If structured propositions are logical procedures then how are procedures individuated?

Abstract

This paper deals with two issues. First, it identifies structured propositions with logical procedures. Second, it considers various rigorous definitions of the granularity of procedures, hence also of structured propositions, and comes out in favour of one of them. As for the first point, structured propositions are explicated as algorithmically structured procedures. I show that these procedures are structured wholes that are assigned to expressions as their meanings, and their constituents are sub-procedures occurring in executed mode (as opposed to displayed mode). Moreover, procedures are not mere aggregates of their parts; rather, procedural constituents mutually interact. As for the second point, there is no universal criterion of the structural isomorphism of meanings, hence of co-hyperintensionality, hence of synonymy for every kind of language. The positive result I present is an ordered set of rigorously defined criteria of fine-grained individuation in terms of the structure of procedures. Hence procedural semantics provides a solution to the problem of the granularity of co-hyperintensionality.

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Notes

  1. 1.

    See Duží, Jespersen and Materna (2010, Chapters 1 and 2), and also Tichý (1988).

  2. 2.

    Faroldi (2016) makes a similar point.

  3. 3.

    When I say that the two sentences are logically equivalent I tacitly presuppose here that both A and B have a truth-value. In other words, I disregard the possibility of truth-value gaps. In the logic of partial functions, wide-scope and narrow-scope negation may fail to be equivalent. For details, see Duží (2017b).

  4. 4.

    See, for instance, Pickel (2017).

  5. 5.

    Hanks and Soames have recently presented theories of complex acts in a series of articles and books, see, for instance, Hanks (2011, 2015) and Soames (2012, 2014). Complex acts with which propositions are identified can differ with respect to different ways of cognizing objects and properties. Distinct complex acts can deal with the same objects, which have the same properties and stand in the same relations. In other words, these theories can handle the cases of distinct propositions expressed by analytically equivalent sentences denoting one and the same PWS-proposition.

  6. 6.

    The paradox of inference was put forward in Cohen and Nagel (1934, p. 173). It goes roughly like this: since the conclusion of a valid argument is contained in the premises, it fails to provide any novel information. Yet Duží (2010) argues that it seems evident that there is something that we learn when deducing the conclusion of a sound argument. We obtain a new piece of analytic information about the procedure the product of which is the proposition (or truth-value, in the case of mathematics) denoted by the conclusion.

  7. 7.

    See Cleland (2002) for discussion.

  8. 8.

    For details, see Duží (2014).

  9. 9.

    Similar arguments can be found also in Tichý (1995, pp. 179–80).

  10. 10.

    In order to terminologically distinguish truth-conditions (understood as functions from possible worlds and times to truth-values) from procedures producing truth-conditions, in what follows I will use the term ‘PWS-propositions’ for the former, and the term ‘hyperproposition’ for structured procedures producing PWS-propositions, or truth-values in the case of mathematics.

  11. 11.

    Tichý’s TIL was developed simultaneously with Montague’s IL. For a critical comparison of TIL and IL, see Duží et al. (2010, §2.4.3).

  12. 12.

    The other source can be a type-theoretically incoherent (‘nonsensical’) way of composing a construction, for instance, by composing the Sun with being a natural number.

  13. 13.

    If there is no such product, the construction is v-improper.

  14. 14.

    For details see Duží (2012).

  15. 15.

    See Russell (1996: §136). For more on interaction between parts, see Jespersen (2017b).

  16. 16.

    The fact that 7 is the number produced is a piece of mathematical knowledge external to the above Composition; 7 is not mentioned above. If we wanted to indicate that this particular number is produced, we would specify the identity [[\(^{0}+\,^{0}\)\(^{0}\)5] = \(^{0}\)7].

  17. 17.

    In this respect, the structure of constructions is similar to the formal mereology introduced in Bennet (2013).

  18. 18.

    For instance, Cotnoir illustrates this problem in (ibid., p. 835): “The classic counterexample to extensionality involves objects (e.g., a statue) and the matter which constitutes them (e.g., a lump of clay). They presumably have different properties: e.g., the clay can survive squashing whereas the statue cannot. They must, therefore, be different objects. Yet every part of one appears to be part of the other. Their structure (insofar as mereology is concerned, anyway) is exactly the same. Another example involves the construction of two objects by a rearrangement of the same parts. Suppose my son builds a house out of some Lego bricks. He then destroys the house (as he often does) and proceeds to build a boat from the same Lego bricks. Is the house identical to the boat? Or are they distinct? Extensionality would seem to force us to identify the two.”

  19. 19.

    A theory of concepts has been worked out in TIL by Materna in (1998, 2004). We explicate concepts as closed constructions in their normal form. For details, see also Duží et al. (2010, §2.2).

  20. 20.

    “die Art,wie diese Theile untereinander verbunden sind“ (1837, §244).

  21. 21.

    By these critical remarks, I do not want to imply that the classical concept theory is not useful. There are many useful applications of this theory. So-called Formal Concept Analysis has been successfully applied in computer science. For details, see, e.g., Ganter and Wille (1999).

  22. 22.

    For the rules of substitution see Duží, Materna (2017), and for the rules of existential quantification see Duží and Jespersen (2015).

  23. 23.

    Church (1951, n. 15) offers various examples of non-propositional attitudes, including the famous example of Ponce de León searching the fountain of youth.

  24. 24.

    It might be debatable whether the argument is invalid, and whether ‘brother of’ and ‘male sibling of’ are not synonymous. As for the latter, the two terms are equivalent rather than strictly synonymous, as I explain above. As for the former, true, on its intensional reading the argument would be valid. In such a case Tilman would be related to the property of being Tilman’s brother or Tilman’s male sibling, regardless of the way in which this property is conceptualised. Yet, from a strictly logical point of view, if it is just possible that the premise was true without the conclusion being true as well, the argument should not be considered valid. For this reason, I opt for the hyperintensional reading and analysis of the above argument.

