distribution side (not full logic). We propose using probabilistic logic to represent natural language semantics combining the expressivity and the automated inference of logic, and the gradedness of distributional representations. We evaluate this semantic representation on two tasks, Recognizing Textual Entailment (RTE) and Semantic Textual Similarity (STS). Doing RTE and STS better is an indication of a better semantic understanding. Our system has three main components, 1. Parsing and Task Representation, 2. Knowledge Base Construction, and 3. Inference. The input natural sentences of the RTE/STS task are mapped to logical form using Boxer which is a rule based system built on top of a CCG parser, then they are used to formulate the RTE/STS problem in probabilistic logic. Then, a knowledge base is represented as weighted inference rules collected from different sources like WordNet and on-the-fly lexical rules from distributional semantics.

The famous van Benthem theorem states that modal logic corresponds exactly to the fragment of first-order logic that is invariant under bisimulation. In this article we prove an exact analogue of this theorem in the framework of modal dependence logic (MDL) and team semantics. We show that Modal Team Logic (MTL) extending MDL by classical negation captures exactly the FO-definable bisimulation invariant properties of Kripke structures and teams. We also compare the expressive power of MTL to most of the variants and extensions of MDL recently studied in the area.

In this paper, I focus on some examples of semantic under-determinacy, which give rise to the view that language involves some kind of context-sensitivity. Jason Stanley and others have argued that such context-sensitivity could be traced to the terms’ syntax, or logical form. I offer an argument against this contention and defend a view in which context-sensitivity is traced to purely conceptual (semantic) constraints. I then explore the implications of this view on a conception of lexical meaning, urging that we should embrace a form of molecularism about the semantics of words.