Advertisement

\(\boldsymbol{\forall }\) and \(\boldsymbol{\omega }\)

  • Elia ZardiniEmail author
Part of the Synthese Library book series (SYLI, volume 373)

Abstract

I first briefly rehearse the two substructural solutions that I’ve elsewhere proposed to the semantic and vagueness paradoxes. I then ask what the correct principle of universal generalisation is. The traditional answer to this question is represented by the familiar principle to the effect that, provided that τ does not occur free in either Γ, Δ or φ, if \(\varGamma \vdash \varDelta,\varphi _{\tau /\xi }\) holds, \(\varGamma \vdash \varDelta,\forall \xi \varphi\) holds. I argue for interpreting such principle as in effect licencing the inference from ‘anything’ to ‘everything’. I then proceed to offer five arguments against that inference. The first three arguments rely on considerations concerning the preface paradox, the failure of agglomeration for counterfactual implication and free-choice permission respectively. The last two arguments connect back with the semantic and vagueness paradoxes. I show how the inference from ‘anything’ to ‘everything’ would wreak havoc for the workings both of my non-contractive solution to the semantic paradoxes and of my non-transitive solution to the vagueness paradoxes. I then inquire into what a more adequate generalisation principle should be, and argue in favour of a suitably generalised version of the ω-rule, defending it from several prominent objections. I then trace back the quantificational phenomena studied in the paper, in particular those most directly related to the semantic and vagueness paradoxes, to their sentential root concerning the behaviour of conjunction. I sketch a metaphysical view making sense of the failure of the conjunctive analogue of the traditional generalisation principle, and close by bringing out some positive implications such view has for our logical freedom.

Keywords

Singular Term Universal Quantification Logical Truth Relevant Domain Generalisation Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Earlier versions of the material in this paper have been presented in 2013 at the LOGOS Workshop Substructural Approaches to Paradox (University of Barcelona) and at the 7th Navarre Vagueness Workshop in Pamplona (University of Navarre), where Paul Égré gave a valuable response; in 2014, at the 3rd Colombian Congress on Logic, Epistemology and Philosophy of Science in Bogotá (University of the Andes/University of the Rosario) and at the PERSP Metaphysics Seminar (University of Barcelona). I’d like to thank all these audiences for very stimulating comments and discussions. Special thanks go to Pablo Cobreros, Anamaria Fălăuş, Nissim Francez, John Horden, Dan López de Sa, Pepe Martínez, Giovanni Merlo, Joe Moore, Julien Murzi, Sergi Oms, Francesco Paoli, Paloma Pérez-Ilzarbe, Dave Ripley, Sven Rosenkranz, Gonçalo Santos, Lionel Shapiro, Roy Sorensen, Luca Tranchini, Alan Weir and Dan Zeman. I’m also grateful to the editor Alessandro Torza for inviting me to contribute to this volume and for his support and patience throughout the process. At different stages during the writing of the paper, I’ve benefitted from the FP7 Marie Curie Intra-European Research Fellowship 301493 on A Non-Contractive Theory of Naive Semantic Properties: Logical Developments and Metaphysical Foundations (NTNSP) and from the FCT Research Fellowship IF/01202/2013 on Tolerance and Instability: The Substructure of Cognitions, Transitions and Collections (TI), as well as from partial funds from the project CONSOLIDER-INGENIO 2010 CSD2009-00056 of the Spanish Ministry of Science and Innovation on Philosophy of Perspectival Thoughts and Facts (PERSP), from the FP7 Marie Curie Initial Training Network 238128 on Perspectival Thoughts and Facts (PETAF), from the project FFI2011-25626 of the Spanish Ministry of Science and Innovation on Reference, Self-Reference and Empirical Data and from the project FFI2012-35026 of the Spanish Ministry of Economy and Competition on The Makings of Truth: Nature, Extent, and Applications of Truthmaking.

