In proof-theoretic semantics the meaning of an atomic sentence is usually determined by a set of derivations in an atomic system which contain that sentence as a conclusion (see, in particular, Prawitz, 1971, 1973). The paper critically discusses this standard approach and suggests an alternative account which proceeds in terms of subatomic introduction and elimination rules for atomic sentences. A simple subatomic normal form theorem by which this account of the semantics of atomic sentences and the terms from which they (...) are composed is underpinned, shows moreover that the proof-theoretic analysis of first-order logic can be pursued also beneath the atomic level. (shrink)

The operators of first-order logic, including negation, operate on whole formulae. This makes it unsuitable as a tool for the formal analysis of reasoning with non-sentential forms of negation such as predicate term negation. We extend its language with negation operators whose scope is more narrow than an atomic formula. Exploiting the usefulness of subatomic proof-theoretic considerations for the study of subatomic inferential structure, we define intuitionistic subatomic natural deduction systems which have several subatomic operators and an additional operator for (...) formula negation at their disposal. We establish normalization and subexpression property results for the systems. The normalization results allow us to formulate a proof-theoretic semantics for formulae composed of the subatomic operators. We illustrate the systems with applications to reasoning with combinations of sentential negation, predicate term negation, subject term negation, and antonymy. (shrink)

A first-order language with a defined identity predicate is proposed whose apparatus for atomic predication is sensitive to grammatical categories of natural language. Subatomic natural deduction systems are defined for this naturalistic first-order language. These systems contain subatomic systems which govern the inferential relations which obtain between naturalistic atomic sentences and between their possibly composite components. As a main result it is shown that normal derivations in the defined systems enjoy the subexpression property which subsumes the subformula property with respect (...) to atomic and identity formulae as a special case. The systems admit a proof-theoretic semantics which does not only apply to logically compound but also to atomic and identity formulae—as well as to their components. The potential of the defined systems for a meticulous first-order analysis of natural inferences whose validity crucially depends on expressions of some of the aforementioned categories is demonstrated. (shrink)

The paper presents an alternative substitutional semantics for first-order modal logic which, in contrast to traditional substitutional (or truth-value) semantics, allows for a fine-grained explanation of the semantical behavior of the terms from which atomic formulae are composed. In contrast to denotational semantics, which is inherently reference-guided, this semantics supports a non-referential conception of modal truth and does not give rise to the problems which pertain to the philosophical interpretation of objectual domains (concerning, e.g., possibilia or trans-world identity). The paper (...) also proposes the notion of modality de nomine as an alternative to the denotational notion of modality de re. (shrink)

The analysis of atomic sentences and their subatomic components poses a special problem for proof-theoretic approaches to natural language semantics, as it is far from clear how their semantics could be explained by means of proofs rather than denotations. The paper develops a proof-theoretic semantics for a fragment of English within a type-theoretical formalism that combines subatomic systems for natural deduction [20] with constructive (or Martin-Löf) type theory [8, 9] by stating rules for the formation, introduction, elimination and equality of (...) atomic propositions understood as types (or sets) of subatomic proof-objects. The formalism is extended with dependent types to admit an interpretation of non-atomic sentences. The paper concludes with applications to natural language including internally nested proper names, anaphoric pronouns, simple identity sentences, and intensional transitive verbs. (shrink)

The paper develops a proof-theoretic semantics for belief reports by extending the constructive type-theoretical formalism presented in Więckowski with a specific kind of set-forming operator suited for the representation of belief attitudes. The extended formalism allows us to interpret constructions which involve, e.g., iteration of belief, quantifying into belief contexts, and anaphora in belief reports. Moreover, constructive solutions to canonical instances of the problem of hyperintensionality are suggested. The paper includes a discussion of Ranta’s constructive account of belief reports.

We define intuitionistic subatomic natural deduction systems for reasoning with elementary would-counterfactuals and causal since-subordinator sentences. The former kind of sentence is analysed in terms of counterfactual implication, the latter in terms of factual implication. Derivations in these modal proof systems make use of modes of assumptions which are sensitive to the factuality status of the formula that is to be assumed. This status is determined by means of the reference proof system on top of which a modal proof system (...) is defined. The introduction and elimination rules for counterfactual (resp. factual) implication draw on this status. It is shown that derivations in the systems normalize and that normal derivations have the subexpression/subformula property. An intuitionistically acceptable proof-theoretic semantics is formulated in terms of canonical derivations. The systems are applied to so-called counterpossibles and to related constructions. (shrink)

It is argued that the intuitionistic conception of negation as implication of absurdity is inadequate for the proof-theoretic semantic analysis of negative predication and distinctness. Instead, it is suggested to construe negative predication proof-theoretically as subatomic derivation failure, and to define distinctness—understood as a qualified notion—by appeal to negative predication. This proposal is elaborated in terms of intuitionistic bipredicational subatomic natural deduction systems. It is shown that derivations in these systems normalize and that normal derivations have the subexpression (incl. subformula) (...) property. A proof-theoretic semantics for negative predication and distinctness is defined, and an intuitionistic conception of truth based on canonical derivations is proposed. An application to the doctrine of the Trinity, to which negative predication and distinctness are central, illustrates the systems and their proof-theoretic semantics. (shrink)

This dissertation develops a substitutional semantics for first-order (modal) logic which, unlike truth-value semantics, allows a fine-grained analysis of the semantical behaviour of the terms and predicates from which atomic formulae are composed. Moreover, it proposes a nondenotational philosophical foundation for the semantics of substitutional quantified (modal) logic.