Skip to main content Accessibility help
×
Hostname: page-component-5db58dd55d-mhzq2 Total loading time: 0 Render date: 2026-06-01T16:11:25.504Z Has data issue: false hasContentIssue false
Series:   Elements in Philosophy and Logic

Classical First-Order Logic

Published online by Cambridge University Press:  26 April 2022

Stewart Shapiro  and
Teresa Kouri Kissel
Stewart Shapiro
Affiliation:
Ohio State University
Teresa Kouri Kissel
Affiliation:
Old Dominion University, Virginia

Summary

One is often said to be reasoning well when they are reasoning logically. Many attempts to say what logical reasoning is have been proposed, but one commonly proposed system is first-order classical logic. This Element will examine the basics of first-order classical logic and discuss some surrounding philosophical issues. The first half of the Element develops a language for the system, as well as a proof theory and model theory. The authors provide theorems about the system they developed, such as unique readability and the Lindenbaum lemma. They also discuss the meta-theory for the system, and provide several results there, including proving soundness and completeness theorems. The second half of the Element compares first-order classical logic to other systems: classical higher order logic, intuitionistic logic, and several paraconsistent logics which reject the law of ex falso quodlibet.
Get access

Keywords

classical logicfirst-order logicmodel theoryproof theorymeta-theory higher-order logicintuitionistic logicparaconsistent logic

Information

Type
Element
Information
Series: Elements in Philosophy and Logic
Online ISBN: 9781108982009
Publisher: Cambridge University Press
Print publication: 19 May 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Element purchase

