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Connexive logic is a class of non-classical logics designed to exclude the paradoxes of material implication. The characteristic that separates connexive logic from other non-classical logics is its acceptance of Aristotle's thesis, i.e. the formula,
as a logical truth. Aristotle's thesis asserts that no statement follows from its own denial. Stronger connexive logics also accept Boethius' thesis,
which states that if a statement implies one thing, it does not imply its opposite.
Relevance logic is another logical theory that tries to avoid the paradoxes of material implication.
Connexive logic is arguably one of the oldest approaches to logic. Aristotle's thesis is named after Aristotle because he uses this principle in a passage in the Prior Analytics .
It is impossible that the same thing should be necessitated by the being and the not-being of the same thing. I mean, for example, that it is impossible that B should necessarily be great if A is white, and that B should necessarily be great if A is not white. For if B is not great A cannot be white. But if, when A is not white, it is necessary that B should be great, it necessarily results that if B is not great, B itself is great. But this is impossible. An. Pr. ii 4.57b3.
The sense of this passage is to perform a reductio ad absurdum proof on the claim that two formulas, (A → B) and (~A → B), can be true simultaneously. The proof is:
1 | (A → B) | hypothesis (reductio) |
2 | (~A → B) | hypothesis (reductio) |
3 | ~B | hypothesis |
4 | ~A | 1, 3, modus tollens |
5 | B | 2, 4, modus ponens |
6 | (~B → B) | 3, 5, conditional proof |
Aristotle then declares the last line to be impossible, completing the reductio. But if it is impossible, its denial, ~(~B → B), is a logical truth.
Aristotelian syllogisms (as opposed to Boolean syllogisms) appear to be based on connexive principles. For example, the contrariety of A and E statements, "All S are P," and "No S are P," follows by a reductio ad absurdum argument similar to the one given by Aristotle.
Later logicians, notably Chrysippus, are also thought to have endorsed connexive principles. By 100 BCE logicians had divided into four or five distinct schools concerning the correct understanding of conditional ("if...then...") statements. Sextus Empiricus described one school as follows:
And those who introduce the notion of connexion say that a conditional is sound when the contradictory of its consequent is incompatible with its antecedent.
The term "connexivism" is derived from this passage (as translated by Kneale and Kneale).
It is believed that Sextus was here describing the school of Chrysippus. That this school accepted Aristotle's thesis seems clear because the definition of the conditional,
requires that Aristotle's thesis be a logical truth, provided we assume that every statement is compatible with itself, which seems fairly fundamental to the concept of compatibility.
The medieval philosopher Boethius also accepted connexive principles. In De Syllogismo Hypothetico, he argues that from, "If A, then if B then C," and "If B then not-C," we may infer "not-A," by modus tollens. However, this follows only if the two statements, "If B then C," and "If B then not-C," are considered incompatible.
Since Aristotelian logic was the standard logic studied until the 19th century, it could reasonably be claimed that connexive logic was the accepted school of thought among logicians for most of Western history. (Of course, logicians were not necessarily aware of belonging to the connexivist school.) However, in the 19th century Boolean syllogisms, and a propositional logic based on truth functions, became the standard. Since then, relatively few logicians have subscribed to connexivism. These few include Everett J. Nelson and P. F. Strawson.
The objection that is made to the truth-functional definition of conditionals is that there is no requirement that the consequent actually follow from the antecedent. So long as the antecedent is false or the consequent true, the conditional is considered to be true whether there is any relation between the antecedent and the consequent or not. Hence, as the philosopher Charles Sanders Peirce once remarked, you can cut up a newspaper, sentence by sentence, put all the sentences in a hat, and draw any two at random. It is guaranteed that either the first sentence will imply the second, or vice versa. But when we use the words "if" and "then" we generally mean to assert that there is some relation between the antecedent and the consequent. What is the nature of that relationship? Relevance (or relevant) logicians take the view that, in addition to saying that the consequent cannot be false while the antecedent is true, the antecedent must be "relevant" to the consequent. At least initially, this means that there must be at least some terms (or variables) that appear in both the antecedent and the consequent. Connexivists generally claim instead that there must be some "real connection" between the antecedent and the consequent, such as might be the result of real class inclusion relations. For example, the class relations, "All men are mortal," would provide a real connection that would warrant the conditional, "If Socrates is a man, then Socrates is mortal." However, more remote connections, for example "If she apologized to him, then he lied to me." (suggested by Bennett) still defy connexivist analysis.
In classical logic, disjunctive syllogism is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.
In propositional logic, modus ponens, also known as modus ponendo ponens, implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q.P is true. Therefore, Q must also be true."
In propositional logic, modus tollens (MT), also known as modus tollendo tollens and denying the consequent, is a deductive argument form and a rule of inference. Modus tollens is a mixed hypothetical syllogism that takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.
Chrysippus of Soli was a Greek Stoic philosopher. He was a native of Soli, Cilicia, but moved to Athens as a young man, where he became a pupil of the Stoic philosopher Cleanthes. When Cleanthes died, around 230 BC, Chrysippus became the third head of the Stoic school. A prolific writer, Chrysippus expanded the fundamental doctrines of Cleanthes' mentor Zeno of Citium, the founder and first head of the school, which earned him the title of the Second Founder of Stoicism.
A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.
The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India, China, and Greece. Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic.
Deductive reasoning is the process of drawing valid inferences. An inference is valid if its conclusion follows logically from its premises, meaning that it is impossible for the premises to be true and the conclusion to be false.
In classical logic, a hypothetical syllogism is a valid argument form, a deductive syllogism with a conditional statement for one or both of its premises. Ancient references point to the works of Theophrastus and Eudemus for the first investigation of this kind of syllogisms.
In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or biimplication or bientailment, is the logical connective used to conjoin two statements and to form the statement " if and only if ", where is known as the antecedent, and the consequent.
In classical logic, intuitionistic logic and similar logical systems, the principle of explosion, or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a contradiction. That is, from a contradiction, any proposition can be inferred; this is known as deductive explosion.
The material conditional is an operation commonly used in logic. When the conditional symbol is interpreted as material implication, a formula is true unless is true and is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum.
In logic, a strict conditional is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal logic. For any two propositions p and q, the formula p → q says that p materially implies q while says that p strictly implies q. Strict conditionals are the result of Clarence Irving Lewis's attempt to find a conditional for logic that can adequately express indicative conditionals in natural language. They have also been used in studying Molinist theology.
The barbershop paradox was proposed by Lewis Carroll in a three-page essay titled "A Logical Paradox", which appeared in the July 1894 issue of Mind. The name comes from the "ornamental" short story that Carroll uses in the article to illustrate the paradox. It existed previously in several alternative forms in his writing and correspondence, not always involving a barbershop. Carroll described it as illustrating "a very real difficulty in the Theory of Hypotheticals". From the viewpoint of modern logic, it is seen not so much as a paradox than as a simple logical error. It is of interest now mainly as an episode in the development of algebraic logical methods when these were not so widely understood, although the problem continues to be discussed in relation to theories of implication and modal logic.
In logic, the logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called well-formed formulas.
Stoic logic is the system of propositional logic developed by the Stoic philosophers in ancient Greece.