Abstract
A first-order language with a defined identity predicate is proposed whose apparatus for atomic predication is sensitive to grammatical categories of natural language (e.g., common nouns, verbs, adjectives, adverbs, modifiers). Subatomic natural deduction systems are defined for this naturalistic first-order language. These systems contain subatomic systems which govern the inferential relations which obtain between naturalistic atomic sentences and between their possibly composite components. As a main result it is shown that normal derivations in the defined systems enjoy the subexpression property which subsumes the subformula property with respect to atomic and identity formulae as a special case. The systems admit a proof-theoretic semantics which does not only apply to logically compound but also to atomic and identity formulae—as well as to their components. The potential of the defined systems for a meticulous first-order analysis of natural inferences whose validity crucially depends on expressions of some of the aforementioned categories is demonstrated.
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Notes
Digression: In contrast to Więckowski (2011, forthcoming), the present elimination rules for atomic sentences are selective in so far as their conclusion contains only the term assumptions for a single component of the atomic sentence rather than compactly the term assumptions for each of its components. As a result subatomic derivations receive a tree structure which is more familiar. Furthermore, the present eliminative term assumptions for atomic sentences do not contain exactly the eliminated atomic sentence. Accordingly, they are not singletons and they are not set up in such a way that they do not contain anything beyond what is licensed by the internal structure of the premiss of the elimination rule. An advantage of this relaxation is that there is no need to appeal to a rule for supplementation of eliminative term assumptions (see Więckowski forthcoming) in order to pass from one atomic sentence to another within the subatomic system, as no contents are lost from term assumptions in the course of applications of the elimination rules for atomic sentences. It is not possible to derive an atomic sentence from another one inside subatomic systems of the kind proposed in Więckowski (2011). In the subatomic natural deduction systems presented in that paper the derivational relations between distinct atomic sentences are governed externally by meaning postulates which contain logical vocabulary.
These rules elaborate on those presented in Więckowski (2011) which, in turn, have been modeled on the \(\ddot{=}\)-axioms of Więckowski (2010). A type-theoretical variant of \(\ddot{=}\)-rules is presented in Więckowski (2012, 2015). It might be interesting to consider \(\ddot{=}\)-rules for infinite \(\mathcal {P}\), but for the present purposes we may put such considerations aside.
Perhaps the \(\ddot{=}\)I/E-rules are also more satisfactory than other currently available rules for identity (see, for instance, Read 2004 and the discussion in Griffiths 2014). In particular, as derivation (14) suggests, one may readily derive an ids of the form \(\alpha _{1} \ddot{=} \alpha _{2}\) (where \(\alpha _{1} \not \equiv \alpha _{2}\)) in an I(\(\mathcal {S}^{\ddot{=}}\))-system (as a thesis) without deriving it either from other idss of this kind or from absurdity.
This strategy is essentially an adaptation of the strategy suggested in Francez (2015a, 335).
References
Davidson, D. (2001). The logical form of action sentences. In D. Davidson, Essays on actions and events (pp. 105–122). Oxford: Clarendon Press. (Originally published in N. Rescher (ed.), The Logic of Decision and Action, Pittsburg, University of Pittsburg Press, 1967.).
Davies, R., & Pfenning, F. (2001). A modal analysis of staged computation. Journal of the ACM, 48(3), 555–604.
Dummett, M. (1991). The logical basis of metaphysics. Cambridge, MA: Harvard University Press.
Fitch, F. B. (1973). Natural deduction rules for English. Philosophical Studies, 24(2), 89–104.
Forbes, G. (2013). Intensional transitive verbs. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Fall 2013 Edition). http://plato.stanford.edu/archives/fall2013/entries/intensional-trans-verbs/.
Francez, N. (2015a). Proof-theoretic semantics. London: College Publications.
Francez, N. (2015b). On the notion of canonical derivations from open assumptions and its role in proof-theoretic semantics. The Review of Symbolic Logic, 8(2), 296–305.
Francez, N., & Ben-Avi, G. (2015). Proof-theoretic reconstruction of generalized quantifiers. Journal of Semantics, 32(3), 313–371.
Francez, N., & Dyckhoff, R. (2010). Proof-theoretic semantics for a natural language fragment. Linguistics and Philosophy, 33(6), 447–477.
Francez, N., & Dyckhoff, R. (2012). A note on harmony. The Journal of Philosophical Logic, 41(3), 613–628.
Francez, N., Dyckhoff, R., & Ben-Avi, G. (2010). Proof-theoretic semantics for subsentential phrases. Studia Logica, 94(3), 381–401.
Francez, N., & Więckowski, B. (forthcoming). A proof theory for first-order logic with definiteness. IfCoLog Journal of Logics and their Applications.
Frege, G. (1879). Begriffsschrift. Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle an der Saale: Verlag von Louis Nebert. (Reprint: I. Angelelli (ed.), Hildesheim, Georg Olms Verlag, 1998).
Frege, G. (1893/1903). Grundgesetze der Arithmetik I/II. Jena: Verlag von Hermann Pohle. (Reprint: C. Thiel (ed.), Hildesheim, Georg Olms Verlag, 2009).
Gamut, L. T. F. (1991). Logic, language, and meaning: Intensional logic and logical grammar (Vol.2). Chicago: The University of Chicago Press.