Temporarily unavailable

References

Anderson, A. R. , and Belnap, N. D. (1975). Entailment: The Logic of Relevance and Necessity, Vol. I. Princeton: Princeton University Press.Google Scholar
Anderson, A. R. , Belnap, N. D., and Dunn, M. (1992). Entailment: The Logic of Relevance and Necessity, Vol. II. Princeton: Princeton University Press.Google Scholar
Avron, A. (1990). Relevance and paraconsistency—a new approach. Journal of Symbolic Logic 55(2), 707–732.Google Scholar
Barrio, E. A. , Pailos, F., and Szmuc, D. (2021). Substructural logics, pluralism and collapse. Synthese 198, 4991–5007.CrossRefGoogle Scholar
Barwise, J. (1985). Model-theoretic logics: Background and aims. In Barwise, J. and Feferman, S. eds.), Model-theoretic Logics, pp. 3–23. New York: Springer-Verlag.Google Scholar
Barwise, J. , and Perry, J. (1983). Situations and Attitudes. Cambridge, Massachusetts, and London: MIT Press.Google Scholar
Beall, J. (2019). FDE as the One True Logic. In Omori, H. and Wansing, H. (eds.), New Essays on Belnap-Dunn Logic. Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science), vol 418, pp. 115–125. Cham: Springer.Google Scholar
Beall, J. , Brady, R., Dunn, J. M., Hazen, A. P., Mares, E., Meyer, R. K., Priest, G., Restall, G., Ripley, D., Slaney, J., and Sylvan, R. (2012). On the ternary relation and conditionality. Journal of Philosophical Logic 41(3), 595–612.CrossRefGoogle Scholar
Beall, J. , and Restall, G. (2006). Logical Pluralism. Oxford: Clarendon Press.Google Scholar
Benacerraf, P. , and Putnam, H. (1983). Philosophy of Mathematics. Cambridge: Cambridge University Press.Google Scholar
Boolos, G. (1984). To be is to be a value of a variable (or to be some values of some variables). Journal of Philosophy 81(8), 430–449.Google Scholar
Boolos, G. (1985). Nominalist platonism. Philosophical Review 94, 327–344.CrossRefGoogle Scholar
Boolos, G. , Burgess, J., and Jeffrey, R. (2002). Computability and Logic, 4th ed., Cambridge: Cambridge University Press.Google Scholar
Brouwer, L. E. J. (1964a). Consciousness, philosophy and mathematics. In Benacerraf, P. and Putnam, H. (eds.), Philosophy of Mathematics: Selected Readings, pp. 90–96. Englewood Cliffs, NJ: Cambridge University Press.Google Scholar
Brouwer, L. E. J. (1964b). Intuitionism and Formalism. In Benacerraf, P. and Putnam, H. (eds.), Philosophy of Mathematics: Selected Readings, pp. 77–89. Englewood Cliffs, NJ: Cambridge University Press.Google Scholar
Burgess, J. (1992). Proofs about proofs: A defense of classical logic. In Detlefsen, M. (ed.), Proof, Logic and Formalization, pp. 8–23. London: Routledge.Google Scholar
Cocchiarella, N. (1988). Predication versus membership in the distinction between logic as language and logic as calculus. Synthese 77, 37–72.Google Scholar
Cook, R. T. (2002). Vagueness and mathematical precision. Mind 111(442), 225–247.Google Scholar
Corcoran, J. (1973). Gaps between logical theory and mathematical practice. In Bunge, M. A. (ed.), The Methodological Unity of Science, pp. 23–50. Boston: Reidel.Google Scholar
Davidson, D. (1984). Inquiries into Truth and Interpretation. Oxford: Oxford University Press.Google Scholar
Davis, M. (1965). The Undecidable. Hewlett, New York: The Raven Press.Google Scholar
Dean, W. (forthcoming). Skolem’s Paradox and Non-absoluteness. Cambridge: Cambridge University Press.Google Scholar
Dedekind, R. (1888). Was Sind und Was Sollen die Zahlen? F. Vieweg.Google Scholar
Dummett, M. (1978a). The Justification of Deduction. In Truth and Other Enigmas, pp. 290–318. Cambridge, MA: Harvard University Press.Google Scholar
Dummett, M. (1978b). The Philosophical Basis of Intuitionistic Logic. In Truth and Other Enigmas, pp. 215–247. Cambridge, MA: Harvard University Press.Google Scholar
Dummett, M. (2000). Elements of Intuitionism. Oxford: Clarendon Press.Google Scholar
Dunn, J. M. (2015). The relevance of relevance to relevance logic. In Banerjee, M. and Krishna, S. N. (eds.), Logic and Its Applications, pp. 11–29. Berlin Heidelberg: Springer.Google Scholar
Eklund, M. (1996). How logic became first-order. Nordic Journal of Philosophical Logic 1, 147–167.Google Scholar
Feferman, S. (2006). Predicativity. In Shapiro, S. (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, pp. 590–624. Oxford: Oxford University Press.Google Scholar
Frege, G. (1879). Begriffsschrift: Eine der Arithmetischen Nachgebildete Formelsprache des Reinen Denkens. Halle a.d.S.: Louis Nebert.Google Scholar
Gilmore, P. (1957). The monadic theory of types in the lower-predicate calculus. In Summaries of Talks Presented at the Summer Institute of Symbolic Logic at Cornell, pp. 309–312. Princeton, NJ: Institute for Defense Analysis.Google Scholar
