Abstract
Some authors, most notably Luciano Floridi, have recently argued for a notion of “strongly” semantic information, according to which information “encapsulates” truth (the so-called “veridicality thesis”). We propose a simple framework to compare different formal explications of this concept and assess their relative merits. It turns out that the most adequate proposal is that based on the notion of “partial truth”, which measures the amount of “information about the truth” conveyed by a given statement. We conclude with some critical remarks concerning the veridicality thesis in connection with the role played by truth and information as relevant cognitive goals of inquiry.
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Notes
- 1.
As a terminological remark, note that while “misinformation” simply denotes false or incorrect information, “disinformation” is false information deliberately intended to deceive or mislead.
- 2.
- 3.
Nothing substantial, in what follows, depends on such assumption.
- 4.
In the literature, it is usual to say that Eqs. (1) and (2) define the (amount of) “substantive information” or “information content” of \(A\), as opposed to the “unexpectedness” or “surprise value” of \(A\), which is defined as
[4, p. 20]. On this distinction, see for instance Hintikka [18, p. 313] and Kuipers [22, p. 865]. - 5.
- 6.
This does not mean that this assumption is the only culprit. As noted during discussion at the Trends in Logic XI conference, a way of avoiding BCP would be to adopt a non-classical logic according to which contradictions do not entail everything and hence are not maximally informative. Systems of this kind are provided by those “connexive logics” that reject the classical principle ex contradictione quodlibet (for all \(A\), \(\bot \) entails \(A\)) in favor of the (Aristotelian) intuition that ex contradictione nihil sequitur (cf. [33, Sect. 1.3]).
- 7.
In the following, we replace, without any significant loss of generality, Floridi’s talk of “infons”—“discrete items of factual information qualifiable in principle as either true or false, irrespective of their semiotic code and physical implementation” [10, p. 199]—by talk of sentences or statements in the given language \(\fancyscript{L}_{n}\).
- 8.
In logical parlance, a c-statement is a statement in conjunctive normal form such that each of its clauses is a single literal. Following Oddie [26, p. 86], a c-statement may be also called a “quasi-constituent”, since it can be conceived as a “fragment” of a constituent.
- 9.
- 10.
One may note that measure \( vs _{\phi }\) is not normalized, and varies between \(-\phi \) and \(1\). A normalized measure of the verisimilitude of \(A\) is \(( vs _{\phi }(A)+\phi )/(1+\phi )\), which varies between 0 and 1.
- 11.
Proof note that, among c-statements, \(A\) entails \(B\) iff \( b (A)\supseteq b (B)\). If both are true, this implies \( t (A,C_{\star })\supseteq t (B,C_{\star })\) and hence \( vs _{\phi }(A)= cont _{t}(A,C_{\star })\ge cont _{t}(B,C_{\star })= vs _{\phi }(B)\). For discussion of this Popperian requirement, see [24, pp. 186–187, 233, 235–236]. Note also that \( vs _{\phi }\) satisfies the stronger requirement that among true theories, the one with the greater degree of (true) basic content is more verisimilar than the other; i.e., if \(A\) and \(B\) are true and \( cont _{t}(A,C)> cont _{t}(B,C)\) then \( vs _{\phi }(A)> vs _{\phi }(B)\). \(\square \)
- 12.
Proof if \(A\) entails \(B\) and both are completely false, then \( f (A,C_{\star })\supseteq f (B,C_{\star })\) and hence \( cont _{f}(A,C_{\star })\ge cont _{f}(B,C_{\star })\). Since \(\phi \) is positive, it follows that \( vs _{\phi }(A)=-\phi cont _{f}(A,C_{\star })\le -\phi cont _{f}(B,C_{\star })= vs _{\phi }(B)\). \(\square \)
- 13.
Proof If \(A\) is true, then \( vs _{\phi }(A)= cont _{t}(A,C_{\star })\); if \(B\) is completely false, then \( vs _{\phi }(B)=-\phi cont _{f}(B,C_{\star })\); since \(\phi \) is positive, it follows that \( vs _{\phi }(A)> vs _{\phi }(B)\).
- 14.
Proof sketch When \(A\) is a c-statement, the constituents in its range are \(2^{n-k_A}\); it follows that \( vac (A)=2^{n-k_A} / 2n\), i.e., \(1 / 2^{k_A}\). Thus, among true c-statements, \( cont _{S}(A)=1-1 / 2^{k_A}\) co-varies with the degree of basic content of \(A\), \( b (A)=k_A / n\). As far as false c-statements are concerned, since \( inacc (A)=1- acc (A)\), \( cont _{S}\) co-varies with the accuracy of \(A\). \(\square \)
- 15.
Note that, by definition, a c-statement can not be contradictory; hence, \( cont _{S}^{*}\) is undefined for contradictions. Of course, it is always possible to stipulate, as Floridi does, that contradictions have a minimum degree of SSI.
- 16.
Note again that \( cont _{t}\) is undefined for contradictions, which can be assigned a minimum degree of SSI by stipulation. Interestingly, an argument to this effect was already proposed by Hilpinen [17, p. 30].
- 17.
I thank Gerhard Schurz for raising this point in discussion during the Trends in Logic XI conference.
- 18.
Usually, \(\varDelta _{}(C_i,C_j)\) is identified with the so-called normalized Hamming distance (or Dalal distance, as it is also known in the field of AI), i.e., with the number of literals on which \(C_i\) and \(C_j\) disagree, divided by the total number \(n\) of atomic sentences.
- 19.
Proof Note that, if \(A\) is a c-statement, all constituents \(C_i\) in the range of \(A\) (which are c-statements themselves) are “completions” of \(A\) in the sense that \( b (A)\subset b (C_i)\). The constituent in \(R(A)\) farthest from \(C_{\star }\) will be the one which makes all possible additional mistakes besides the mistakes already made by \(A\): this means that \(\varDelta _{ max }(A,C_{\star })=1-\frac{t_A}{n}\). It follows by (7) that \( pt (A)=1-(1-\frac{t_A}{n})= cont _{t}(A,C_{\star })\). \(\square \)
- 20.
This is still clearer if one consider the generalization of the definition above to arbitrary (non-conjunctive) statements \(A\). Given (18), the misinformation conveyed by \(A\) is given by \(1- pt (A)=\varDelta _{ max }(A,C_{\star })\), that reduces to \( misinf (A)\) as far as c-statements are concerned (the proof is straightforward, see footnote 19).
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Acknowledgments
This paper is based on presentations delivered at the Fourth Workshop on the Philosophy of Information (University of Hertfordshire, 10–11 May 2012) and at the Trends in Logic XI conference (Ruhr University Bochum, 3–5 June 2012). I thank the participants in those meetings, and in particular Luciano Floridi and Gerhard Schurz, for valuable feedback. This work was supported by Grant CR 409/1-1 from the Deutsche Forschungsgemeinschaft (DFG) as part of the priority program New Frameworks of Rationality (SPP 1516) and by the Italian Ministry of Scientific Research within the FIRB project Structures and dynamics of knowledge and cognition (Turin unit: D11J12000470001).
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Cevolani, G. (2014). Strongly Semantic Information as Information About the Truth. 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_5
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