Abstract
The approach of Transparent Intensional Logic to truth differs significantly from rivalling approaches. The notion of truth is explicated by a three-level system of notions whereas the upper-level notions depend on the lower-level ones. Truth of possible world propositions lies in the bottom. Truth of hyperintensional entities—called constructions—which determine propositions is dependent on it. Truth of expressions depends on truth of their meanings; the meanings are explicated as constructions. The approach thus adopts a particular hyperintensional theory of meanings; truth of extralinguistic items is taken as primary. Truth of expressions is also dependent, either explicitly or implicitly, on language (its notion is thus also explicated within the approach). On each level, strong and weak variants of the notions are distinguished because the approach employs the Principle of Bivalence which adopts partiality. Since the formation of functions and constructions is non-circular, the system is framed within a ramified type theory having foundations in simple theory of types. The explication is immune to all forms of the Liar paradox. The definitions of notions of truth provided here are derivation rules of Pavel Tichý’s system of deduction.
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Notes
- 1.
Such gradual construction was in fact suggested by Tichý in his remarks on truth [13, Chaps. 11 and 12]. There, certain (verbal) definitions of the notions can be found. Tichý’s investigations surely inspired my approach. The present paper is an extract from a large manuscript on truth; some of my results have been published in Raclavský [10].
- 2.
It is in the spirit of intensional explication of our conceptual scheme to say that propositions can be construed as facts and our world can be construed as a collection of (actual) facts. Then, the proposal of TIL confirms a sort of correspondence theory of truth (true sentences correspond to facts that obtain). However, these issues cannot be discussed here.
- 3.
The Principle of Bivalence adopted here reads as: for any proposition \(P\), \(P\) has at most one of the two truth-values T and F in a given \(W\) and \(T\). In other words, a proposition can be gappy; for instance, the proposition “The king of France is bald” is gappy in the actual \(W\) and present \(T\). (Note that I use single quotation marks for quotation of expressions or, sometimes, for indication of a shift in meaning; double quotation marks are used for indication of propositions and other extralinguistic entities.)
- 4.
In TIL, possible world intensions (i.e. propositions, properties, relations-in-intension, etc.) are total or partial functions from world-time couples to certain entities (viz. truth-values, classes of objects, classes of \(n\)-tuples of objects, etc.). Among non-intensions one can find in TIL, e.g., the well-known classical truth-functions \(\lnot \), \(\rightarrow \), \(\wedge \), and \(\vee \), the well-known subclasses of classes of \(\xi \)-objects \(\exists ^{\xi }\) and \(\forall ^{\xi }\) (for any type \(\xi \); the indication ‘\(^{\xi }\)’ will be usually suppressed), or the well-known identity relation between \(\xi \)-objects, =\(^{\xi }\). Constructions are also non-intensions.
- 5.
One of the notorious arguments for adoption of hyperintensions is that due to intensional analysis, beliefs which are equivalent but non-identical are merged to one. On such use of possible world propositions, an argument that one believes that \(1+1=2\) thus one believes Fermat’s Last Theorem is wrongly rendered as valid.
- 6.
The stratification of entities into such hierarchy is justified by four Vicious Circle Principles [10], each of them being entailed by the Principle of Specification: you cannot fully specify an entity by means of the entity itself.
- 7.
I view definitions as certain \(\Leftrightarrow \)-rules (both \(\Rightarrow \) and \(\Leftrightarrow \) concern satisfiability of sequents). Two constructions flanking \(\Leftrightarrow ^{\xi }\) are \(v\)-congruent for any \(v\); the type of the object \(v\)-constructed by both constructions will be indicated nearby ‘\(\Leftrightarrow \)’. Definitions can also be viewed as proposing an explication of the intuitive notion whose rigorous correlate occurs in the left hand side of the definition; its right hand side shows in which sense the notion ‘is meant’, which objects ‘fall under’ it, cf. [10].
- 8.
‘\(C_{wt}\)’ abbreviates ‘[[\(C \; w\)] \(t\)]’.
- 9.
It is just this notion which should be deployed in appropriate reformulations of classical laws in order to be valid within a framework adopting partiality.
- 10.
For that purpose a bit richer type basis is needed.
- 11.
Some of them might be defined also by other theoreticians (assuming here translatability of their results to the present framework).
- 12.
Truth of expressions’ tokens can be defined as dependent on truth of expressions. It is entirely omitted in this paper.
- 13.
To ask for an expression’s meaning in a hierarchy of codes amounts to ask for its meaning in the (virtually) highest code of the hierarchy, i.e. \(L^n\).
- 14.
The typing technique within Tichý’s type theory is similar to that in Russellian ramified type theories.
- 15.
Recall that if properly closed by lambdas, both constructions flanking \(\Leftrightarrow ^\mathrm{o } v\)-construct one and the same property. It can be proved that the property cannot be discussed by \(\mathrm L ^n\), cf. our discussion below.
- 16.
- 17.
In insufficiently expressive codes-languages, a partial truth-predicate can be meaningful without a risk of the Liar paradox (cf. [11]).
- 18.
Sufficient richness was an original Tarski’s condition, cf. his [12].
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Raclavský, J. (2014). Explicating the Notion of Truth Within Transparent Intensional Logic. In: Ciuni, R., Wansing, H., Willkommen, C. (eds) Recent Trends in Philosophical Logic. Trends in Logic, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-319-06080-4_12
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