Gödel, K. (1930). Die vollständigkeit der axiome des logischen funktionenkalkuls. Montatshefte für Mathematik und Physik 37, 349–360. Translated as “The completeness of the axioms of the functional calculus of logic,” in Van Heijenoort (1967), pp. 582–591.CrossRefGoogle Scholar
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter systeme i. Montatshefte für Mathematik und Physik 38(1), 173–198. Translated as “On formally undecidable propositions of the Principia Mathematica,” in Davis (1965), pp. 4–35, Van Heijenoort (1967), pp. 596–616, and Gödel (1986), 144–195.Google Scholar
Gödel, K. (1986). Collected Works I. Oxford: Oxford University Press.Google Scholar
Gottwald, S. (2020). Many-valued logic. In Zalta, E. N. (ed.), The Stanford Encyclopedia of Philosophy (Summer 2020 ed.). <https://plato.stanford.edu/archives/sum2020/entries/logic-manyvalued/>.Google Scholar
Haack, S. (1978). Philosophy of Logic. Cambridge: Cambridge University Press.Google Scholar
Haack, S. (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism. Chicago: University of Chicago Press.Google Scholar
Henkin, L. (1950). Completeness in the theory of types. The Journal of Symbolic Logic 15(2), 81–91.CrossRefGoogle Scholar
Heyting, A. (1956). Intuitionism. Amsterdam: North Holland.Google Scholar
Hughes, G. E. , and Cresswell, M. (1996). A New Introduction to Modal Logic. United Kingdom: Routledge.Google Scholar
Kerr, A. D. (2021). A plea for KR. Synthese 198, 3047–3071.CrossRefGoogle Scholar
Kripke, S. (1965). Semantical analysis of intuitionistic logic I. In Crossley, J. and Dummett, M. (eds.), Formal Systems and Recursive Functions, pp. 92–130. Amsterdam: North Holland.CrossRefGoogle Scholar
Lewis, C. I. , and Langford, C. H. (1932). Symbolic Logic. New York: Dover Publications.Google Scholar
Lindström, P. (1969). On extensions of elementary logic. Theoria 35(1), 1–11.Google Scholar
Link, G. (1998). Algebraic Semantics in Language and Philosophy. Stanford, CA: SCLI Publications.Google Scholar
Linnebo, O. (2017). Plural quantification. In Edward, N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). https://plato.stanford.edu/archive/sum2017/entries/plural-quant.Google Scholar
Logan, Shay & Graham, Leach-Krouse (2021). On Not Saying What We Shouldn’t Have to Say. _Australasian Journal of Logic_ 18(5):524–568.CrossRefGoogle Scholar
Löwenheim, L. (1915). Über möglichkeiten im relativkalkül. Mathematische Annalen 76, 447–470.CrossRefGoogle Scholar
Marcus, R. (1995). Modalities: Philosophical Essays. New York: Oxford University Press.Google Scholar
Mares, E. (2004). Relevant Logic: A Philosophical Interpretation. Cambridge: Cambridge University Press.Google Scholar
Mares, E. (2020). Relevance logic. In Zalta, E. N. (ed.), The Stanford Encyclopedia of Philosophy (Summer 2020 ed.). Stanford: Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/entries/logic-relevance/Google Scholar
Martin, C. J. (1986). William’s machine. The Journal of Philosophy 83(10), 564–572.CrossRefGoogle Scholar
Montague, R. (1974). Formal Philosophy: Selected Papers of Richard Montague. New Haven: Yale University Press.Google Scholar
Moore, G. H. (1980). Beyond first-order logic, the historical interplay between logic and set theory. History and Philosophy of Logic 1, 95–137.Google Scholar
Moore, G. H. (1982). Zermelo’s Axiom of Choice: Its Origins, Development, and influence. Journal of Symbolic Logic 49(2), 659–660. Springer-Verlag.Google Scholar
Moore, G. H. (1988). The emergence of first-order logic. In Aspray, W. and Kitcher, P. (eds.), History and Philosophy of Modern Mathematics, pp. 95–135. Minneapolis: University of Minnesota Press. Minnesota Studies in the Philosophy of Science Volume 11.Google Scholar
Mortensen, C. (2013). Inconsistent Mathematics. Mathematics and Its Applications. Netherlands: Springer.Google Scholar
Øgaard, T. F. (2016). Paths to triviality. Journal of Philosophical Logic 45(3), 237–276.Google Scholar
Peano, G. (1889). Arithmetices principia: nova methodo exposita. Bocca brothers.Google Scholar
Prawitz, D. (1974). On the idea of a general proof theory. Synthese 27(1), 63–77.Google Scholar
Prawitz, D. (2008). Meaning and proofs: On the conflict between classical and intuitionistic logic. Theoria 43(1), 2–40.Google Scholar
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic 8(1), 219–241.CrossRefGoogle Scholar
Priest, G. (1987). Doubt Truth to Be a Liar. New York: Oxford University Press.Google Scholar
Priest, G. (2001). An Introduction to Non-Classical Logic. Cambridge: Cambridge University Press.Google Scholar
