Skip to main content
Log in

Grounding from a Syntactic Point of View: A Sentential-Logical Approach

  • Original Research
  • Published:
Erkenntnis Aims and scope Submit manuscript

Abstract

We define the term \(\ulcorner \)a set T of sentential-logical formulae grounds a sentential-logical formula A from a syntactic point of view\(\urcorner \) in such a way that A is a syntactic sentential-logical consequence of T, and specific additional syntactic requirements regarding T and A are fulfilled. These additional requirements are developed strictly within the syntactics of sentential-logical languages, the three most important being new, namely: to be atomically minimal, to be minimal in degree, and not to be conjunction-like. Our approach is independent of any specific sentential-logical calculus.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We are aware that the predicate \(\ulcorner x\) grounds \(y \urcorner \) is nowadays often understood in one or other metaphysical sense of \(\ulcorner \)fact y obtains in virtue of fact \(x \urcorner \). See for instance, Fine (2012). For a critical overview, see Raven (2015). Please note that our use of the verb >to ground< bears no relation to any of its metaphysical meanings. Our use of >to ground< also differs from its usage in English translations of Bolzano’s pertinent work (see for example, Tatzel 2002).

  2. The word >relevant< is taken in its intuitive sense, which does not completely coincide with its technical meanings within relevance logic.

  3. For earlier works see Kleinknecht (2006) and OS CorGlaReiSie (2010).

  4. Reasons for filtering classical logic are given in, for example, Weingartner (2000).

  5. Although our approach is based on new ideas, it stands in a tradition whose beginnings were already sketched by Weingartner over 30 years ago (cf. Weingartner 1988, in particular subsection 3.1 “Different Approaches to Relevance”, pp. 169–170).

  6. Of course, \(\{\sim , \cdot , \vee , \supset , \equiv \}\) would also suffice. The important point is not the shape of the connectives, but their rough pragmatic correspondence to >not<, >and<, >or<, >if, then< and >if and only if<.

  7. Since \(\ulcorner R\) is a sentential-logical consequence relation\(\urcorner \) is here defined by a syntactic definition, it may be called >a syntactic sentential-logical consequence relation<, although this can be misleading. Strictly speaking, it is not the sentential-logical consequence relation which is syntactic, but it is its definition.

  8. We did without the monotonicity condition, since monotonicity follows logically from condition (2) and condition (3).

  9. Where A is sentential-logically equivalent to B iff (1) \(\{A\} \vartriangleright B\) and (2) \(\{B\} \vartriangleright A\).

  10. Where A is sentential-logically tautologous iff \(\emptyset \vartriangleright A\).

  11. Where A is sentential-logically consistent iff there is no B such that \(\{A\} \vartriangleright B\) and \(\{A\} \vartriangleright \lnot B\).

  12. Where A is sentential-logically indeterminate iff (1) A is not sentential-logically tautologous and (2) A is sentential-logically consistent.

  13. Where T is sentential-logically equivalent\(^*\) to \(T'\) iff \(\{A : T \vartriangleright A\} = \{A : T' \vartriangleright A\}\).

  14. Where T is sentential-logically consistent\(^*\) iff there is no A such that \(T \vartriangleright A\) and \(T \vartriangleright \lnot A\).

  15. Where A and B are sentential-logically independent of each other iff (1) \(\{A\} \ntriangleright B\) and (2) \(\{B\} \ntriangleright A\).

  16. However, we are aware that the idea of sentential-logical conjunction-likeness could be explicated in many ways which are slightly different from Definition 16.

  17. Please note that our sentential-logical language \(\Sigma \) has \(\{\lnot , \wedge , \vee , \rightarrow , \ \leftrightarrow \}\) as its set of sentential-logical connectives (cf. Sect. 2.1). If a different sentential-logical language \(\Sigma ^*\) with, say, the set of sentential-logical connectives \(\{\lnot , \wedge , \vee \}\) had been chosen, and p and q were \(\Sigma ^*\) formulae, then \(\lnot p \vee q\) would be sentential-logically minimal in degree within \(\Sigma ^*\), but not within \(\Sigma \). The language \(\Sigma \) has been chosen here because its set of connectives \(\{\lnot , \wedge , \vee , \rightarrow , \leftrightarrow \}\) is the one generally used in the literature on formalisation.

  18. Also, the most prominent formulae among those which are called >paradoxes of strict implication< are not sentential-logically minimal in degree, for instance \((p \wedge \lnot p) \rightarrow q\), \(p \rightarrow (q \rightarrow q)\) and \(p \rightarrow (q \vee \lnot q)\). These formulae are all sentential-logically equivalent to, for instance, \(r \rightarrow r\), which contains fewer occurrences of sentential-logical connectives than any of them.

