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Combinatory Intensional Logic: Formal foundations

Clarence Protin

Abstract

Combinatory Intensional Logic (CIL) is a general framework for a formal theory of natural language meaning and reasoning, including intensional logic. What sets this approach apart is a syntax close to the logico-semantic mechanisms of natural language and being compatible with logical realism, the view that properties, relations and propositions are entities in their own right as well as furnishing the senses of linguistic expressions. CIL models, which formalize a realm of interweavings of senses, are not based on possible world semantics or set-theoretic function spaces. Truth-values and references of senses are extensions determined by states-of-affairs, an idea that goes back to the Stoics. CIL was initially inspired by Bealer's project in Quality and Concept (1983). It was subsequently found that CIL is a good tool to address the shortcomings and gaps present in Bealer's approach, in particular with regards to the soundness proofs and the problem of unifying intensional and modal logic. In this paper we focus exclusively on the formal and mathematical foundations of CIL including a sketch of the main soundness result. A second paper will delve into the philosophical motivation, context and justification as well as further technical development regarding philosophical problems involving definite descriptions, proper names, individual concepts and definitions.

Combinatory Intensional Logic: Formal foundations

Abstract

Combinatory Intensional Logic (CIL) is a general framework for a formal theory of natural language meaning and reasoning, including intensional logic. What sets this approach apart is a syntax close to the logico-semantic mechanisms of natural language and being compatible with logical realism, the view that properties, relations and propositions are entities in their own right as well as furnishing the senses of linguistic expressions. CIL models, which formalize a realm of interweavings of senses, are not based on possible world semantics or set-theoretic function spaces. Truth-values and references of senses are extensions determined by states-of-affairs, an idea that goes back to the Stoics. CIL was initially inspired by Bealer's project in Quality and Concept (1983). It was subsequently found that CIL is a good tool to address the shortcomings and gaps present in Bealer's approach, in particular with regards to the soundness proofs and the problem of unifying intensional and modal logic. In this paper we focus exclusively on the formal and mathematical foundations of CIL including a sketch of the main soundness result. A second paper will delve into the philosophical motivation, context and justification as well as further technical development regarding philosophical problems involving definite descriptions, proper names, individual concepts and definitions.
Paper Structure (9 sections, 17 theorems, 55 equations, 10 figures)

This paper contains 9 sections, 17 theorems, 55 equations, 10 figures.

Table of Contents

  1. Preliminaries
  2. Combinatory Intensional Logic
  3. From BL to CIL
  4. Graphical representation of CIL terms
  5. The sense-laws and reduction to the canonical form
  6. The logical rules of CIL
  7. CIL models
  8. Soundness of CIL
  9. Further developments and problems

Key Result

Lemma 1.2

Given $A \in fix (p)$ we can write $p$ in a unique way as the composition $p_1 p_2$ in which $p_2x = px$ for $x \in A$ and $p_2x = x$ otherwise and $p_1$ is the identity permutation when restricted to $A$.

Figures (10)

  • Figure 3:
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  • Figure 5:
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  • ...and 5 more figures

Theorems & Definitions (38)

  • Definition 1.1
  • Lemma 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5
  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 28 more