Gentzen, G. (1934). Untersuchungen über das logische Schließen I, II. Mathematische Zeitschrift, 39, 176–210, 405–432
Griffiths, O. (2014). Harmonious rules for identity. The Review of Symbolic Logic, 7(3), 499–510.
Jaśkowski, S. (1934). On the rules of suppositions in formal logic. Studia Logica, 1, 5–32.
Maienborn, C. (2011). Event semantics. In C. Maienborn, K. von Heusinger, & P. Portner (Eds.), Semantics: An international handbook of natural language meaning (Vol. 1, pp. 802–829). Berlin: de Gruyter Mouton.
Montague, R. (1974). Formal philosophy. Selected papers of Richard Montague, R. H. Thomason (Ed.). New Haven: Yale University Press.
Pfenning, F., & Davies, R. (2001). A judgmental reconstruction of modal logic. Mathematical Structures in Computer Science, 11, 511–540.
Piecha, T., de Campos Sanz, W., & Schroeder-Heister, P. (2015). Failure of completeness in proof-theoretic semantics. Journal of Philosophical Logic, 44(3), 321–335.
Prawitz, D. (1965). Natural deduction. A proof-theoretical study. Stockholm: Almqvist & Wiksell. (Reprint: Mineola/NY, Dover Publications, 2006.).
Prawitz, D. (1971). Ideas and results in proof theory. In J. E. Fenstad (Ed.), Proceedings of the second Scandinavian Logic symposium (Oslo 1970) (pp. 235–309). Amsterdam: North-Holland.
Prawitz, D. (1973). Towards a foundation of a general proof theory. In P. Suppes (Ed.), Logic, methodology and philosophy of science IV (pp. 225–250). Amsterdam: North-Holland.
Prawitz, D. (2006). Meaning approached via proofs. Synthese, 148(3), 507–524. Special issue on Proof-Theoretic Semantics edited by R. Kahle and P. Schroeder-Heister.
Quine, W. V. (1969). Existence and quantification. In W. V. Quine (Ed.), Ontological relativity and other essays (pp. 94–96). New York: Columbia University Press.
Read, S. (2004). Identity and harmony. Analysis, 64, 113–119. (Revised version: Identity and harmony revisited, 2014).
Read, S. (2015). General elimination harmony and higher-level rules. In H. Wansing (Ed.), Dag Prawitz on proofs and meaning (pp. 293–312). Berlin: Springer.
Sandqvist, T. (2015). Hypothesis-discharging rules in atomic bases. In H. Wansing (Ed.), Dag Prawitz on proofs and meaning (pp. 313–328). Berlin: Springer.
Schroeder-Heister, P. (2013). Proof-theoretic semantics. In Zalta, E. N. (Ed.), The Stanford encyclopedia of philosophy (Spring 2013 Edition). http://plato.stanford.edu/archives/spr2013/entries/proof-theoretic-semantics/.
Schroeder-Heister, P. (2015). Harmony in proof-theoretic semantics: A reductive analysis. In H. Wansing (Ed.), Dag Prawitz on proofs and meaning (pp. 329–358). Berlin: Springer.
Steinberger, F. (2013). On the equivalence conjecture for proof-theoretic harmony. Notre Dame Journal of Formal Logic, 54(1), 79–86.
Tarski, A. (1995). Pisma Logiczno-Filozoficzne: Tom 1. Prawda, J. Zygmunt (Ed.), Warszawa: Wydawnictwo Naukowe PWN.
Tranchini, L. (2015). Harmonising harmony. The Review of Symbolic Logic, 8(3), 411–423.
Troelstra, A. S., & Schwichtenberg, H. (2000). Basic proof theory (2nd ed.). Cambridge: Cambridge University Press.
van Dalen, D. (2004). Logic and structure (4th ed.). Berlin: Springer.
Wansing, H. (2001). The idea of a proof-theoretic semantics and the meaning of the logical operations. Studia Logica, 64(1), 3–20.
Wansing, H. (2015). Dag Prawitz on proofs and meaning: Outstanding contributions to logic (Vol. 7). Berlin: Springer.
Więckowski, B. (2010). Associative substitutional semantics and quantified modal logic. Studia Logica, 94(1), 105–138.
Więckowski, B. (2011). Rules for subatomic derivation. The Review of Symbolic Logic, 4(2), 219–236.
Więckowski, B. (2012). A constructive type-theoretical formalism for the interpretation of subatomically sensitive natural language constructions. Studia Logica, 100(4), 815–853. Special issue on Logic and Natural Language edited by N. Francez and I. Pratt-Hartmann.
Więckowski, B. (2015). Constructive belief reports. Synthese, 192(3), 603–633. Special section on Hyperintensionality edited by M. Duží and B. Jespersen.
Więckowski, B. (forthcoming). Refinements of subatomic natural deduction, Journal of Logic and Computation, (first published online: August 20, 2014).
Acknowledgments
I would like to thank Nissim Francez for discussions on proof-theoretic semantics and an anonymous referee for her/his feedback. This work was supported by the DFG (Grant WI 3456/2-1).
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Więckowski, B. Subatomic Natural Deduction for a Naturalistic First-Order Language with Non-Primitive Identity. J of Log Lang and Inf 25, 215–268 (2016). https://doi.org/10.1007/s10849-016-9238-7
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DOI: https://doi.org/10.1007/s10849-016-9238-7