  19. If we had not presupposed in Sect. 2.1 that a certain sentential-logical language and a certain sentential-logical consequence relation have been chosen, we would have to use instead of the binary predicate \(\ulcorner T\) sentential-logically grounds A from a syntactic point of view\(\urcorner \) a quartenary predicate \(\ulcorner T\) sentential-logically grounds A from a syntactic point of view with regard to L and \(C \urcorner \), where \(> L<\) is a variable for sentential-logical languages and \(> C<\) is a variable for sentential-logical consequence relations.

  20. In other words, no formula in \(T \cup \{A\}\) is sentential-logically conjunction-like.

  21. In other words, each formula in \(T \cup \{A\}\) is sentential-logically minimal in degree.

  22. Note that \(\emptyset \) fulfils condition (8), since \(\emptyset \) is atomically minimal\(^*\) (cf. Theorem 14 in Sect. 2.2).

  23. Where T is sentential-logically weaker\(^*\) than \(T'\) iff (1) \(T' \vartriangleright ^* T\) and (2) \(T \ntriangleright ^* T'\).

  24. There would indeed be more to say on overlaps and differences between conditions in our main definition, as well as on axioms and derivation rules in relevance-logic systems, as for example in comparison with the relevance-logic system in Schamberger (2016). Work in this direction is desirable, but calls for separate papers.

  25. Concerns regarding condition (6) are treated in Sect. 4.4.

  26. Conditions (2) and (8) overlap in many cases, such as: \(\{p \vee q, p \rightarrow q\} \nsucc q\), \(\{q \rightarrow p, p \rightarrow \lnot q\} \nsucc \lnot q\), \(\{p \rightarrow (q \vee r), (p \wedge q) \rightarrow r\} \nsucc p \rightarrow r\) and \(\{(q \wedge r) \rightarrow \lnot p, (p \wedge r) \rightarrow \lnot q\} \nsucc p \rightarrow \lnot r\).

  27. Because \(\{p\}\) is sentential-logically equivalent\(^*\) to \(\{r \vee p, r \rightarrow p\}\), and \(af^*(\{r \vee p, r \rightarrow p\}) \nsubseteq af^*(\{p\})\).

  28. Because \(\{r \wedge q\}\) is sentential-logically equivalent\(^*\) to \(\{p \vee q, r, r \rightarrow (p \rightarrow q)\}\), and \(af^*(\{p \vee q, r, r \rightarrow (p \rightarrow q)\}) \nsubseteq af^*(\{r \wedge q\})\).

  29. Condition (8) is twice violated. First, because \(af^{*}(\{p\rightarrow (q\vee s), (p\wedge s)\rightarrow q\}) \nsubseteq af^{*} (\{p\rightarrow q\})\) and \(\{p\rightarrow q\}\) is sentential-logically equivalent* to \(\{p\rightarrow (q\vee s), (p\wedge s)\rightarrow q\}\). Second, because \({af}^{*}(\{{q}\rightarrow ({r } \vee {t}), {q}\rightarrow ({t}\rightarrow r)\})\nsubseteq af^{*} (\{q\rightarrow r\})\) and \(\{q\rightarrow r\}\) is sentential-logically equivalent* to \(\{{q}\rightarrow ({r}\vee {t}), {q}\rightarrow ({t}\rightarrow {r})\}.\)

  30. Condition (2) requires that there be no proper subset \(T'\) of T such that A is a sentential-logical consequence of \(T'\). Condition (6) requires that neither A nor any formula of T be sentential-logically conjunction-like. Condition (7) requires A and each formula of T to be sentential-logically minimal in degree. Condition (8) requires that each subset of T be atomically minimal\(^*\).

  31. For a detailed example, see Sect. 4.3.1.

  32. Our definition of grounding stated that T sentential-logically grounds A from a syntactic point of view iff it holds:

    1. (1)

      \(T \vartriangleright A\),

    2. (2)

      there is no proper subset \(T'\) of T such that \(T' \vartriangleright A\),

    3. (3)

      A and at least one B in T are sentential-logically independent of each other,

    4. (4)

      each atomic formula of A is also an atomic formula of at least one formula in T,

    5. (5)

      T is finite,

    6. (6)

      no formula in \(T \cup \{A\}\) is sentential-logically conjunction-like,

    7. (7)

      each formula in \(T \cup \{A\}\) is sentential-logically minimal in degree,

    8. (8)

      each subset of T is atomically minimal\(^*\).

  33. This theorem states that each atomically-minimal\(^*\) T is also sentential-logically consistent\(^*\).

  34. This theorem states that if \(\{A\}\) is atomically minimal\(^*\), then A is atomically minimal.

  35. This definition states: A is sentential-logically conjunction-like iff there is at least one B and at least one C such that:

    1. (1)

      B and C are sentential-logically independent of each other, and

    2. (2)

      A is sentential-logically equivalent to \(B \wedge C\), and

    3. (3)

      \(B \wedge C\) is atomically minimal.

