Abstract
In this paper we elaborate a conception of entailment based on what we call the Ackermann principle, which explicates valid entailment through a logical connection between (sets of) sentences depending on their informational content. We reconstruct Dunn’s informational semantics for entailment on the basis of Restall’s approach, with assertion and denial as two independent (primary) speech acts, by introducing the notion of a ‘position description’. We show how the machinery of position descriptions can effectively be used to define the positive and the negative information carried by sentences of a given language and to present a formalized version of the Ackermann principle as an inclusion relationship between the informational contents of the conclusions and the premises of a valid entailment. Being so interpreted, the entailment relation exhibits certain properties, including the property of transitivity (and, more generally, admissibility of the cut rule). Whereas properties such as Anderson and Belnap’s variable sharing property or Parry’s proscriptive principle are normally presented as imposing a relevance requirement on valid entailment, the suggested formalization of the Ackermann principle supports all of Gentzen’s structural rules, including weakening, a rule that is normally given up in sequent-style proof systems for relevance logics. In this way we propose an Ackermann-inspired explication of the nature of entailment as a relation between the informational contents of sentences.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
15 April 2020
A Correction to this paper has been published: https://doi.org/10.1007/s11229-020-02640-0
Notes
Or with the Ackermann principle of steering, due to Rudolph Ackermann.
There are also approaches in which the entities connected by the entailment relation are sequences, multisets, or so-called ‘Gentzen terms’, but we shall here consider only sets as relata of the entailment relation.
This concerns, in particular, Stephen Read’s account developed in Read (1988, p. 133), where entailment is defined through the notion of fusion: A entails B iff A cannot be true and (fuse) B false. Fusion, in its turn, is used to explicate the idea of relevance as a special connection between sentences, which holds regardless of their truth values.
Of course, this does not mean that Restall’s general idea is incompatible with the Ackermann principle. The Ackermann principle is also not incompatible with the notion of logical compatibility explicated by Everett Nelson in Nelson (1930).
Note that Ferguson (2017, p. 7) draws a distinction between \(\rightarrow \)-Parry and \(\vdash \)-Parry systems. If \(\vdash _{\mathsf {L}}\) is the consequence relation of a given logic \(\mathsf L\) with a language containing an implication connective, \(\rightarrow \), then \(\mathsf L\) is \(\rightarrow \)-Parry iff \(\vdash _{\mathsf {L}} (A \rightarrow B) \text{ implies } \mathbf {At}(B) \subseteq \mathbf {At}(A).\) The logic \(\mathsf L\) is \(\vdash \)-Parry iff \(\Gamma \vdash _{\mathsf {L}} A \text{ implies } \mathbf {At}(A) \subseteq \bigcup \{\mathbf {At}(B)\mid B \in \Gamma \}.\) The relation of strict entailment endorsed by Wessel (1998, p. 144) is \(\vdash \)-Parry for single premises.
From a classical perspective, a constantly true sentence may be seen to carry minimal information insofar as it does not rule out any situation, whereas a constantly false sentence may be taken to carry maximal information insofar as it rules out every situation whatsoever.
But see the conception of ‘intuitionistic state descriptions’ in Shramko (2000), which makes use of a metalanguage (factual) negation.
A critical discussion of rejectivism can be found in Martin (2016).
Note that conditions \(\Gamma ^a \cup \Delta ^a = \mathcal{A}\) and \(\Gamma ^a \cap \Delta ^a =\varnothing \) generally are not taken, i.e., having some position description, an atomic sentence can well be neither asserted nor denied in it (the position description is then incomplete), or both asserted and denied in this position description (which is then over-complete or inconsistent). Taking into account the underlying intuitive understanding, the allowing of such non-classical situations seems quite natural.
In a broad sense, it can be said that a position description implies an assertion or a denial if and only if it is compatible with such an assertion or a denial. More strictly, such an implying can be determined by certain assertion and denial conditions for compound sentences, as it is done below for a specific sentential language.
For ‘Conjunction, Disjunction, Negation’.
Let \(Ass(A,\mathcal {PD})\) be the set of position descriptions in \(\mathcal {PD}\) where A is asserted and \(Den(A,\mathcal {PD})\) be the set of position descriptions in \(\mathcal {PD}\) where A is denied. Then, an immediate corollary of Definition 2.1 is that \(A \models B\) if and only if for every \(\mathcal {PD}\): \(Ass(A, \mathcal{PD}) \subseteq Ass(B,\mathcal {PD})\), and \(Den(B,\mathcal{PD}) \subseteq Den(A,\mathcal {PD})\).