References

  1. 1.
    Ackermann, W. 1928. Zum Hilbertschen Aufbau der reellen Zahlen. Mathematische Annalen 99: 118–133.CrossRefGoogle Scholar
  2. 2.
    Adams, E. 1970. Subjunctive and indicative conditionals. Foundations of Language 6: 89–94.Google Scholar
  3. 3.
    Anderson, A. 1956. The formal analysis of normative systems. New Haven: Yale University Press.Google Scholar
  4. 4.
    Armstrong, D. 1997. A world of states of affairs. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  5. 5.
    Esenin-Vol’pin, A. 1970. The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics. In Intuitionism and proof theory, ed. A. Kino, J. Myhill, and R. Vesley, 3–45. Amsterdam: North-Holland.Google Scholar
  6. 6.
    Evans, G. 1978. Can there be vague objects? Analysis 38: 208.CrossRefGoogle Scholar
  7. 7.
    Fiengo, R. 2007. Asking questions. Oxford: Oxford University Press.CrossRefGoogle Scholar
  8. 8.
    Fine, K. 1983. A defence of arbitrary objects. Proceedings of the Aristotelian Society Supplementary Volume 57: 55–77.Google Scholar
  9. 9.
    Heidegger, M. 1929. Was ist Metaphysik? Bonn: Friedrich Cohen.Google Scholar
  10. 10.
    Kyburg, H. 1961. Probability and the logic of rational belief. Middletown: Wesleyan University Press.Google Scholar
  11. 11.
    Leonard, H. 1956. The logic of existence. Philosophical Studies 7: 49–64.CrossRefGoogle Scholar
  12. 12.
    Makinson, D. 1965. The paradox of the preface. Analysis 25: 205–207.CrossRefGoogle Scholar
  13. 13.
    McGee, V. 1985. A counterexample to modus ponens. The Journal of Philosophy 82: 462–471.CrossRefGoogle Scholar
  14. 14.
    Paoli, F. 2005. The ambiguity of quantifiers. Philosophical Studies 124: 313–330.CrossRefGoogle Scholar
  15. 15.
    Read, S. 1981. What is wrong with disjunctive syllogism? Analysis 41: 66–70.CrossRefGoogle Scholar
  16. 16.
    Russell, B. 1903. The principles of mathematics. Cambridge: Cambridge University Press.Google Scholar
  17. 17.
    Vogel, J. 2000. Reliabilism leveled. The Journal of Philosophy 97: 602–623.CrossRefGoogle Scholar
  18. 18.
    von Wright, G. 1951. Deontic logic. Mind 60: 1–15.CrossRefGoogle Scholar
  19. 19.
    Weir, A. 2005. Naive truth and sophisticated logic. In Deflationism and paradox, ed. B. Armour-Garb and J.C. Beall, 218–249. Oxford: Oxford University Press.Google Scholar
  20. 20.
    Weir, A. 2010. Truth through proof. Oxford: Oxford University Press.CrossRefGoogle Scholar
  21. 21.
    Wittgenstein, L. 1921. Logisch-philosophische Abhandlung. Annalen der Naturphilosophie 14: 185–262.Google Scholar
  22. 22.
    Wright, C. 1982. Strict finitism. Synthese 51: 203–282.CrossRefGoogle Scholar
  23. 23.
    Zardini, E. 2008a. A model of tolerance. Studia Logica 90: 337–368.CrossRefGoogle Scholar
  24. 24.
    Zardini, E. 2008b. Living on the slippery slope. The nature, sources and logic of vagueness. Ph.D. thesis, Department of Logic and Metaphysics, University of St. Andrews.Google Scholar
  25. 25.
    Zardini, E. 2009. Towards first-order tolerant logics. In Philosophy, mathematics, linguistics: Aspects of interaction, ed. O. Prozorov, 35–38. St. Petersburg: Russian Academy of Sciences Press.Google Scholar
  26. 26.
    Zardini, E. 2011. Truth without contra(di)ction. The Review of Symbolic Logic 4: 498–535.CrossRefGoogle Scholar
  27. 27.
    Zardini, E. 2013a. It is not the case that [P and ‘It is not the case that P’ is true] nor is it the case that [P and ‘P’ is not true]. Thought 1: 309–319.Google Scholar
  28. 28.
    Zardini, E. 2013b. Naive modus ponens. Journal of Philosophical Logic 42: 575–593.CrossRefGoogle Scholar
  29. 29.
    Zardini, E. 2013c. Closed without boundaries. MS.Google Scholar
  30. 30.
    Zardini, E. 2014a. Confirming the less likely, discovering the unknown. Dogmatisms: Surd and doubly surd, natural, flat and sharp. In Scepticism and perceptual justification, ed. D. Dodd and E. Zardini, 33–70. Oxford: Oxford University Press.Google Scholar
  31. 31.
    Zardini, E. 2014b. Evans tolerated. In Vague objects and vague identity, ed. K. Akiba and A. Abasnezhad, 327–352. Berlin: Springer.CrossRefGoogle Scholar
  32. 32.
    Zardini, E. 2014c. Naive truth and naive logical properties. The Review of Symbolic Logic 7: 351–384.CrossRefGoogle Scholar
  33. 33.
    Zardini, E. 2014d, Forthcoming. The opacity of truth. Topoi.Google Scholar
  34. 34.
    Zardini, E. 2015. Breaking the chains. Following-from and transitivity. In Foundations of logical consequence, ed. C. Caret and O. Hjortland, 221–275. Oxford: Oxford University Press.CrossRefGoogle Scholar
  35. 35.
    Zardini, E. 2015. Getting one for two, or the contractors’ bad deal. Towards a unified solution to the semantic paradoxes. In Unifying the philosophy of truth, ed. T. Achourioti, K. Fujimoto, H. Galinon, and J. Martínez, 461–493. Berlin: Springer.CrossRefGoogle Scholar
  36. 36.
    Zardini, E. Forthcoming. És la veritat una mentida? Perspectives sobre les paradoxes semàntiques. Anuari de la Societat Catalana de Filosofia.Google Scholar
  37. 37.
    Zardini, E. Forthcoming. First-order tolerant logics. The Review of Symbolic Logic.Google Scholar
  38. 38.
    Zardini, E. Forthcoming. Restriction by non-contraction. Notre Dame Journal of Formal Logic.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.LanCog, Language, Mind and Cognition Research GroupCentro de filosofia, Universidade de LisboaLisbonPortugal

Personalised recommendations