Modal Calculus of Illocutionary Logic

The aim of illocutionary logic is to explain how context can affect the meaning of certain special kinds of performative utterances. Recall that performative utterances are understood as follows: a speaker performs the illocutionary act (e.g. act of assertion, of conjecture, of promise) with the illocutionary force (resp. assertion, conjecture, promise) named by an appropriate performative verb in the way of representing himself as performing that act. In the paper I proposed many-valued interpretation of illocutionary forces understood as modal operators. As a result, I built up a non-Archimedean valued logic for formalizing illocutionary acts. A formal many-valued approach to illocutionary logic was offered for the first time.

💡 Research Summary

The paper “Modal Calculus of Illocutionary Logic” proposes a novel formal framework for illocutionary logic by treating illocutionary forces (the pragmatic forces conveyed by performative verbs such as “promise,” “order,” “think,” etc.) as modal operators within a many‑valued, non‑Archimedean logical system. The author argues that conventional approaches to illocutionary logic—most notably the semantic‑phenomenological model of Searle and Vanderveken—are limited because they either rely on a Fregean compositional principle (the meaning of a complex expression is a function of the meanings of its parts) or they do not provide a systematic way to capture the “success” or “failure” of an illocutionary act in a formal semantics.

The core of the proposal is a four‑valued matrix M = ⟨{1, ½, 0, –½}, {1}, ¬, F, ⇒, ∨, ∧⟩. The set {1, 0} are the classical truth values for propositional content; the additional values ½ and –½ represent, respectively, a successful performance of an illocutionary act and an unsuccessful one. The unary operator F denotes the illocutionary force of a performative verb; for the illustrative case of the verb “think” the author defines F’s truth‑functional behavior together with negation ¬. Important algebraic properties are proved: (1) a ≥ F(a), (2) ¬a ≥ ¬F(a), (3) F(a)∧F(b) ≥ F(a∧b), (4) F(a)∨F(b) ≤ F(a∨b), (5) F(a⇒b) ≥ F(a⇒b), (6) idempotence F(F(a)) = F(a), and (7) commutation with negation ¬F(a) = F(¬a). These conditions capture the intuition that a successful “think” of Φ entails the truth of Φ, while the converse does not hold, and they generate a series of illocutionary tautologies such as F(Φ)⇒Φ and ¬F(Φ)⇒¬Φ.

The paper distinguishes two kinds of composite expressions. When a composite evaluates to 0 or 1 it follows the Fregean pattern: the whole’s meaning is determined inductively from its parts. When it evaluates to ½ or –½, the composition is non‑Fregean: the ordering of truth values is reversed (conjunction behaves like supremum, disjunction like infimum), so the meaning of the whole can alter the meanings of the components. This duality allows the system to model both ordinary propositional reasoning and the pragmatic “success” dimension of speech acts within a single algebraic structure.

To move beyond the single verb “think,” the author introduces a partial order ≤ on the (potentially infinite) set of all illocutionary forces. Ve(F₁(Φ)) ≤ Ve(F₂(Ψ)) holds exactly when, in every context, performing F₁(Φ) necessarily entails performing F₂(Ψ). This ordering makes it possible to define notions of opposition and contradiction among forces. The paper then constructs a square of opposition for illocutionary acts: given two forces F₁ and F₂ that are contrary (they cannot both be successful on the same proposition), their negations ¬F₁ and ¬F₂ complete the classic four‑corner diagram (contraries, subcontraries, contradictories, subalternations). Concrete linguistic examples such as “order” vs. “forbid” or “bless” vs. “damn” illustrate how the square captures semantic incompatibility and mutual exclusivity of certain performative acts.

Strengths of the work include (i) a clear formalization of the success/failure dimension of speech acts, (ii) an elegant integration of Fregean and non‑Fregean composition within a single many‑valued algebra, and (iii) the introduction of an ordering and opposition structure that mirrors classical logical squares while being tailored to pragmatic forces. The approach also opens the door to a systematic treatment of all performative verbs, something that previous phenomenological models handled only informally.

However, the paper has notable limitations. The four‑valued matrix is illustrated only for “think”; no explicit matrices are provided for other verbs, leaving the generalization somewhat abstract. The choice of ½ and –½ as success markers is motivated intuitively but lacks empirical validation—real conversational data might exhibit more graded or context‑sensitive notions of success. Moreover, the paper does not address meta‑logical properties such as completeness, soundness, or decidability of the resulting system, nor does it discuss how the non‑Archimedean ordering interacts with standard modal logics. The presentation suffers from typographical errors in the formal definitions, which hampers readability and reproducibility.

In summary, the article makes a significant conceptual contribution by proposing a modal‑calculus‑style, many‑valued logic for illocutionary acts, uniting pragmatic success with propositional truth, and by introducing an ordering and opposition framework for performative forces. Future work should flesh out concrete semantics for a broader set of verbs, provide rigorous meta‑theoretical results, and test the model against real‑world linguistic corpora.