Of course, this claim is based on the assumption that one cannot coherently assert anything together with its negation. Indeed, even paraconsistent logics have often been motivated not by a desire to tolerate or excuse contradictions, but rather by the intention to retain the possibility of drawing inferences in a “sensible fashion” even in inconsistent situations, see, e.g., Priest (2002, p. 288).
Cf. the following remark by Peter Schroeder-Heister: “Structural rules may be looked upon as axiomatizing a consequence relation in Tarski’s sense” (Schroeder-Heister 2002, p. 247). This does not mean, of course, that all the structural properties of a consequence relation of some logical system are exhausted by the initial structural rules explicitly accepted in the system (and the rules derived thereof). One should also take into account the so called admissible rules, which cannot be derived in the system, but which nevertheless can hold in it, expressing thus further properties of the corresponding consequence relation (for a recent detailed account of the notion of admissibility in the setting of consequence relation consult Iemhoff 2016).
Recall that in Gentzen’s LK the relata of the derivability relation are finite sequences of formulas, although the structural rules justify considering these sequences as sets. As Došen (1997, p. 293) explains with respect to single conclusion sequents, “in general, it makes sense to extend Gentzen’s approach to sequents \(\Gamma \vdash A\) with \(\Gamma \) infinite. Most logicians shun such infinitary sequents, as if they had finitistic scruples. However, one may expect to find such sequents useful in second-order or higher-order logic, in logic with infinitary connectives, in arithmetic with the \(\omega \)-rule, or whenever finite axiomatizability fails.”
For validation of \(\lnot \)left (as well as \(\lnot \)right, dual to it), the incomplete and inconsistent position descriptions should be excluded from consideration, i.e., one should admit as acceptable only those members \([\Gamma ^a\), \(\Delta ^a]\) of a set of position descriptions which are subject to the restrictions \(\Gamma ^a \cup \Delta ^a = \mathcal{A}\) and \(\Gamma ^a \cap \Delta ^a =\varnothing \). Thus, if we wish that entailment statements also comprise expressions with Boolean negation, we should impose these restrictions when determining the informational content of such expressions. It must be recognized, however, that information about the completeness and consistency of the background descriptions will then be implicitly added to the informational content of the expressions as such.
Note that the deducibility of this sequent shows that Tennant’s core logic is not a containment logic in the sense of Parry’s proscriptive principle.
Another recent development of a notion of non-transitive entailment can be found in many-valued logic. The notion in question is that of so-called ‘p-entailment’ (plausible entailment) as introduced by Frankowski (2004), see also Shramko and Wansing (2011), and independently arrived at by Cobreros et al. (2012). An entailment \(\Gamma \models A\) is valid in the sense of p-entailment iff it holds that if every premise from \(\Gamma \) is designated, then A is not anti-designated. (For a detailed discussion of designated and anti-designated values see Shramko and Wansing (2011, Chapter 8).) A two-dimensional notion of entailment, called ‘B-entailment’, that covers p-entailment and other kinds of entailment is presented in Blasio et al. (2017). This framework reveals that there is a sense in which p-entailment is transitive. Since B-entailment and the notions of entailment covered by it are defined in terms of truth values, they are beyond the scope of the Ackermann principle.
References
Ackermann, W. (1956). Begründung einer strengen Implikation. Journal of Symbolic Logic, 21, 113–128.
Almukdad, A., & Nelson, D. (1984). Constructible falsity and inexact predicates. Journal of Symbolic Logic, 49, 231–233.
Anderson, A. R., & Belnap, N. D. (1975). Entailment: The logic of relevance and necessity (Vol. I). Princeton, NJ: Princeton University Press.
Ashby, W. R. (1956). An introduction to cybernetics. New York: Wiley.
Bar-Hillel, Y., & Carnap, R. (1953). Semantic information. British Journal for the Philosophy of Science, 4, 147–157.
Barwise, J., & Perry, J. (1985). Shifting situations and shaken attitudes: An interview with Barwise and Perry. Linguistics and Philosophy, 8, 105–161.
Baylis, C. A. (1931). Implication and subsumption. Monist, 41, 392–399.
