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Strong Kleene Metainferential Logics

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Metainferential Logics

Part of the book series: Trends in Logic ((TREN,volume 61))

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Abstract

In this chapter, we will introduce the most studied family of pure and impure mixed metainferential logics of level 1: the Strong Kleene metainferential logics. We will only focus on some of them: the ones that can be characterized through the inferential logics \(\textbf{K3}, \textbf{LP}, \textbf{ST }\), and \(\textbf{TS }\), which will be introduced below.

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Notes

  1. 1.

    Though strictly speaking, boolean valuations are not Strong Kleene valuations, they are obviously equivalent to these Strong Kleene valuations.

  2. 2.

    Although we won’t expand on this point, it is important to mention it since it helps to deeply understand the connection between \(\textbf{ST }\) and \(\textbf{CL }\). See e.g. [1] for more details.

  3. 3.

    Meaning that \(\langle x, y\rangle \le _{lex} \langle x', y'\rangle \) if and only if \(x\le x'\) and \(y\le y'\).

  4. 4.

    In the sense that it is locally invalid in \(\textbf{ST }/\textbf{ST }\).

  5. 5.

    For an extensive presentation of \(\textbf{ST }\), see Cobreros, Ripley, Egré and van Rooij [4], Ripley [5, 6] and Cobreros, Ripley, Egré and van Rooij [7]. Though originally applied to vagueness and semantic paradoxes by these authors, Hlobil [8] offers an interpretation of this logic in terms of truth-makers.

  6. 6.

    \(\textbf{TS }\) is discussed by e.g., Cobreros, Ripley, Egré and van Rooij [4], and also by Chemlá, Egré and Spector [11]. Moreover, it was also discussed by Malinowski [12] as a tool to model empirical inference, and more recently was stressed by Rohan French [13], in connection with the paradoxes of self-reference.

  7. 7.

    Shramko and Wansing offer in [14] a way to read these two kinds of logics. While some logics, like \(\textbf{ST }\), take as valid derivations of conclusions whose degree of strength (i.e., the conviction in its truth) is smaller than the one of the premises, other logics, like \(\textbf{TS }\), are designed to qualify as valid derivations of true sentences from non-refuted premises (understood as hypotheses).

  8. 8.

    Barrio, Rosenblatt and Tajer [15], Dicher and Paoli [16] and Pynko [17] have shown that—through some suitable translation—the set of valid inferences in \(\textbf{LP}\) coincides with the set of valid metainferences in \(\textbf{ST }\). Moreover, French [13], have conjectured that—again, through some suitable translation—the set of valid inferences in \(\textbf{K3}\) coincides with the set of valid metainferences in \(\textbf{TS }\).

  9. 9.

    As we have already mentioned, a similar move can be made using different notions of being a counterexample, as Ripley has recently done in [18]. Nevertheless, we will stick to the way they have been initially presented in Pailos [19].

  10. 10.

    This means that we will be explicit about the four metainferential mixed but pure logics about which Pailos [19] says nothing about—i.e., \(\mathbf {ST/ST}\), \(\mathbf {TS/TS}\), \(\mathbf {LP/LP}\) and \(\mathbf {K_3/K_3}\). His silence might be explained by the (presumed) fact that he is identifying them with the inferential and well-known logics \(\textbf{ST }\), \(\textbf{LP}\), \(\textbf{TS }\), and \(\textbf{K3}\), respectively. But this identification can and has been, resisted. See, in particular, what Ripley says in [18]. There, Ripley defends the view that inferential logics are silent about higher-order behavior. (And, more generally, that an n-logic is silent about what happens in higher-than-n metainferential levels).

  11. 11.

    A detailed review of each of these impure logics can be found in Pailos [19].

  12. 12.

    In order to make sense to these rules we would need to take the collections of formulas as lists or sequences of formulas, in the case of Exchange, and as multisets in the case of Contraction.

  13. 13.

    First established in [19], and whose consequences are explored, e.g., in Barrio, Pailos, and Szmuc [21], Pailos [21] or Scambler [22].

  14. 14.

    Nevertheless, it is not true that a valuation v is a counterexample in \(\textbf{CL }\) to an inference \(\Gamma \Rightarrow \Delta \) if and only if v is a counterexample to that inference in \(\textbf{ST }\). Consider for example the \(\textbf{SK}\)-valuation \(v^{**}\) such that \(v^{**}(p)=1\) and \(v(q)=\frac{1}{2}\). The valuation \(v^{**}\) is a counterexample in \(\textbf{ST }\) to the inference \(\psi \Rightarrow \), but it surely is not a counterexample in \(\textbf{CL }\) to that inference, because \(v^{**}\) is not a bivaluation. A similar point can be made with respect to the relation between \(\textbf{TS }\) and \(\textbf{CL }\). That’s why it is important to keep in mind that we are transforming bivaluations into trivaluations, and vice versa.

  15. 15.

    Recall that, as we explained at the beginning of the chapter, this notion is defined over the partial (information) order \(\le \) of this schema, which is defined by stipulating that \(\frac{1}{2}\le 0\) and \(\frac{1}{2}\le 1\).

  16. 16.

    Though this is not mandatory. If these two things are regarded as different, then logics that validate one but not the other may be admitted.

  17. 17.

    The previous footnote can be adapted here for inferences and metainferences, and, in general, for metainferences of level n and metainferences of level \(n+1\). See, in particular, Ripley’s position in [18].

  18. 18.

    But it has many valid metainferences of higher levels, as we will soon show.

  19. 19.

    Therefore, it is not completely non-Tarskian.

  20. 20.

    Similarly, Identity is valid in \(\mathbf {LP/K3 }\) because it is valid in \(\textbf{K3}\).

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Authors and Affiliations

  1. IIF-SADAF-CONICET, University of Buenos Aires, Buenos Aires, Argentina

    Federico Pailos & Bruno Da Ré

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  1. Federico Pailos
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Correspondence to Federico Pailos .

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Pailos, F., Da Ré, B. (2023). Strong Kleene Metainferential Logics. In: Metainferential Logics. Trends in Logic, vol 61. Springer, Cham. https://doi.org/10.1007/978-3-031-44381-7_3

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