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

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


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.


Singular Term Universal Quantification Logical Truth Relevant Domain Generalisation Principle 
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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.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

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

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