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A timeline of philosophy: from Confucius, teaching a practical philosophy, to Karl Popper publishing The Open Society and its Enemies
‘David Hume’ from the Encyclopedia of Aesthetics
Logical symbols from The Oxford Dictionary of Philosophy
In your opinion, which is the most fascinating entry in your Dictionary and why?
I believe that the notion of constructible falsity—first described by the mathematician and logician David Nelson—is one of the most intriguing notions in philosophical logic. Nelson introduced this notion in the 1940s as a criticism of how proponents of intuitionistic logic view the distinction between truth and falsity.
The school of intuitionistic logic really springs from L. E. J. Brouwer’s philosophy of mathematics, in which mathematical objects—integers, sets, vectors—are treated as constructions carried out in the minds of mathematicians, rather than as Platonic objects with independent existence. The mind-dependence of mathematical objects, Brouwer argued, requires that some classical assumptions concerning logic must be set aside when reasoning about mathematics. In particular, intuitionistically, the assertion that a sentence A is true is interpreted as the possession of a proof of A. Consequently, intuitionistic logic enjoys the disjunction property that whenever a sentence of the form ‘A or B’ is provable, one of the disjuncts—either ‘A’ or ‘B’—must be explicitly provable as well. In other words, when a disjunction is true, one must be able to establish which disjunct is ‘responsible’ for its truth.
Nelson observed, however, that in intuitionistic logic a conjunction ‘A and B’ may be regarded as false—that is, disprovable—without determining which conjunct contributes to its falsity. For example, the contradiction ‘A and not-A’ is considered to be disprovable even when one possesses neither a disproof of ‘A’ nor a disproof of ‘not-A.’ This fact reveals that there is an asymmetry between the intuitionistic notions of truth and falsity. Nelson offered this property of constructible falsity in order to resolve this asymmetry; I find that the restoration of symmetry to the intuitionistic theory leads to a very beautiful and elegant theory of inference. I think others are starting to agree. Nearly seventy years on, the central elements of Nelson’s work are beginning to be taken more seriously, with connections and applications to a number of fields being identified in the literature.
What would you say is the most unusual/obscure term in your subject area?
Judging by the contents of texts on philosophical logic, the term ‘connexive logic’ is probably the most obscure term in the dictionary, a distinction that I think is entirely undeserved. Connexive logics compose a family of theories of deduction that make related assumptions about inference—not only concerning what can be correctly inferred but also what cannot be inferred. Most important among these is the so-called Aristotle’s thesis, which insists that no proposition entails its own negation. In other words, Aristotle’s thesis asserts that it cannot be that the mere supposition that some sentence is true provides sufficient grounds to reject that sentence.
Now, these sorts of systems radically disagree with classical logic in important ways. For example, classical logic assumes the principle of explosion, according to which the hypothesis that a contradiction is true permits one to infer that everything is true, including the negation of that very contradiction. So connexive logics require the abandonment of the principle of explosion. The principle of explosion is thought by many to be a very plausible thesis, and assumption of other connexive principles—such as Boethius’ thesis—frequently require the rejection of further plausible theses of classical logic. Consequently, the costs of these theses have been traditionally judged as too steep.
Interestingly, though, these connexive assumptions are effervescent in the history of logic. They appear not only in antiquity in the works of Aristotle and Boethius, but in the modern era as well, playing implicit roles in the development of modal logic and conditional logic. Experimental research has shown that these principles are in fact well-entrenched in the beliefs of the layperson, leading to an intriguing question of why views about inference held by so many people have such severe consequences.
What do you think is the most commonly held misconception in your subject area?
The blog post that Graham and I recently wrote to mark the publication of the dictionary describes the orthodox view of logic as the position that the logical theses defended by Gottlob Frege and Bertrand Russell provide an absolutely correct canon of inference. Consequently, theories of reasoning that deviate from this line are often regarded as undeserving of study or consideration. That alternative theories of deduction are unworthy of the time of students is an unfortunate misconception, I think. This is a view that dominates the study of logic in most mathematics, philosophy, and computer science departments, and this dominance entails that many students of logic end up missing much of the variety of themes that have been raised during the development of modern logic.
The breadth of points that have been disputed in the history of logic is exhilarating, to be honest, and reflects the subtle and difficult problems that emerge while investigating how to capture correct reasoning. I have already mentioned several of these themes—for example, the question of whether some propositions entail their own falsehood. Many more appear at various episodes in logic’s history—for example, on its face, the assumption that valid inferences require some kind of relevance between hypotheses and conclusions is plausible. The family of theories of inference known as relevant logic captures this intuition by imposing certain requirements on formal inferences. Classical logic disagrees with the need for relevance—from a contradictory assumption ‘A and not-A,’ one may infer anything at all—meaning that relevant logics are considered ‘deviant.’
To merely ignore that such intuitions exist—or to dismiss them merely as instances of the naive mistakes of the layperson—deprives students of the opportunity to really explore the philosophical foundations of logic. Now, I consider myself to be a ‘classical logician’ in the sense that I lean towards metaphysical assumptions suggesting the correctness of classical logic—in other words, I think that Russell pretty much got it right. But this alone doesn’t discount the importance of his competitors. After all, philosophers still find it important to read and teach Descartes despite their general rejection of his notion of substance dualism. Likewise, even if one assumes that we have arrived at a correct theory of reasoning, studying how we got to this point remains a worthwhile and important endeavor.
The enduring evolution of logic
Thomas Ferguson and Graham Priest explore the development of Western logic.
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