Abstract
In this paper, we will be interested in the notion of a computable proposition. It allows for feasible computational semantics of empirical sentences, despite the fact that it is in general impossible to get to the truth value of a sentence through a series of effective computational steps. Specifically, we will investigate two approaches to the notion of a computable proposition based on constructive type theory and transparent intensional logic. As we will see, the key difference between them is their accounts of denotations of empirical sentences.
Work on this paper was supported by Grant Nr. 19-12420S from the Czech Science Foundation, GA ČR.
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Notes
- 1.
For simplicity, we assume we are working directly with the mathematical objects, thus ignoring the syntactic layer for the sake of the semantic one. For example, consider the difference between a mathematical expression ‘\(2 + 2\)’ and a mathematical object \(2 + 2\): while the former can be reduced to the numeral ‘4’, the latter can be computed to the number 4.
- 2.
- 3.
This example as well as the whole idea of computing propositions to canonical forms not to truth values was proposed in [15], a paper presented at a workshop on Frege at the University of Leiden in August 25, 2001, transcribed by Bjørn Jespersen.
- 4.
Since we are mainly interested in the notion of a computable proposition, we intentionally choose very basic examples of propositions such as ‘\(\lnot A = A \supset \bot \)’ or ‘Charles is a bachelor’ to keep the focus on the computability aspect rather than on the propositional aspect. To learn how to analyze more complex sentences in CTT, consult, e.g., [2, 12, 25]. For TIL, see, e.g., [7, 21, 23].
- 5.
This investigation can be viewed as a follow up to the paper [18] that explores how these two systems approach mathematical and logical propositions.
- 6.
- 7.
- 8.
For an excellent overview, see, e.g., [31].
- 9.
For simplicity, we omit the Haskell Curry’s side of the discovery – the fact that combinators from combinatory logic correspond to axioms in Hilbert-style calculus. For example, the combinator \(\mathsf {K}\) (i.e., \(\mathsf {K} x y = x\)) corresponds to the axiom \(A~\rightarrow ~B~\rightarrow ~A\).
- 10.
Or more precisely, ‘What is a constructive set?’ since in CTT the word ‘type’ is typically reserved for the higher-order presentation of CTT, which we will not use here. In CTT, one typically starts with the notion of a set and defines a proposition in terms of it. The category of sets is then identified with the category of propositions, which is the way the propositions as types principle is adopted in CTT. For more, see, e.g., [16].
- 11.
The natural number \(2 + 2\) is considered non-canonical because it does not follow the forms prescribed by the introduction rules for the type N, i.e., it is neither 0 nor does it have the form s(n).
- 12.
See, e.g., [20], pp. 21–26. This, as we will see, is in contrast with TIL, where meanings and denotations are kept apart and viewed as entities of distinct kinds.
- 13.
- 14.
For a proper specification, see [9], especially sections §4. Noncanonical sets and elements and §5. Nominal definitions. It is also worth to note that non-canonical propositions/sets are already considered in [13], however, as opposed to [9], no dedicated proposition-computability judgments of the form \(A \Rightarrow B : prop\) are used.
- 15.
Of course, more complex reductions for other logical and/or mathematical propositions can be introduced. For example, [24] (pp. 41–42) shows how we can in CTT define, and thus reduce the propositional function prime(x) (assuming x : N) into more basic concepts. Formulating the corresponding computation rule based on the provided definition is then straightforward.
- 16.
Note that our approach has nothing further to say about the meaning of the conjuncts of this canonical proposition. More specifically, the meaning of atomic empirical propositions such as man(Charles) is assumed to be given externally.
- 17.
From this perspective, TIL types are much closer to categories in CTT (i.e., types in proper CTT terminology), but even a category is a stricter notion as it has to come with a criterion of application and identity, which is not the case with TIL types.
- 18.
In standard TIL terminology, the term ‘hyperproposition’ is used instead and the term ‘proposition’ is reserved for functions from possible worlds and time moments to truth values. We will diverge from this terminology.
- 19.
In standard TIL, denotations of propositions are understood as functions from possible world and time moments to truth values, however, we omit the time parameter for simplicity here.
- 20.
The purpose of the bold font is, simply put, to distinguish between the constructional level and non-constructional level. Let us take, e.g., \(\mathbf{2} \) and 2. What is the difference between them? The former is a construction, the latter is a constructed object. In other words, \(\mathbf{4} \) can be understood as a trivial computation that yields the number 4 as a result. Furthermore, we now switch to the standard TIL notation with square brackets to better distinguish its expressions from those of CTT.
- 21.
Strictly speaking, in TIL we can have higher-order constructions that yield lower-order constructions as their denotations, but for simplicity of presentation, we omit these cases here.
- 22.
Note that in comparison with CTT’s semantic scheme, here we have three levels of objects: syntactic (the expression ‘\(2+2\)’), semantic (the construction \([\mathbf{2} \; \mathbf + \; \mathbf{2} ]\)), and denotational (the natural number 4). Recall that in CTT, there are only two levels: syntactic and semantic. In other words, in TIL the semantic level is distinguished further into constructional and denotational levels.
- 23.
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Pezlar, I. (2021). Meaning and Computing: Two Approaches to Computable Propositions. In: Silva, A., Wassermann, R., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2021. Lecture Notes in Computer Science(), vol 13038. Springer, Cham. https://doi.org/10.1007/978-3-030-88853-4_7
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