The Rule of Existential Generalisation and Explicit Substitution

The present paper offers the rule of existential generalization (EG) that is uniformly applicable within extensional, intensional and hyperintensional contexts. In contradistinction to Quine and his followers, quantification into various modal contexts and some belief attitudes is possible without obstacles. The hyperintensional logic deployed in this paper incorporates explicit substitution and so the rule (EG) is fully specified inside the logic. The logic is equipped with a natural deduction system within which (EG) is derived from its rules for the existential quantifier, substitution and functional application. This shows that (EG) is not primitive, as often assumed even in advanced writings on natural deduction. Arguments involving existential generalisation are shown to be valid if the sequents containing their premises and conclusions are derivable using the rule (EG). The invalidity of arguments seemingly employing (EG) is explained with recourse to the definition of substitution.

Pure type systems with explicit substitutions

We introduce a new formulation of pure type systems (PTSs) with explicit substitution and de Bruijn indices and formally prove some of its meta-theory. Using techniques based on Normalisation by Evaluation, we prove that untyped conversion can be typed for predicative PTSs. Although this equivalence was settled by Siles and Herbelin for the conventional presentation of PTSs, we strongly conjecture that our proof method can also be applied to PTSs with η.

Complex Syntactic Whole: the Use of Grammatical Forms of the Predicate for Expressing Communicative Intentions

The article considers the role of the grammatical forms of the predicate in expressing communicative intentions of the speaker within complex syntactic whole. Analyzing the specifics of substitution, the author points out that implicit substitution, or deletion, is made with the help of zero substitute, while explicit substitution or replacement, implies the use of the verb ‘to do’. This type of substitution exists as a separate grammatical phenomenon based on the opposition to the explicit form of the predicate. Auxiliary and modal verbs serve as markers of implicit (zero) substitution of the lexical part of the predicate. The substitute verb ‘to do’, due to its wide semantics, often replaces ‘action verbs’. In most cases, these two types of predicate substitution and the related phenomena are to be considered in a wide context – a complex syntactical whole. An analysis of opposition of grammatical forms of the predicate, with the aim of defining communicative intentions of the speaker or new implications relevant for both communicants and ways of effecting the addressee, shows that such an opposition helps to build the expressive and pragmatic potential of the statement.