Skip to main content
SpringerLink
Log in

Hyper-Slingshot. Is Fact-Arithmetic Possible?

  • Published:
Foundations of Science Aims and scope Submit manuscript

Abstract

The paper presents a new argument supporting the ontological standpoint according to which there are no mathematical facts in any set theoretic model (world) of arithmetical theories. It may be interpreted as showing that it is impossible to construct fact-arithmetic. The importance of this conclusion arises in the context of cognitive science. In the paper, a new type of slingshot argument is presented, which is called hyper-slingshot. The difference between meta-theoretical hyper-slingshots and conventional slingshots consists in the fact that the former are formulated in the semantic meta-language of mathematical theories without the use of the iota-operator or the lambda-operator as the abstractor, whereas the latter require for their expression at least one of these non-standard term-operators. Hyper-slingshots implement simpler language tools in comparison with those used in conventional slingshots.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Similar content being viewed by others

Notes

  1. The slingshot argument was probably first presented by Kurt Gödel as an attempt to prove Frege’s thesis that sentences denote their truth values (Gödel 1944). In the same time, Church in his review of Carnap’s Introduction to Semantics, presented the slingshot formally. Church’s argument was intended as an attempt to defend the Fregean category of propositional senses (Church 1943). Quine and Davidson invented their own versions against any category of propositional entities (Davidson 1967; Føllesdal 1983; Barwise and Perry 1981; Neale 1995). The expression ‘truth-makers’ is introduced to the philosophical language in Mulligan et al. (1984). The notion of entities which make sentences true stems from two philosophical traditions. The first comprises the philosophies of Brentano, Meinong, Husserl, Twardowski and their pupils, whereas the second is due to Wittgenstein and Russell. Some historians of logic notice that one may find explicit traces of the use of the notion of truth-makers among German idealists like Bergmann and Lotze (Smith 1989). A similar standpoint is expressed in (Putnam 1975).

  2. For some philosophers, the slingshot argument is intended to undermine the correspondence theory of truth (Davidson 1990). Others claim that the acceptance of the slingshot must lead to the rejection of the thesis that there are causal relations obtaining between facts (Baumgartner 2010a). Niiniluoto (2004) compares Tarski’s conception of truth with semantic theories of truth-makers. Neale mentions four main fields of philosophical considerations for which slingshot arguments are relevant (Neale 1995).

  3. For instance, Oppy (1997) asserts that ‘there isn’t even prima facie reason for friends of facts to be afraid of Gödel’s Slingshot, nor of the more familiar Quine-Church-Davidson Slingshots’. Other philosophers, especially such interpreters of Frege’s semantics as Geach (1976) and Dummett (1973), think that the treatment of truth values as referents of sentences is erroneous or even absurd.

  4. Barwise and Perry (1981) say: “Intuitively, situations are complexes of objects and relations /.../. Given this conception, the role of the parts of the sentence is to identify objects and relations out of which the complex is constructed”. This means that if some of objects or relations out of which the complex is constructed are substituted by different objects or relations, the resulting complex will be different from the complex at input. Donaho (1998) also attributes a similar conception to Barwise and Perry.

  5. Donaho (1998), for instance, presents the standpoint of slingshot’s defenders in the following way: ‘if all true sentences represent the same thing then the notion that sentences are representations should not be used in modeling the phenomena of meaning, truth conditions, and understanding’ (p. 34). A similar explication of the slingshot is proposed by Yourgrau (1987). According to this author, the slingshot debate concerns referential functions of sentences; or in other words, the main question is: Do sentences refer to facts? Williamson (1976) notices that the controversy over facts is entangled in the debate over the correspondence theory of truth. Hence, slingshot arguments should be regarded as attempts to undermine correspondence conceptions of truth. Drai (2002) comprehends slingshot arguments as attempts to ‘establish Frege’s thesis that the reference of sentences is their truth value’, and hence that there are no facts as different from truth values which are denoted by sentences.

