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Encyclopedia > Logical conditional

In propositional calculus, or logical calculus in mathematics, the logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). The operator is denoted using a right-arrow "→". The hypothesis is sometimes also called sufficient condition for the conclusion, while the conclusion may be called necessary condition for the hypothesis. Jump to: navigation, search In mathematical logic the propositional calculus or sentential calculus is a formal deduction system whose atomic formulas are propositional variables. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ... Jump to: navigation, search A hypothesis (foundation from ancient Greek hupothesis where hupo = under and thesis = placing) is a proposed explanation for a phenomenon. ... In logic, a conclusion is a proposition inferred from premises. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ...

It must be emphasized that in logic there is no single interpretation of the conditional; it represents a variety of closely-related concepts, which have more specific names and often separate symbols (such as ⇒ and ⊃). Jump to: navigation, search Logic (from Classical Greek Î»ÏŒÎ³Î¿Ï‚ (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy amongst philosophers (see below). ...

A conditional statement, or simply a conditional for short, is an "if-then" statement, written in the form: 'if P, then Q'. Here, 'P' is the antecedent (the "if" part of the statement) and 'Q' is the consequent (the "then" part). For example, in "If you give me ten dollars, then I will be your best friend," the claim "you give me ten dollars" is the antecedent of the conditional, and "I will be your best friend" is the consequent.

In traditional logic, a statement if A then B is true if and only if either A is false or B is true, or both are false or both A and B are both true. There have been attempts in areas such as modal logic to find a formal definition that is closer to the 'intuitive' meaning: in the traditional logic interpretation "If it is raining now, then I am a unicorn." is true provided it is not raining now. Jump to: navigation, search A modal logic, or (less commonly) intensional logic, is a logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, necessarily, eventually, etc. ... The gentle and pensive virgin has the power to tame the unicorn, in this fresco in Palazzo Farnese, Rome, probably by Domenichino, ca 1602 The unicorn is a legendary creature shaped like a horse, but slender and with a single â€” usually spiral â€” horn growing out of its forehead. ...

Conditional Statement: Is an “if… then…” statement; for example: If all girls turn into women, then all boys turn into men.

Material conditional

The truth value of expressions involving the logical conditional "if-then" is often (but by no means always) defined by the following truth table: Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

p q pq
F F T
F T T
T F F
T T T

Note that this is logically equivalent to the statement $neg p vee q$. This particular conditional is called material implication or the material conditional, and is properly represented by the symbol ⊃ rather than the less specific symbol $rightarrow$.

The truth value of expressions involving the logical conditional "(then)-if" is often (but by no means always) defined by the following truth table: Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

p q pq
F F T
F T F
T F T
T T T

Note that this is logically equivalent to the statement $p vee neg q$. This particular conditional is properly represented by the symbol ⊂ rather than the less specific symbol $leftarrow$.

Discrepancies between the material conditional and everyday if-then reasoning

The material conditional does not always function in accordance with everyday if-then reasoning. Therefore there are drawbacks with using the material conditional to represent if-then statements.

One problem is that the material conditional allows implications to be true even when the antecedent and the consequent have no logical connection. For example, it's commonly accepted that the sun is made of gas, on one hand, and that 3 is a prime number, on the other. The standard definition of implication allows us to conclude that: since the sun is made of gas, 3 is a prime number. This is arguably synonymous to the following: the sun's being made of gas makes 3 be a prime number. Many people intuitively think that this is false, because the sun and the number three simply have nothing to do with one another. Logicians have tried to address this concern by developing alternative logics, e.g. relevant logic. Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. ...

For a related problem, see vacuous truth. Informally, a logical statement is vacuously true if it is true but doesnt say anything; examples are statements of the form everything with property A also has property B, where there is nothing with property A. It is tempting to dismiss this concept as vacuous or silly. ...

Another issue is that the material conditional is not designed to deal with counterfactuals and other cases that people often find in if-then reasoning. This has inspired people to develop modal logic. A counterfactual conditional (sometimes called a subjunctive conditional) is a logical conditional statement whose antecedent is (ordinarily) taken to be contrary to fact by those who utter it. ... Jump to: navigation, search A modal logic, or (less commonly) intensional logic, is a logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, necessarily, eventually, etc. ...

A further problem is that the material conditional is such that P AND ¬P → Q, regardless of what Q is taken to mean. That is, a contradiction implies that absolutely everything is true. Logicians concerned with this have tried to develop paraconsistent logics. A paraconsistent logic is a logic which attempts to deal with contradictions. ...

Connection with other concepts

The logical conditional, and particularly the material conditional, is closely related to inclusion (for sets), subsumption (for concepts), or implication (for propositions). It also has formal properties analogous to those of the mathematical relation less than or more exactly $leq$, especially the relation of not being symmetrical. Jump to: navigation, search A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Symbolic Logic - See minor premise Object-Oriented Programming - See Liskov Substitution Principle ... A concept is an abstract, universal idea, notion, or entity that serves to designate a category or class of entities, events, or relations. ... In logic, material implication is a binary operator. ... Proposition is a term used in logic to describe the content of assertions, content which may be taken as being true or false, and which are a non-linguistic abstraction from the linguistic sentence that constitutes an assertion. ... For the socioeconomic meaning, see social inequality. ...

In the conceptual interpretation, when a and b denote concepts, the relation $a in b$ signifies that the concept a is subsumed under the concept b; that is, it is a species with respect to the genus b. From the extensive point of view, it denotes that the class of a's is contained in the class of b's or makes a part of it; or, more concisely, that "All a's are b's". From the comprehensive point of view it means that the concept b is contained in the concept a or makes a part of it, so that consequently the character a implies or involves the character b. Example: "All men are mortal"; "Man implies mortal"; "Who says man says mortal"; or, simply, "Man, therefore mortal".

In the propositional interpretation, when a and b denote propositions, the relation $a Rightarrow b$ signifies that the proposition a implies or involves the proposition b, which is often expressed by the hypothetical judgement, "If a is true, b is true"; or by "a implies b"; or more simply by "a, therefore b". We see that in both interpretations the relation may be translated approximately by "therefore".

Remark. -- Such a relation is a proposition, whatever may be the interpretation of the terms a and b.

Consequently, whenever a $Rightarrow$ relation has two like relations (or even only one) for its members, it can receive only the propositional interpretation, that is to say, it can only denote an implication.

A relation whose members are simple terms (letters) is called a primary proposition; a relation whose members are primary propositions is called a secondary proposition, and so on.

From this it may be seen at once that the propositional interpretation is more homogeneous than the conceptual, since it alone makes it possible to give the same meaning to the copula in both primary and secondary propositions.

In computer science and in computer programming, statements in pseudocode or in a program are normally obeyed (or executed) one after the other in the order in which they are written (sequential flow of control). ... A counterfactual conditional (sometimes called a subjunctive conditional) is a logical conditional statement whose antecedent is (ordinarily) taken to be contrary to fact by those who utter it. ... In logic, a strict conditional is a material conditional that is acted upon by the necessity operator from modal logic. ...

Results from FactBites:

 logical implication: Information from Answers.com (1461 words) In mathematics and mathematical logic, the concept of logical implication encompasses a specific logical function, a specific logical relation, and the various symbols that are used to denote the function or the relation. Assuming that the conditional is true, then the truth of the antecedent is a sufficient condition for the truth of the consequent, while the truth of the consequent is a necessary condition for the truth of the antecedent. The material conditional and logical implication are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if the first operand is true and the second operand is false.
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