Modal logic is not truth conditional, and so it has often been proposed as a non-classical logic. However, modal logic is normally formalized with the principle of the excluded middle, and its relational semantics is bivalent, so this inclusion is disputable. What is the epistemological status of the laws of logic? What sort of argument is appropriate for criticizing purported principles of logic? In an influential paper entitled "Is Logic Empirical?" Hilary Putnam, building on a suggestion of W. V.

Paradoxes of material implication

paradoxes of the material conditionalparadoxical propositions
A Philosophical Guide to Conditionals. Oxford: Clarendon Press. 2003. Conditionals, ed. Frank Jackson. Oxford: Oxford University Press. 1991. Etchemendy, J. The Concept of Logical Consequence. Cambridge: Harvard University Press. 1990. Sanford, D. If P, Then Q: Conditionals and the Foundations of Reasoning. New York: Routledge. 1989. Priest, G. An Introduction to Non-Classical Logic, Cambridge University Press. 2001.

Outline of logic

Logicoutlinelist of topics in logic
Classical logic Non-classical logic Modal logic Mathematical logic – History of logic * Association for Symbolic Logic Analytic-synthetic distinction. Antinomy. A priori and a posteriori. Definition. Description. Entailment. Identity (philosophy). Inference. Logical form. Logical implication. Logical truth. Logical consequence. Name. Necessity. Material conditional. Meaning (linguistic). Meaning (non-linguistic). Paradox (list). Possible world. Presupposition. Probability. Quantification. Reason. Reasoning. Reference. Semantics. Strict conditional. Syntax (logic). Truth. Truth value. Validity. Argument. Argument map. Accuracy and precision. Ad hoc hypothesis. Ambiguity. Analysis.

Logical truth

necessarily truenecessary truthlogical necessity
Non-classical logic is the name given to formal systems which differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth. ca:Valor vertader Contradiction. False (logic). Satisfiability. Tautology (logic) (for symbolism of logical truth). Theorem. Validity.

Logical connective

logical operatorconnectivesconnective
These correspond to possible choices of binary logical connectives for classical logic. Different implementations of classical logic can choose different functionally complete subsets of connectives. One approach is to choose a minimal set, and define other connectives by some logical form, as in the example with the material conditional above. The following are the minimal functionally complete sets of operators in classical logic whose arities do not exceed 2: See more details about functional completeness in classical logic at Functional completeness in truth function.

Monotonicity of entailment

Linear logic which disallows arbitrary Idempotency of entailment. Contraction. Exchange rule. Substructural logic. No-cloning theorem.

Modal logic

Strict conditional. Two dimensionalism. This article includes material from the Free On-line Dictionary of Computing, used with permission under the GFDL. Barcan-Marcus, Ruth JSL 11 (1946) and JSL 112 (1947) and "Modalities", OUP, 1993, 1995. Beth, Evert W., 1955. " Semantic entailment and formal derivability", Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, N.R. Vol 18, no 13, 1955, pp 309–42. Reprinted in Jaakko Intikka (ed.) The Philosophy of Mathematics, Oxford University Press, 1969 (Semantic Tableaux proof methods).

Alfred Tarski

TarskiTarski, AlfredTarskian
Tarski's 1936 article "On the concept of logical consequence" argued that the conclusion of an argument will follow logically from its premises if and only if every model of the premises is a model of the conclusion. In 1937, he published a paper presenting clearly his views on the nature and purpose of the deductive method, and the role of logic in scientific studies. His high school and undergraduate teaching on logic and axiomatics culminated in a classic short text, published first in Polish, then in German translation, and finally in a 1941 English translation as Introduction to Logic and to the Methodology of Deductive Sciences.

Boolean algebra

Booleanboolean logiclogic
The antecedent is interpreted as the conjunction of its propositions, the succedent as the disjunction of its propositions, and the sequent itself as the entailment of the succedent by the antecedent. Entailment differs from implication in that whereas the latter is a binary operation that returns a value in a Boolean algebra, the former is a binary relation which either holds or does not hold. In this sense entailment is an external form of implication, meaning external to the Boolean algebra, thinking of the reader of the sequent as also being external and interpreting and comparing antecedents and succedents in some Boolean algebra.

Kurt Gödel

GödelGödel, Kurt Gödel, Kurt
He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic. Gödel was born April 28, 1906, in Brünn, Austria-Hungary (now Brno, Czech Republic) into the German family of Rudolf Gödel (1874–1929), the manager of a textile factory, and Marianne Gödel (née Handschuh, 1879–1966). Throughout his life, Gödel would remain close to his mother; their correspondence was frequent and wide-ranging. At the time of his birth the city had a German-speaking majority which included his parents. His father was Catholic and his mother was Protestant and the children were raised Protestant.