  6. For instance, Searle’s reading of the slingshot refers to the version according to which each fact corresponds to any true statement. According to him, the slingshot argument is invalid because of the use of the rule that statements of the form ‘S* corresponds to the fact that S’ are preserved under substitution of logically equivalent statements (Searle 1996, pp. 223–224). Another defender of facts denies that in contexts with fact identity operator, the principle of substitution for singular terms and the principle of substitution for descriptions are valid inference rules (Oppy 1997).

  7. One of the logics which serves to build various formal theories of facts is the non-Fregean logic created by Suszko (1968) and motivated by ideas from Wittgenstein’s Tractatus (Suszko 1968). In non-Fregean logic, the standard principle of extensionality is adopted. Hence, this logic is interpreted as non-intensional (Omyła 1986). Non-intensionality of non-Fregean logic is defended in (Wójtowicz 2007, pp. 109–113, 128). The project of non-Fregean logic is criticized as trivial by Wójcicki (1984). In recent years, Fine (2012) has constructed the pure logic of ground (PLG) which attempts to formalize the notion of truth-making. This logic does not, however, comprise its predicative calculus. At any rate, Fine’s proposal of building the logic of facts is less mature than Suszko’s logic. Correia (2010), utilizing Fine’s ideas, presents the sentential which comprises axioms of Suszko’s basic non-Fregean logic. The author does not cite Suszko’s papers.

  8. Cummins and Gottlieb (1972) present two possible interpretations of slingshot arguments formulated with the use of class-abstractions. One of these interpretations assumes that class abstractions are not referential terms. If classes are treated as objects denoted by class-abstractions, then an error occurs in the slingshot. It consists in a mistaken attribution of the relation of logical equivalence to formulas \(\upalpha \) and \(\{\hbox {x}: \hbox {x} =x \wedge \upalpha \} = \{\hbox {x}: \hbox {x}=\hbox {x}\}\). They argue that the specified formulas cannot be logically equivalent because they have different ontological commitments. If classes, in turn, are not considered as objects to which class abstractions refer, then each logical transformation of a sentential formula with class-abstractions should first require the use of the contextual definition of the abstractor and, in result, the elimination of this operator from a formula under formal analysis. Wójtowicz (2007, pp. 160–170) also proposes to analyze descriptions and class-abstractions in slingshot arguments in the contextual way. A similar approach to slingshot arguments is proposed in Baumgartner (2010b). The author constructs a method of avoiding slingshot arguments by an application of Quine’s strategy of shallow analysis of sentences formulated in ordinary languages.

  9. This means that expressions formed out of such term-operators as descriptors and abstractors, for example, cannot be used in inferential transformations of logical formulas before reducing them, in virtue of appropriate definitions, to canonical forms. The logical properties of such formulas may only appear in their canonical shapes. In light of this view, term-operators are only language tools which shorten long formulas. They are not primitive operators of the language of classical logic. It is difficult to accept this solution because class-abstractions are understood in mathematical practice as referential expressions (Biłat 2009).

  10. Such an argument is constructed by Rodriguez-Pereyra (1998). In this case, however, it is required to use an inference rule of the following shape: \(\upalpha \equiv \upbeta (\upalpha )\vdash \Omega (\upbeta )\), where \(\Omega \) is any sentential context in which \(\upalpha \) occurs. This rule is not accepted in many intensional logics. That is why any attempt to block Rodriguez-Pereyra’s slingshot requires the use of a special meta-rule which limits the application of the rule of substitutivity for material equivalence. Meta-rules which limit ranges of applications of logical rules are extra-logical inference-tools.

  11. A similar reconstruction of Quine’s slingshot is offered in Lycan (1974). Widerker (1983) also analyses Quine’s standpoint.

  12. In Krysztofiak (2013), I argue in detail that fact-operators which enable us to speak of facts are intensional. Semantic conditions imposed in a meta-theory upon an object-language with fact-operators imply a meta-theorem according to which the existence of the a multiplicity of facts in the actual world entails the existence of a multiplicity of possible worlds in any semantic model of a given theory.

