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On Consequence and Rejection as Operators

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Abstract

This paper is devoted to the concepts of consequence and rejection, formulated as operators on a nonempty set of sentences, which may initially be unstructured. One of the issues that we pay attention to is the “cyclicity” of these concepts when they are defined one through the other. In addition, we explore this cyclicity, when the set of all sentences acquires some structure, or we can assume some structure of sentences in the sense that the operation of substitution can be applied to them.

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Notes

  1. An alternative approach to rejection, based on Tarski’s consequence, is given in [9]; see also [7].

  2. Cf. [8], chapters III and V; or modern accounts, such as [10], Sect. 1.2.1, or [4], Sect. 4.2.

  3. Another feature of \({\mathcal {S}}\), which is of no importance to us here, was the limitation of the cardinality of \({\mathcal {S}}\); namely, Tarski considered \(\overline{\overline{{\mathcal {S}}}}\le \aleph _0\) by definition.

  4. A rare exception is the works of pupils and followers of Słupecki and Wójcicki, such as, for example, [11, 12] and [7].

  5. The only exception I know of is [3].

  6. At least definition DCn\(^+\) offers this rationale, because as soon as Cn\(^+\) is defined, its relation to \({\textbf{C}}{}\) immediately arises. What happens if recovery does not occur will be discussed at the end of Sect. 4.

  7. See Remark 3.6 about the differences between Cn\(^+\) and \(\textbf{C}_{\textbf{R}_{\textbf{C}}}\).

  8. In a way, this is similar to Carnap’s rule (2) from [2], §26, definition D26-6.

  9. Written in Polish, [11] is a publication of the author’s doctoral dissertation. On the other hand, in [12] \((\text {r}_{6})\) is only mentioned but not proved; there is no mention that \((\text {r}_{6})\) was proved in [11].

  10. This is the minimum condition that can be imposed on \({\textbf{C}}{}\).

  11. I am indebted to Alex Citkin, who drew my attention to this issue. His point of view is categorical: the rejection operator must not be structural (see definition below).

  12. As Wójcicki believed, calling the consequence structural and logical in a synonymous way; in his own words,

    [\(\ldots \)] a logic of formulas is not just a set of formulas closed under substitutions but the set of all ‘logically true’ formulas, a logic of inferences is not just a structural consequence operation (or the set of inferences sound for it) but the set of all ‘logically valid’ inferences ([10], p. 45).

References

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  1. Louisiana Scholars’ College, Northwestern State University, Natchitoches, LA, 71495, USA

    Alexei Muravitsky

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Muravitsky, A. On Consequence and Rejection as Operators. Log. Univers. 17, 443–460 (2023). https://doi.org/10.1007/s11787-023-00334-y

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