  25. 25.

    For details on property modifiers, see Jespersen (2015b, §4). Here we deal with a subsective attribute modifier that assigns to an attribute sibling of (somebody) a new attribute male sibling of(somebody).

  26. 26.

    I agree with Church’s view, see his (1956, p. 8, n. 20). Church argues that in “Schliemann sought the site of Troy’ it is a concept of a location, not the location itself, that guides Schliemann’s and every other seeker’s search for the site of Troy.

  27. 27.

    Here I analyse de dicto seeking; Tilman wants to find out who his brother is. It can be the case, for instance, in such a situation where Tilman receives information that his parents might have had another son of whom he had no idea before. For more details on the difference between de dicto and de re seeking, see Duží (2003), or Duží et al. (2010, § 5.2.2).

  28. 28.

    For simplicity, I am ignoring the anaphoric reference ‘his’ and stick to the result of resolving this reference by substituting \(^{0}\)Tilman for the anaphoric variable denoted by ‘his’. More on the logic of dynamic discourse and anaphora resolution, see Duží (2017a).

  29. 29.

    See Jespersen (2015b, §5) for the parallel example of ‘is a bachelor’, ‘is an unmarried man’.

  30. 30.

    For details on analytic information see Duží (2010).

  31. 31.

    Valid rules for existential quantification into hyperintensional context have been specified in Duží and Jespersen (2015).

  32. 32.

    Constructions C, D are analytically equivalent if and only if for any valuation v C and Dv-construct the same object or are both v-improper.

  33. 33.

    See Duží et al. (2010, § 5.1.1) for discussion of Mates’s (1952) puzzle. The authors argue here that if ‘is a woodchuck’ and ‘is a groundhog’ are synonymous predicates then there is no room for even the slightest hyperintensional distinction. The Trivializations \(^{0}\)Woodchuck and \(^{0}\)Groundhog are not two constructions, but one and the same construction.

  34. 34.

    This holds also for terms of different languages. Recall the example from the outset of this paper. The sentences ‘Schnee ist weiß’ and ‘Snow is white’ are synonymous, because the terms ‘Schnee’ and ‘Snow’ are atomic references to the same property; the same holds for the terms ‘weiß’ and ‘white’. Hence, the respective Trivializations \(^{0}\)Schnee, \(^{0}\)Snow and \(^{0}\)Weiß, \(^{0}\)White are identical constructions, and the respective Closures \(\lambda w\lambda t\) [\(^{0}\)Weiß\(_{wt }{}^{0}\)Schnee], \(\lambda w\lambda t\) [\(^{0}\)White\(_{wt }{}^{0}\)Snow] are also identical.

  35. 35.

    For the discussion on the issue of synonymy of complex expressions see also Duží (2014b).

  36. 36.

    By this I do not want to suggest that variables x and y (or any other different variables) are one and the same procedure and substitutable in all contexts in any language. They are different constructions, and in a logical or programming language this difference may turn out to be significant. See also Pickel and Rabern (2016) on ‘the antinomy of the variable’.

  37. 37.

    For details see Anderson (1998) and Church (1993).

  38. 38.

    For discussion of Salmon’s arguments, see Jespersen (2015a).

  39. 39.

    For details see Duží (2017b).

  40. 40.

    See Duží (2014b).

  41. 41.

    For details on \(\upbeta \)-conversion in \(\lambda \)-calculi see Duží and Kosterec (2017).

  42. 42.

    In programming languages, the difference between conversion by name and by value revolves around the programmer’s choice of evaluation strategy. Historically, call-by-value and call-by-name date back to Algol 60, a language designed in the late 1950s. The difference between call-by-name and call-by-value is often called passing by reference versus passing by value, respectively. Call-by-value is not a single evaluation strategy, but rather a cluster of evaluation strategies in which a function’s argument is evaluated before being passed to the procedure. In call-by-reference evaluation (also referred to as call-by name or pass-by-reference), a calling procedure receives an implicit reference to the argument sub-procedure. This typically means that the calling procedure can modify the argument sub-procedure. A call-by-reference language makes it more difficult for a programmer to track the effects of a procedure call, and may introduce subtle bugs. The notion of conversion strategy in the \(\lambda \)-calculi is similar to the evaluation strategy in programming languages.

  43. 43.

    For the proofs see, for instance, Duží and Jespersen (2015) or Duží and Kosterec (2017).

  44. 44.

    This proposal can be found in Duží (2014).

  45. 45.

    An important role of non-synonymous definitions of the number \(\uppi \) is examined in Duží et al. (2009).

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Acknowledgements

The research reported here in was supported by the Grant Agency of the Czech Republic, Project No. GA15-13277S, Hyperintensional logic for natural language analysis, and by the internal grant agency of VSB-TU Ostrava, Project SGS No. SP2017/133, “Knowledge modelling and its applications in software engineering III”. Versions of this paper were read at the Barcelona Workshop on Reference 9 (BW9): Unity and Individuation of Structured Propositions, Barcelona, 22–24 June 2015. I want to thank Bjørn Jespersen for great comments along the way, and not least two anonymous referees for Synthese.

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Duží, M. If structured propositions are logical procedures then how are procedures individuated?. Synthese 196, 1249–1283 (2019). https://doi.org/10.1007/s11229-017-1595-5

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Keywords

  • Procedural semantics
  • Transparent intensional logic
  • Structured propositions
  • Mereology of structured procedures
  • Unity of propositions
  • synonymy
  • Co-hyperintensionality
  • Procedural isomorphism