  13. Explicit conventional slingshot arguments are proofs of the following thesis: If \(\upalpha \) is materially equivalent to \(\upbeta \), then the fact that \(\upalpha \) is identical to the fact that \(\upbeta \). This thesis may be expressed in an appropriate non-Fregean object language as well as in meta-theories which admit the semantic function of propositional denotation. In some extensions of PCI, slingshot arguments may be constructed. In Wójtowicz (2005), a slingshot argument is formally reconstructed on the ground of WBQ-logic enriched with some special sentential schema for the description operator. According to Shramko and Wansing (2009), the slingshot thesis is a theorem in the WBQ-logic enriched with the appropriate definition of the description operator (or, alternatively, the abstractor). Wójtowicz’s schema is not a necessary formal tool responsible for the generation of slingshot arguments. Omyła (2009) constructs a proof according to which the axiom of Frege is false in some semantic models. WBQ-logic results from PCI (predicate logic with sentential identity) by adding axioms which fall under the schema \(\hbox {CL}_{\mathrm{P}} \vdash (\upalpha \equiv \upbeta ) \rightarrow \hbox {WB} \vdash (\upalpha \cdot \upbeta )\), where \(\hbox {CL}_{\mathrm{P}}\) is classical predicate logic, \(\equiv \) is material equivalence and \(\cdot \) is sentential identity.

  14. In PA founded upon non-Fregean logic, for instance, the sentence ‘the fact that \(1+1=2\) is not identical to the fact that \(1+2=3\)’ would be a well-formed sentence. If this sentence is true, it should have a proof. Such proofs require consistent criteria of the non-identity of mathematical facts. On the basis of Peano’s axioms alone, these criteria cannot be formulated. To express them, bridge-axioms must be constructed which say that such and such properties of natural numbers occurring in mathematical facts make these facts non-identical (or, alternatively, identical). So, the following question arises: How can such bridge-axioms be expressed?

  15. In Krysztofiak (2007b), formal ways of construing various holistic principles of compositionality are presented. These principles say how denotations of contexts determine denotations of constituent expressions of a given context.

  16. In Krysztofiak (2007a), truth-makers are treated as entities which inhabit, not only the actual world, but also purely conceptual worlds. On the ground of such a construction, I try to justify the philosophical standpoint of epistemological fragmentary idealism, according to which some fragments of the actual worlds, understood as objects of scientific cognition, require for their constitution the existence of purely intentional states of affairs in some possible worlds. In my study, I use the method of semantic paraphrase developed by Carnap, Ajdukiewicz and Woleński.

References

  • Anscombe, G. E. M. (1969). Causality and extensionality. The Journal of Philosophy, 66, 152–159.

    Article  Google Scholar 

  • Barwise, J., & Perry, J. (1981). Semantic innocence and uncompromising situations. Midwest Studies in Philosophy, 6, 387–403.

    Article  Google Scholar 

  • Baumgartner, M. (2010a). Causal slingshots. Erkenntnis, 72, 11–133.

    Article  Google Scholar 

  • Baumgartner, M. (2010b). Shallow analysis and the slingshot argument. Journal of Philosophical Logic, 39, 531–556.

    Article  Google Scholar 

  • Biłat, A. (2009). Filozoficzne podstawy logiki sytuacji’ (‘Philosophical Foundations of logic of situations’). In A. Biłat (Ed.), Aporie ontologii sytuacji (pp. 95–153). Lublin: Wydawnictwo Uniwersytetu Marii Curie-Skłodowskiej.

    Google Scholar 

  • Church, A. (1943). Carnap’s introduction to semantics. The Philosophical Review, 52, 298–304.

    Article  Google Scholar 

  • Correia, F. (2010). Grounding and truth-functions. Logique et Analyse, 53(211), 251–279.

    Google Scholar 

  • Cummins, R., & Gottlieb, D. (1972). On an argument for truth-functionality. American Philosophical Quarterly, 9, 265–269.

    Google Scholar 

  • Davidson, D. (1967). Truth and meaning. Synthese, 17, 304–323.