Priest, G. (2006). In Contradiction: A Study of the Transconsistent, 2nd ed., Oxford: Oxford University Press.Google Scholar
Priest, G. , Tanaka, K., and Weber, Z. (2018). Paraconsistent logic. In Zalta, E. N. (ed.), The Stanford Encyclopedia of Philosophy (Summer 2018 ed.). Research Lab, Stanford University. https://plato.stanford.edu/entries/logic-paraconsistent/MetaphysicsGoogle Scholar
Quine, W. V. O. (1986). Philosophy of Logic. Cambridge, MA: Harvard University Press.Google Scholar
Rayo, A. , and Yablo, S. (2001). Nominalism through de-nominalization. Noûs 35(1), 74–92.Google Scholar
Read, S. (1988). Relevant Logic: A Philosophical Examination of Inference. Oxford: Wiley-Blackwell.Google Scholar
Resnik, M. (1988). Second-order logic still wild. Journal of Philosophy 85, 75–87.Google Scholar
Resnik, M. (1996). Ought there to be but one logic? In Copeland, B. J. (ed.), Logic and Reality: Essays on the Legacy of Arthur Prior, pp. 489–517. Oxford: Oxford University Press.Google Scholar
Restall, G. (2000). An Introduction to Substructural Logic. New York: Routledge.Google Scholar
Ripley, D. (2015). Naive set theory and nontransitive logic. The Review of Symbolic Logic 8(3), 553–571.Google Scholar
Routley, R. , and Meyer, R. K. (1973). The semantics of entailment. In Leblanc, H. (ed.), Truth, Syntax, and Modality: Proceedings of the Temple University Conference on Alternative Semantics, pp. 199–243. North Holland.Google Scholar
Routley, R. , and Routley, V. (1972). The semantics of first degree entailment. Noûs 6(4), 335–359.Google Scholar
Rumfitt, I. (2015). The Boundary Stones of Thought: An Essay in the Philosophy of Logic. Oxford: Oxford University Press.Google Scholar
Russell, B. (1908). Mathematical logic as based on a theory of types. American Journal of Mathematics 30, 222–262.Google Scholar
Russell, B. (1973). Essays in Analysis. London: George Allen and Unwin Ltd.Google Scholar
Shapiro, S. (1991). Foundations without Foundationalism. Oxford: Clarendon Press.Google Scholar
Shapiro, S. (1993). Modality and ontology. Mind 102, 455–481.Google Scholar
Shapiro, S. (1996). (ed.) The Limits of Logic: Higher-order Logic and the Löwenheim-Skolem Theorem. United Kingdom: Routledge.Google Scholar
Shapiro, S. (1998). Logical consequence: Models and modality. In Schirn, M. (ed.), The Philosophy of Mathematics Today, pp. 131–156. Oxford: Clarendon Press.Google Scholar
Shapiro, S. , and Kouri Kissel, T. (2020). Classical logic. In Zalta, E. N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2020 ed.). Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/entries/logic-classical/Google Scholar
Stebbing, L. S. (1939). Thinking to Some Purpose. London: Penguin Books.Google Scholar
Tarski, A. (2002). On the concept of following logically. History and Philosophy of Logic 23, 155–196.Google Scholar
Tedder, A. (2021). Information flow in logics in the vicinity of BB. The Australasian Journal of Logic 18, 1–24.Google Scholar
Tennant, N. (1997). The Taming of the True. New York: Oxford University Press.Google Scholar
Tennant, N. (2005). Relevance in reasoning. In Shapiro, S. (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, pp. 696–726. Oxford: Oxford University Press.Google Scholar
Tennant, N. (2015). A new unified account of truth and paradox. Synthese 124, 571–605.Google Scholar
Tennant, N. (2017). Core Logic. Oxford: Oxford University Press.Google Scholar
Urquhart, A. (1972). Semantics for relevant logics. The Journal of Symbolic Logic 37(1), 159–169.Google Scholar
Van Heijenoort, J. (1967). From Frege to Gödel. Cambridge, MA: Harvard University Press.Google Scholar
Wang, H. (1974). From Mathematics to Philosophy. London: Routledge and Kegan Paul.Google Scholar
Weber, Z. (2011). A paraconsistent model of vagueness. Mind 119(476), 1025–1045.Google Scholar
Whitehead, A. N. , and Russell, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press.Google Scholar
Woods, C. (2021). An introduction to reasoning. https://sites.google.com/site/anintroductiontoreasoning/ Accessed: March 2nd 2021.Google Scholar

Accessibility standard: Unknown

Why this information is here

This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.

Accessibility Information

Accessibility compliance for the HTML of this element is currently unknown and may be updated in the future.

Save element to Kindle

To save this element to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Classical First-Order Logic
Available formats
×

Save element to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Classical First-Order Logic
Available formats
×

Save element to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Classical First-Order Logic
Available formats
×