Belnap, N. D. (1982). Display logic. Journal of Philosophical Logic, 11, 375–417.
Blanshard, B. (1939). The nature of thought. London: Allen and Unwin.
Blasio, C., Marcos, J., & Wansing, H. (2017). An inferentially many-valued two-dimensional notion of entailment. Bulletin of the Section of Logic, 46, 233–262.
Brillouin, L. (1964). Scientific uncertainty and information. New York: Academic Press.
Burgess, J. (2009). Philosophical logic. Princeton: Princeton University Press.
Carnap, R., & Bar-Hillel, Y. (1952). An outline of a theory of semantic information. MIT Research Laboratory of Electronics, technical report, no. 247.
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41, 347–385.
Cockett, J. R. B., & Seely, R. A. G. (2017). Proof theory of the cut rule. In E. Landry (Ed.), Categories for the working philosopher (pp. 223–261). Oxford: Oxford University Press.
De, M., & Omori, H. (2018). There is more to negation than modality. Journal of Philosophical Logic, 47, 281–299.
Došen, K. (1988). Sequent systems and groupoid models I. Studia Logica, 47, 353–389.
Došen, K. (1997). Logical consequence: A turn in style. In M. L. D. Chiara, et al. (Eds.), Logic and scientific methods: Volume one of the tenth international congress of logic, methodology and philosophy of science (pp. 289–312). Dordrecht: Springer.
Duncan-Jones, A. E. (1935). Is strict implication the same as entailment? Analysis, 2, 70–78.
Dunn, J. M. (1966). The algebra of intensional logics. Doctoral dissertation. University of Pittsburgh. Ann Arbor, University Microfilms.
Dunn, J. M. (1976). Intuitive semantics for first-degree entailment and coupled trees. Philosophical Studies, 29, 149–168.
Dunn, J. M. (1986). Relevance logic and entailment. In D. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic, volume III. Alternatives to classical logic. Synthese library (Vol. 166, pp. 117–224). Dordrecht: D. Reidel Publishing Company.
Dunn, J. M., & Zhou, C. (2005). Negation in the context of gaggle theory. Studia Logica, 80, 235–264.
Epstein, R. (1979). Relatedness and implication. Philosophical Studies, 29, 149–168.
Ferguson, T. M. (2017). Meaning and proscription in formal logic. Variations on the propositional logic of William T. Parry. Dordrecht: Springer.
Font, J. M. (1997). Belnap’s four-valued logic and De Morgan lattices. Logic Journal of the IGPL, 5, 413–440.
Frankowski, S. (2004). Formalization of a plausible inference. Bulletin of the Section of Logic, 33, 41–52.
Geach, P. T. (1958). Entailment. Proceedings of the Aristotelian Society, 32, 157–172.
Gentzen, G. (1935). Untersuchungen über das logische Schließen. I. Mathematische Zeitschrift, 39, 176–210.
Hacking, I. (1979). What is logic? Journal of Philosophy, 76, 285–319.
Hertz, P. (1929). Über Axiomensysteme für beliebige Satzsysteme. Mathematische Annalen, 101, 457–514.
Horn, L. R., & Wansing, H. (2017). Negation. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Spring 2017 Edition). https://plato.stanford.edu/archives/spr2017/entries/negation/.
Humberstone, L. (2000). The revival of rejective negation. Journal of Philosophical Logic, 29, 331–381.
Iemhoff, R. (2016). Consequence relations and admissible rules. Journal of Philosophical Logic, 45, 327–348.
Ilić, M. (2016). The proof-theoretical analysis of contraction-less relevant logics. Bulletin of the International Mathematical Virtual Institue, 6, 1–11.
Israel, D., & Perry, J. (1990). What is information? In P. P. Hanson (Ed.), Information, language, and cognition (pp. 1–19). Vancouver: University of British Columbia Press.
Kleene, S. C. (1967). Mathematical logic. New York: Wiley.
Kneale, W., & Kneale, M. (1962). The development of logic. London: Duckworth.
Lewis, C. I. (1917). The issues concerning material implication. Journal of Philosophy, 14, 350–356.
Lewy, C. (1958). Entailment. Proceedings of the Aristotelian Society, 32, 123–142.
Łoś, J., & Suszko, R. (1958). Remarks on sentential logics. Indagationes mathematicae, 20, 177–183.