    Article  Google Scholar 

  • Davidson, D. (1969). True to the facts. The Journal of Philosophy, 66, 748–764.

    Article  Google Scholar 

  • Davidson, D. (1990). The structure and content of truth. The Journal of Philosophy, 87, 279–328.

    Article  Google Scholar 

  • Donaho, S. (1998). Are declarative sentences representations? Mind, 107, 33–57.

    Article  Google Scholar 

  • Drai, D. (2002). The slingshot argument: An improved version. Ratio (new series), 15, 194–204.

    Google Scholar 

  • Dummett, M. (1973). Frege: Philosophy of language. London: Duckworth.

    Google Scholar 

  • Fine, K. (2012). The pure logic of ground. The review of symbolic logic, 5, 1–25.

    Article  Google Scholar 

  • Føllesdal, D. (1983). Situation semantics and the “slingshot” argument. Erkenntnis, 19, 91–98.

    Article  Google Scholar 

  • Geach, P. T. (1976). Saying and showing in Frege and Wittgenstein. In J. Hintikka (Ed.), Essays on Wittgenstein in Honour of G.H. von Wright (pp. 54–70). Amsterdam: Acta Philosophica Fennica, 28.

    Google Scholar 

  • Golińska-Pilarek, J., & Huuskonen, T. (2005). Number of extensions of non-Fregean logics. Journal of Philosophical Logic, 34, 193–206.

    Article  Google Scholar 

  • Gödel, K. (1944). Russell’s mathematical logic. In P. A. Schlipp (Ed.), The Philosophy of Bertrand Russell (pp. 125–153). Evanston and Chicago: Northwestern University Press.

    Google Scholar 

  • Krysztofiak, W. (2007a). The phenomenological idealism controversy in light of possible worlds semantics. Axiomathes, 17, 75–97.

    Article  Google Scholar 

  • Krysztofiak, W. (2007b). Spór o ontologię sytuacji jako spór o zasadę kompozycyjności. Argument z metafory (The controversy over ontology of situations as the controversy over the compositionality principle. The argument of a metaphor). Filozofia Nauki, 15, No 4(60), pp. 51–70.

    Google Scholar 

  • Krysztofiak, W. (2009). Co wyraża argument slingshot? (What does the slingshot argument express?). In A. Biłat (Ed.), Aporie ontologii sytuacji (Aporias of ontology of situations) (pp. 37–73). Lublin: Wydawnictwo Uniwersytetu Marii Curie-Skłodowskiej.

    Google Scholar 

  • Krysztofiak, W. (2011). Fakty matematyczne w świetle logiki niefregowskiej (Mathematical facts in light of non-Fregean logic). Filozofia Nauki, 19, No 4(76), 83–101.

    Google Scholar 

  • Krysztofiak, W. (2013). Do we need mathematical facts? History and philosophy of logic. http://www.tandfonline.com/doi/full/10.1080/01445340.2013.828517

  • Lycan, W. D. (1974). The extensionality of cause, space and time. Mind, 83, 498–511.

    Article  Google Scholar 

  • Mulligan, K., Simons, P., & Smith, B. (1984). Truth-makers. Philosophy and the Phenomenological Research, 44, 287–321.

    Article  Google Scholar 

  • Niiniluoto, I. (2004). Tarski’s definition and truth-makers. Annals of Pure and Applied Logic, 126, 57–76.

    Article  Google Scholar 

  • Nowak, M. (2008). The logics of analytic equivalence. Bulletin of the Section of Logic, 37(3/4), 265–272.

    Google Scholar 

  • Neale, S. (1995). The philosophical significance of Gödel’s slingshot. Mind, 104, 761–825.

    Article  Google Scholar 

  • Omyła, M. (1986). Zarys logiki niefregowskiej (The outline of non-Fregean logic). Warszawa: PWN.

    Google Scholar 

  • Omyła, M. (2009), ‘Deskrypcje w logice niefregowskiej’ (‘Descriptions In non-Fregean logic’), [w:] (red. A. Biłat), Aporie ontologii sytuacji (Aporias of ontology of situations), Lublin 2009, Wydawnictwo Uniwersytetu Marii Curie-Skłodowskiej, s. 75–93.