Propositional calculus

propositional logicpropositionalsentential logic
Even when the logic under study is intuitionistic, entailment is ordinarily understood classically as two-valued: either the left side entails, or is less-or-equal to, the right side, or it is not. Similar but more complex translations to and from algebraic logics are possible for natural deduction systems as described above and for the sequent calculus. The entailments of the latter can be interpreted as two-valued, but a more insightful interpretation is as a set, the elements of which can be understood as abstract proofs organized as the morphisms of a category. In this interpretation the cut rule of the sequent calculus corresponds to composition in the category.

Principle of explosion

ex falso quodlibetexplosionabsurdity constant
Paradox of entailment – a seeming paradox derived from the principle of explosion. Reductio ad absurdum – concluding that a proposition is false because it produces a contradiction. Trivialism – the belief that all statements of the form "P and not-P" are true.

Non-classical logic

non-Aristotelian logicnon-classicalnon-classical" logic
Linear logic rejects idempotency of entailment as well. Modal logic extends classical logic with non-truth-functional ("modal") operators. Paraconsistent logic (e.g., relevance logic) rejects the principle of explosion, and has a close relation to dialetheism. Quantum logic. Relevance logic, linear logic, and non-monotonic logic reject monotonicity of entailment.

Indicative conditional

conditionalif-thenBehavioral experiment (conditional reasoning)
When participants are given counterfactual conditionals, they make both the modus ponens and the modus tollens inferences (Byrne, 2005). Counterfactual conditional. Logical consequence. Material conditional. Strict conditional. Byrne, R.M.J. (2005). The Rational Imagination: How People Create Counterfactual Alternatives to Reality. Cambridge, MA: MIT Press. Edgington, Dorothy. (2006). "Conditionals". The Stanford Encyclopedia of Philosophy, Edward Zalta (ed.). http://plato.stanford.edu/entries/conditionals/. Evans, J. St. B. T., Newstead, S. and Byrne, R. M. J. (1993). Human Reasoning: The Psychology of Deduction. Hove, Psychology Press.


A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if P implies Q, then P is called the antecedent and Q is called the consequent. In some contexts, the consequent is called the apodosis.

First-order logic

predicate logicfirst-orderpredicate calculus
Hodges, Wilfrid (2001); "Classical Logic I: First Order Logic", in Goble, Lou (ed.); The Blackwell Guide to Philosophical Logic, Blackwell. Ebbinghaus, Heinz-Dieter; Flum, Jörg; and Thomas, Wolfgang (1994); Mathematical Logic, Undergraduate Texts in Mathematics, Berlin, DE/New York, NY: Springer-Verlag, Second Edition, ISBN: 978-0-387-94258-2. Stanford Encyclopedia of Philosophy: Shapiro, Stewart; " Classical Logic". Covers syntax, model theory, and metatheory for first-order logic in the natural deduction style. Magnus, P. D.; forall x: an introduction to formal logic. Covers formal semantics and proof theory for first-order logic.

Principle of bivalence

bivalenttwo-valued logicbinary logic
In classical two-valued logic both the law of excluded middle and the law of non-contradiction hold. Many modern logic programming systems replace the law of the excluded middle with the concept of negation as failure. The programmer may wish to add the law of the excluded middle by explicitly asserting it as true; however, it is not assumed a priori. The intended semantics of classical logic is bivalent, but this is not true of every semantics for classical logic.

Law of excluded middle

law of the excluded middleexcluded middletertium non datur
By non-constructive Davis means that "a proof that there actually are mathematic entities satisfying certain conditions would not have to provide a method to exhibit explicitly the entities in question." (p. 85). Such proofs presume the existence of a totality that is complete, a notion disallowed by intuitionists when extended to the infinite—for them the infinite can never be completed: "In classical mathematics there occur non-constructive or indirect existence proofs, which intuitionists do not accept. For example, to prove there exists an n such that P(n), the classical mathematician may deduce a contradiction from the assumption for all n, not P(n).


syllogisticcategorical syllogismsyllogisms
The following problems arise: For example, if it is accepted that AiB is false if there are no As and AaB entails AiB, then AiB has existential import with respect to A, and so does AaB. Further, if it is accepted that AiB entails BiA, then AiB and AaB have existential import with respect to B as well. Similarly, if AoB is false if there are no As, and AeB entails AoB, and AeB entails BeA (which in turn entails BoA) then both AeB and AoB have existential import with respect to both A and B. It follows immediately that all universal categorical statements have existential import with respect to both terms.