Mares, E. (1996). Relevant logic and the theory of information. Synthese, 109, 345–360.
Mares, E., & Paoli, F. (2014). Logical consequence and the paradoxes. Journal of Philosophical Logic, 43, 439–469.
Martin, B. (2016). Rejectivism and the challenge of pragmatic contradictions. Disputatio, 43, 253–268.
McCall, S. (1975). Connexive implication, §29.8. In A. R. Anderson & N. D. Belnap (Eds.), Entailment. The logic of relevance and necessity (Vol. I, pp. 434–446). Princeton: Princeton University Press.
McCall, S. (2012). A history of connexivity. In D. M. Gabbay, et al. (Eds.), Handbook of the history of logic. Logic: A history of its central concepts (Vol. 11, pp. 415–449). Amsterdam: Elsevier.
Nelson, E. J. (1930). Intensional relations. Mind, 39, 440–453.
Paoli, F. (2002). Substructural logics: A primer. New York: Springer.
Priest, G. (2002). Paraconsistent logic. In D. M. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (2nd ed., Vol. 6, pp. 287–393). Dordrecht: Kluwer Academic Publishers.
Read, S. (1988). Relevant logic: A philosophical examination of inference. New York: Wiley.
Restall, G. (1996). Information flow and relevant logics. In J. Seligman & D. Westerståhl (Eds.), Logic, language and computation (pp. 463–477). Stanford: CSLI Publications.
Restall, G. (2000). An introduction to substructural logics. London: Routlege.
Restall, G. (2005). Multiple conclusions. In P. Hájek, L. Valdés-Villanueva, & D. Westerståhl (Eds.), Logic, methodology and philosophy of science: Proceedings of the twelfth international congress (pp. 189–205). London: College Publications.
Restall, G. (2009). Truth values and proof theory. Studia Logica, 92, 241–264.
Restall, G. (2013). Assertion, denial, and non-classical theories. In K. Tanaka, F. Berto, E. Mares, & F. Paoli (Eds.), Paraconsistency: Logic and applications (pp. 81–100). Dordrecht: Springer.
Ripley, D. (2015). Anything goes. Topoi, 34, 25–36.
Routley, R. (1978). Semantics for connexive logics. I. Studia Logica, 37, 393–412.
Routley, R., & Routley, V. (1972). The semantics of first degree entailment. Noûs, 6, 335–359.
Scheffler, U., & Shramko, Y. (1998). Eine Generelle Informationssemantik für Systeme der logischen Folgebeziehung. In K. Wuttich & U. Scheffler (Eds.), Terminigebrauch und Folgebeziehung (pp. 229–247). Berlin: Logos.
Schroeder-Heister, P. (2002). Resolution and the origins of structural reasoning: Early proof-theoretic ideas of Hertz and Gentzen. Bulletin of Symbolic Logic, 8, 246–265.
Shramko, Y. (2000). American plan for intuitionistic logic 1: An intuitive background. In T. Childers (Ed.), The logica yearbook 1999 (pp. 53–64). Prague: Filosofia.
Shramko, Y., & Wansing, H. (2011). Truth and falsehood. An inquiry into generalized logical values. Berlin: Springer.
Tennant, N. (1987). Anti-realism and logic. Oxford: Clarendon Press.
Tennant, N. (2017). Core logic. Oxford: Oxford University Press.
Voishvillo, E. K. (1996). A theory of logical relevance. Logique et Analyse, 155–156, 207–228.
Wansing, H. (2016). Connexive logic. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Spring 2016 Edition). https://plato.stanford.edu/archives/spr2016/entries/logic-connexive/.
Wansing, H. (1993). The logic of information structures. Springer lecture notes in AI 681. Berlin: Springer.
Wansing, H. (1994). Sequent systems for normal modal propositional logics. Journal of Logic and Computation, 4, 125–142.
Wansing, H. (1998). Displaying modal logic. Dordrecht: Kluwer Academic Publishers.
Wessel, H. (1998). Logik (4th ed.). Berlin: Logos Verlag.
Acknowledgements
We would like to thank two anonymous referees for their constructive criticism and useful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
We dedicate this paper to the memory of Carolina Blasio.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shramko, Y., Wansing, H. The nature of entailment: an informational approach. Synthese 198 (Suppl 22), 5241–5261 (2021). https://doi.org/10.1007/s11229-019-02474-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-019-02474-5