  • Oppy, G. (1997). The philosophical insignificance of Gödel’s slingshot. Mind, 106, 121–141.

    Article  Google Scholar 

  • Putnam, H. (1975). Do true assertion correspond to reality. In H. Putnam (Ed.), Mind, language and reality. Philosophical papers (Vol. 2, pp. 70–84). Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • Quine, W. O. (1966). Three grades of modal involvement. In W. O. Quine (Ed.), The ways of paradox (pp. 156–174). New York: Random House.

    Google Scholar 

  • Quine, W. V. (1970). Philosophy of logic. Englewood: Prentice-Hall.

    Google Scholar 

  • Restall, G. (2004). One way to face facts. The Philosophical Quarterly, 54, 420–426.

    Article  Google Scholar 

  • Rodriguez-Pereyra, G. (1998). Searle’s correspondence theory of truth and the slingshot. The Philosophical Quarterly, 48(193) (1998), 513–522.

    Google Scholar 

  • Searle, J. (1996). The construction of social reality. Harmondsworth: Penguin.

    Google Scholar 

  • Shramko, Y., & Wansing, H. (2009). The slingshot argument and sentential identity. Studia Logica, 91, 429–455.

    Article  Google Scholar 

  • Smith, B. (1989). Logic and the sachverhalt. The Monist, 72, 52–69.

    Article  Google Scholar 

  • Suszko, R. (1968). Ontology in the Tractatus of L. Wittgenstein. Notre Dame Journal of Formal Logic, 9(1), 7–33.

    Google Scholar 

  • Suszko, R. (1998). Logika i teorie niefregowskie. In Suszko, R. Wybór pism, Polskie Towarzystwo Semiotyczne: Warszawa. [translated from: Suszko, R. (1968). Non-Fregean logic and theories. Analele Universitatii Bucuresti, Acta logica XI, 105–125].

  • Widerker, D. (1983). The extensionality argument. Noûs, 17, 457–468.

    Article  Google Scholar 

  • Williamson, J. (1976). Facts and truth. The Philosophical Quarterly, 26, 203–216.

    Article  Google Scholar 

  • Wójcicki, R. (1984). Romana Suszki semantyka sytuacyjna (Roman Suszko’s situational semantics). Studia Filozoficzne, 7(224), 3–19.

    Google Scholar 

  • Wójtowicz, A. (2005). The slingshot argument and non-Fregean logic. In A. Brożek, J. J. Jadacki, & W. Strawiński (Eds.), Logic, methodology and philosophy of science in Warsaw (Vol. 2, pp. 185–192). Warszawa: Wydawnictwo Naukowe Semper.

    Google Scholar 

  • Wójtowicz, A. (2007) Znaczenie nazw a znaczenie zdań. W obronie ontologii sytuacji (The meaning of names and the meaning of sentences. In defense of ontology of situations). Warszawa: Wydawnictwo Naukowe Semper.

  • Yourgrau, P. (1987). Frege on truth and reference. Notre Dame Journal of Formal Logic, 28, 132–138.

    Article  Google Scholar 

Download references

Acknowledgments

I would like to thank my student Jason Streitfeld for comments concerning an earlier version of this paper. This paper is written within the project No DEC-2011/01/B/HS1/04029, supported by The National Science Centre. The main ideas of the paper stem from Krysztofiak (2009, 2011) and (2013).

Author information

Authors and Affiliations

  1. Institute of Philosophy, University of Szczecin, Szczecin, Poland

    Wojciech Krysztofiak

Authors
  1. Wojciech Krysztofiak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wojciech Krysztofiak.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Krysztofiak, W. Hyper-Slingshot. Is Fact-Arithmetic Possible?. Found Sci 20, 59–76 (2015). https://doi.org/10.1007/s10699-014-9350-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10699-014-9350-6

Keywords

Search

Navigation