  36. This definition states: A is sentential-logically minimal in degree iff there is no B such that:

    1. (1)

      B is sentential-logically equivalent to A, and

    2. (2)

      B contains fewer occurrences of sentential-logical connectives than A.

  37. Please note: to claim that a tautological formula A cannot be grounded does not imply that a natural-language sentence saying that a given formula A is a tautology cannot be grounded.

  38. This theorem states: if \(T \succ A\), then \(\{A\} \ntriangleright ^* T\).

  39. This theorem states: if \(T \succ A\), then T is sentential-logically consistent\(^*\).

  40. A set M of formulae is sentential-logically independent\(^*\) iff no formula in M is a sentential-logical consequence of the remaining formulae in M; in other words, iff there is no B such that \(B \in M\) and \(M \setminus \{B\} \vartriangleright B\).

  41. This definition states: \(T'\) is a sentential-logical consequence\(^*\) of T iff for each B: if \(B \in T'\), then \(T \vartriangleright B\).

  42. This theorem states: if \(T \vartriangleright ^* T'\) and if \(T' \vartriangleright A\), then \(T \vartriangleright A\).

  43. This theorem states: if \(T \succ A\), then T contains at least two formulae.

  44. Cf. the remarks on the conditions of the main definition in Sects. 3.1 and 4.3.1.

  45. So due to condition (5) of the main definition, \(T_{1}, T_{2}, \dots , T_{n}\) are finite, hence T is finite. All formulae in T as well as \(B_{1}, B_{2}, \dots , B_{n}\) are, due to condition (7) of the main definition, sentential-logically minimal in degree. Furthermore, due to Theorem 27, each of the sets \(T_{1}, T_{2}, \dots , T_{n}\) contains at least two formulae.

  46. Note that this condition cannot be replaced with the following simpler and weaker condition, namely: T is atomically minimal\(^*\). Cf. remarks on condition (8) of the main definition, point (c), Sect. 3.1.

  47. The predicate logic version of this violation of relevance is a well-known paradox in philosophy of science, often called “the paradox of irrelevant explanandum components” (cf. Weingartner 2000, p. 318).

  48. For argument hierarchies see Dorn (2006).

References

  • Dorn, G. (2006). Deskriptive Argumente und Argumenthierarchien. In G. Kreuzbauer & G. Dorn (Eds.), Argumentation in Theorie und Praxis. Philosophie und Didaktik des Argumentierens (pp. 167–201). Wien: Lit-Verlag.

  • Fine, K. (2012). Guide to ground. In F. Correia & B. Schnieder (Eds.), Metaphysical grounding. Understanding the structure of reality (pp. 37–80). Cambridge: Cambridge University Press.

  • Kleinknecht, R. (2006). Deduktive Ableitung und deduktive Begründung. In G. Kreuzbauer & G. Dorn (Eds.), Argumentation in Theorie und Praxis. Philosophie und Didaktik des Argumentierens (pp. 17–29). Wien: Lit-Verlag.

  • OS CorGlaReiSie. (2010). Deduktive Begründung. Zu einem Explikationsvorschlag von Reinhard Kleinknecht. Conceptus: Zeitschrift für Philosophie, 39(95), 31–60.

  • Raven, M. J. (2015). Ground. Philosophy Compass, 10, 322–333.

    Article  Google Scholar 

  • Schamberger, C. (2016). Logik der Umgangssprache. Göttingen: V&R unipress.

    Book  Google Scholar 

  • Schnieder, B. (2011). A logic for ‘because’. The Review of Symbolic Logic, 4, 445–465.

    Article  Google Scholar 

  • Tatzel, A. (2002). Bolzano’s theory of ground and consequence. Notre Dame Journal of Formal Logic, 43, 1–25.

    Article  Google Scholar 

  • Weingartner, P. (1988). Remarks on the consequence-class of theories. In E. Scheibe (Ed.), The role of experience in science. Proceedings of the 1986 conference of the académie internationale de philosophie des sciences (Bruxelles) (pp. 161–180). Berlin: Walter de Gruyter.

    Google Scholar 

  • Weingartner, P. (2000). Reasons for filtering classical logic. In D. Batens, et al. (Eds.), Frontiers in paraconsistent logic. Proceedings of the 1st world congress on paraconsistency, Gent 1997 (pp. 315–327). Baldock: Research Studies Press.

    Google Scholar 

  • Weingartner, P., & Schurz, G. (1986). Paradoxes solved by simple relevance criteria. Logique et Analyse, 113, 3–40.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alexander Zimmermann.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zimmermann, A., Kleinknecht, R. & Dorn, G.J.W. Grounding from a Syntactic Point of View: A Sentential-Logical Approach. Erkenn 87, 717–746 (2022). https://doi.org/10.1007/s10670-019-00215-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10670-019-00215-1

Keywords

Navigation