Abstract
In the present chapter, we discuss the possibility of generalizing the very notion of a truth value by constructing truth values as complex units which may possess a ramified inner structure. We consider some approaches to truth values as structured entities and summarize this point in the notion of a generalized truth value conceived as a subset of some basic set of initial truth values of a lower degree. We are essentially led by the idea which is at the heart of Belnap and Dunn’s useful four-valued logic, where the set \({{\mathbf{2}}} = \{T,F\}\) of classical truth values is generalized to the set \({{\mathbf{4}}} = {{\fancyscript{P}}}({{\mathbf{2}}}) = \{\varnothing,\{T\},\{F\},\{T,F\}\}.\) We argue in favor of extending this process to the set \({{\mathbf{16}}} = {{\fancyscript{P}}}({{\mathbf{4}}}).\) It turns out that this generalization is well-motivated and leads to a notion of a truth value multilattice. In particular, we proceed from the bilattice \(FOUR_{2}\) with both an information and truth-and-falsity ordering to another algebraic structure, namely the trilattice \(SIXTEEN_{3}\) with an information ordering together with a truth ordering \(and\) a (distinct) falsity ordering. We also consider another exemplification of essentially the same structure based on the set of truth values one can find in various constructive logics.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Dunn had already developed this approach in his doctoral dissertation [73] and then presented it in a number of conference talks and publications, most notably in [75] (see also [74]). The reader may consult [79, 80] for a comprehensive account and systematization of Dunn’s (and other) work in this area (cf. also [261]). In the literature, the semantic strategy in question is sometimes called the “American Plan” as opposed to the so-called “Australian Plan”. Both labels were brought into usage by Meyer [173] to contrast the four-valued approach of the “Americans” Dunn and Belnap to the star semantics of the “Australians” Routley and himself.
- 2.
In [231, p. 762] valuations of this kind have been called \(multivaluations\).
- 3.
The idea of the four truth values has been also expressed by Scott in [219], p. 170.
- 4.
Bilattices have been studied by many authors in various contexts, see, e.g., [7, 8, 10, 11, 115, 217], [86–92] and references therein.
- 5.
The information order is sometimes referred to as a “knowledge order” (denoted by \({\leq}_{k}\)), which is not quite accurate from a philosophical point of view when we are taking into account the classical definition of knowledge as justified true belief.
- 6.
As Fitting put it: “[W]e might call a truth value \(t_{1}\) less-true-or-more-false than \(t_{2}\) if \(t_{1}\) contains false but \(t_{2}\) doesn’t, or \(t_{2}\) contains \(true\) but \(t_{1}\) doesn’t” [89, p. 94].
- 7.
As R. Meyer put it: “[I]f we take seriously both true and false and neither true nor false separately, what is to prevent our taking them seriously conjunctively? As in ‘It is both true and false and neither true nor false that snow is white’ ” [173, p. 19]. As we will argue below, Meyer’s own answer to this question—“This way, in the end, lies madness” (i173) appears a bit overhasty.
- 8.
The idea to generalize Belnap’s construction by considering truth values as subsets of a set containing more than two elements (\(T\) and \(F\)) has been also expressed by Karpenko in [145, p. 46].
- 9.
A “Belnap computer” is just a computer that uses Belnap’s four-valued logic. Note also that it is not crucial for our example to have exactly \(four\) Belnap computers. There can well be more of them or fewer—even one would be enough. The main point is that it should be connected to some “higher” computer.
- 10.
Note that we could introduce sets \(x^{-t}\) and \(x^{-f}\) for \(FOUR_{2}\) as well, for example, as follows:
$$ x^{-t} :=\left\{ z\in x\mid z\neq T\right\} ; \quad x^{-f} :=\left\{ z\in x\mid z\neq F\right\} . $$It turns out then, however, that \(x^{-t}=x^{f}\) and \(x^{-f}=x^{t},\) which once again confirms our observation that truth and falsity in \(FOUR_{2}\) are still interdependent.
- 11.
Similarly, T and F from 4 can be viewed as analogues (or representatives) of the classical values \(T\) and \(F.\)
- 12.
Thus, Belnap’s informational interpretation of generalized truth values is not just an incidental façon de parler, but expresses the very essence of his construction. Therefore it is not by chance that this semantics has found so many fruitful applications in theoretical computer science and other areas related to information theory.
- 13.
Note that according to Propositions 3.2 and 3.4, any 16-valuation for an arbitrary formula \(A\) can be unambiguously modeled by a certain combination of the expressions \({\mathbf N }\in v^{16} \left( A\right) , {{\mathbf{F}}}\in v^{16} \left( A\right) , {{\mathbf{T}}}\in v^{16} \left( A\right) , {{\mathbf{B}}}\in v^{16} \left( A\right) \) and their negations. For example \(v^{16}\left ( A\right) = {{\mathbf{NT}}}\) is representable as \({{\mathbf{N}}}\in v^{16} \left(A\right) , {{\mathbf{F}}}\notin v^{16} \left( A\right) , {{\mathbf{T}}}\in v^{16} \left( A\right)\) and \({{\mathbf{B}}}\notin v^{16} \left( A\right),\) etc. This will greatly simplify the whole semantic exposition, see Chap. 5.
- 14.
An approach to a generalization of truth values in intuitionistic logic and formulating on this base the notion of a relevant intuitionistic entailment has been developed in [222–226]. Dunn in [80] presents the results of a similar generalization of Nelson’s logic. The construction below arises from a unification of both approaches into a joint framework. An intuitionistic (constructive) relevant logic has also been developed by some other authors, Garrel Pottinger and Neil Tennant among them, see, for example, [195, 249, 250].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Shramko, Y., Wansing, H. (2011). Generalized Truth Values: From FOUR 2 to SIXTEEN 3 . In: Truth and Falsehood. Trends in Logic, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0907-2_3
Download citation
DOI: https://doi.org/10.1007/978-94-007-0907-2_3
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0906-5
Online ISBN: 978-94-007-0907-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)