Simon D. Vonlanthen

[This is a pre-publication version of a thesis to be published in the near future] This thesis investigates the relationship between natural languages and those of formal logic. It focusses on mismatches, which primarily concern semantic differences between logical constants in formal systems and their designated counterparts in natural language. These mismatches are directly relevant to the goals of philosophical logic, which primarily aims to provide tools for assessing reasoning, the latter which is most often carried out in the vernacular. Furthermore, they are important to various debates in the philosophy of language and the philosophy of logic, not to mention to linguistics and neighbouring fields. Despite this relevance, mismatches themselves are rarely discussed in the literature. This thesis aims to fill this lacuna. Chapter 1 establishes the methodological background, which is that of so-called ‘connective analysis’. It also introduces the main ideas behind use theories of meaning, the central approach to linguistic meaning discussed in this book. Chapter 2 further delves into the topic of meaning by discussing inferentialism. It finds its main variations wanting, and develops a novel version that adopts a middle ground position, thereby combining the advantages of its predecessors. Chapter 3 defends this version against objections from the principle of compositionality. It also discusses the subject matter of formal semantics, thereby clarifying the relationship between use theories and the former. Chapter 4 deals with the mismatches themselves, surveying their different forms as discussed since modern logic’s inception. It argues that there are no ‘qualitative’ mismatches between the two types of languages, and that natural language contains a relation of logical consequence. This vindicates a conception of logic in which the prior analyses of the vernacular under use-theoretic lights is methodologically central. This approach is then applied starting in Chapter 5, which studies the uses of modal expressions and quantifiers in natural language, their interactions in particular. This will lead to the introduction of the so-called ‘quantified argument calculus’ (‘Quarc’), a logic of quantification arguably closer to natural language than the predicate calculus. Afterwards, Chapter 6 introduces two-valued modal extensions of Quarc, and proves soundness and completeness theorems for a family of associated Gentzen-style natural deduction systems. Chapter 7 applies one of these systems to a specific type of categorical reasoning, and discusses an associated polygon of opposition. Finally, Chapter 8 considers three-valued modal extensions of Quarc, motivated by phenomena related to presupposition-failure. This will lead to the introduction of a tableaux system, whose soundness and completeness will also be proven. Thus, overall, this thesis vindicates the use of natural language as tool to study logic, and of logic as a tool to study natural language.

We investigate basic forms of inference involving modal notions and quantifiers, called modal categorical inferences. We do so by extending Quarc, a novel logic that assigns a primary role to quantified phrases, with modalities from the hexagon of opposition. We show that there are two possible readings of de dicto modalities (called symmetric and asymmetric, respectively), as opposed to the unique reading of de re modalities. We focus on the asymmetric reading of de dicto modalities and explore the logical relations that obtain between them. These are proven in a natural deduction system, accompanied by an appropriate syntax and semantics, and graphically represented via a dodecagon of opposition. Moreover, we show that the asymmetric reading, in contrast to the symmetric one, preserves all properties of the hexagon for basic modal notions. Thus, it provides a particularly attractive basis on which to further investigate quantified modal reasoning.

The quantified argument calculus (Quarc) is a novel logic that departs in several ways from mainstream first-order logic. In particular, its quantifiers are not sentential operators attached to variables, but attach to unary predicates to form arguments – quantified arguments – of other predicates. Furthermore, Quarc includes devices to account for anaphora, active-passive-voice distinctions, and sentence- versus predicate-negation. While this base system has already been shown to be sound and complete, modal extensions still lack such results. The present paper fills this lacuna by developing a modal extension of Quarc, including identity. The semantics will invalidate the Quarc-analogues of the Barcan-formula and its converse, as well as treat identity as contingent by default. Furthermore, an unlabelled Gentzen-style natural deduction system will be presented, which includes the full expressive power of Quarc. It will be shown to be strongly sound and complete with respect to relational frames. The paper closes off with considerations on how to extend the system to cover other normal modal logics as well as extensions to suitable three-valued semantics that capture relevant types of presupposition-failure.