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Capturing consequence. (English) Zbl 07063892
Summary: First-order formalisations are often preferred to propositional ones because they are thought to underwrite the validity of more arguments. We compare and contrast the ability of some well-known logics – these two in particular – to formally capture valid and invalid arguments. We show that there is a precise and important sense in which first-order logic does not improve on propositional logic in this respect. We also prove some generalisations and related results of philosophical interest. The rest of the article investigates the results’ philosophical significance. A first moral is that the correct way to state the oft-cited superiority of first-order logic vis-à-vis propositional logic is more nuanced than often thought. The second moral concerns semantic theory; the third logic’s use as a tool for discovery. A fourth and final moral is that second-order logic’s transcendence of first-order logic is greater than first-order logic’s transcendence of propositional logic.
03A05 Philosophical and critical aspects of logic and foundations
03B05 Classical propositional logic
03B10 Classical first-order logic
03B65 Logic of natural languages
Full Text: DOI
[1] Barwise, J.; Feferman, S., Model-Theoretic Logics, (1985), New York: Springer-Verlag, New York · Zbl 0587.03001
[2] Baumgartner, M.; Lampert, T., Adequate Formalization, Synthese, 164, 93-115, (2008) · Zbl 1169.03308
[3] Beall, J. C.; Restall, G., Logical Pluralism, (2006), Oxford: Oxford University Press, Oxford · Zbl 1374.03001
[4] Blok, W.; Pigozzi, D., Algebraizable Logics, 77, Memoirs of the American Mathematical Society, (1989), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0664.03042
[5] Boolos, G., The Journal of Philosophy, 72, On second-order logic, 37-53, (1975), Cambridge, MA: MIT Press, Cambridge, MA
[6] Czelakowski, J., Protoalgebraic Logics, (2001), Dordrecht: Kluwer, Dordrecht · Zbl 0984.03002
[7] Davidson, D.; Rescher, N., The Logic of Decision and Action, The logical form of action sentences, 105-148, (1967), University of Pittsburgh Press: University of Pittsburgh Press, Oxford: Oxford University Press, University of Pittsburgh Press: University of Pittsburgh Press, Oxford
[8] Font, J. M.; Jansana, R.; Pigozzi, D., A survey of abstract algebraic logic, Studia Logica, 74, 13-97, (2003) · Zbl 1057.03058
[9] Givant, S.; Halmos, P., Introduction to Boolean Algebras, (2009), New York: Springer, New York · Zbl 1168.06001
[10] Glanzberg, M.; Caret, C. R.; Hjortland, O. T., Foundations of Logical Consequence, Logical consequence and natural language, 71-120, (2015), New York: Oxford University Press, New York
[11] Griffiths, O.; Paseau, A. C., One True Logic, Oxford University Press
[12] Harman, G., Change in View, (1986), Cambridge, MA: MIT Press, Cambridge, MA
[13] Heim, I.; Kratzer, A., Semantics in Generative Grammar, (1998), Malden, MA: Blackwell, Malden, MA
[14] Higginbotham, J.; Pianesi, F.; Varzi, A., Speaking of Events, (2000), New York: Oxford University Press, New York
[15] Hinman, P. G., Fundamentals of Mathematical Logic, (2005), Wellesley, MA: A.K. Peters, Wellesley, MA · Zbl 1081.03003
[16] Lewitzka, S.; Beziau, J.-Y., Logica Universalis, A topological approach to universal logic: Model-theoretical abstract logics, 35-61, (2007), Basel: Birkhäuser, Basel · Zbl 1143.03355
[17] Monk, J. D.; Bonnet, R., Handbook of Boolean Algebras, 3, (1989), Amsterdam: North-Holland, Amsterdam
[18] Oliver, A. D.; Lear, J.; Oliver, A. D., The Force of Argument: Essays in Honor of Timothy Smiley, The matter of form: Logic’s beginnings, 165-185, (2010), New York: Routledge, New York
[19] Oliver, A. D.; Smiley, T., Plural Logic, (2013), Oxford: Oxford University Press, Oxford · Zbl 1273.03002
[20] Parsons, T., Events in the Semantics of English, (1990), Cambridge, MA: MIT Press, Cambridge, MA
[21] Paseau, A. C., A measure of inferential-role preservation, Synthese, (2015)
[22] Priest, G., Doubt Truth to be a Liar, (2006), Oxford: Oxford University Press, Oxford · Zbl 1274.00014
[23] Quine, W. V., Word and Object, (1960), Cambridge, MA: MIT Press, Cambridge, MA
[24] Quine, W. V., Methods of Logic, (1982), Cambridge, MA: Harvard University Press, Cambridge, MA
[25] Rumfitt, I.; Novák, Z.; Simonyi, A., Truth, Reference and Realism, What is logic?, 125, (2011), Budapest: Central European University Press, Budapest
[26] Rumfitt, I., The Boundary Stones of Thought, (2015), Oxford: Oxford University Press, Oxford
[27] Russell, B., On denoting, Mind, 14, 479-493, (1905)
[28] Shapiro, S., Foundations without Foundationalism, (1991), New York: Clarendon Press, New York · Zbl 0732.03002
[29] Shapiro, S.; Schirn, M., The Philosophy of Mathematics Today, Logical consequence: Models and modality, 131-156, (1998), Oxford: Oxford University Press, Oxford
[30] Shapiro, S., Varieties of Logic, (2014), New York: Oxford University Press, New York · Zbl 1303.03001
[31] Strawson, P. F., On referring, Mind, 59, 320-344, (1950)
[32] Strawson, P. F., Introduction to Logical Theory, (1952), New York: Wiley, New York · Zbl 0049.14602
[33] Väänänen, J., Second order logic or set theory, Bulletin of Symbolic Logic, 18, 91-121, (2012) · Zbl 1252.03024
[34] Yi, B.-U., The logic and meaning of plurals. Part II, Journal of Philosophical Logic, 35, 239-288, (2006) · Zbl 1097.03008
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