logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

, the corresponding conditional of an

argument An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...

(or derivation) is a

material conditional The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol \to is interpreted as material implication, a formula P \to Q is true unless P is true and Q is false. M ...

whose antecedent is the conjunction of the argument's (or derivation's)

premise A premise or premiss is a proposition—a true or false declarative statement—used in an argument to prove the truth of another proposition called the conclusion. Arguments consist of a set of premises and a conclusion. An argument is meaningf ...

s and whose

consequent 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 t ...

is the argument's conclusion. An argument is valid

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

its corresponding conditional is a

logical truth Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement whic ...

. It follows that an argument is valid if and only if the negation of its corresponding conditional is a

contradiction In traditional logic, a contradiction involves a proposition conflicting either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...

. Therefore, the construction of a corresponding conditional provides a useful technique for determining the validity of an argument.

Example

Consider the argument A:

Either it is hot or it is cold
It is not hot
Therefore it is cold

This argument is of the form:

Either P or Q
Not P
Therefore Q

or (using standard symbols of
propositional calculus The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
):

P Q
P
____________
Q

The corresponding conditional C is:

IF ((P or Q) and not P) THEN Q

or (using standard symbols):

((P Q) P) Q

and the argument A is valid just in case the corresponding conditional C is a logical truth. If C is a logical truth then C entails Falsity (The False). Thus, any argument is valid if and only if the denial of its corresponding conditional leads to a contradiction. If we construct a

truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...

for C we will find that it comes out T (true) on every row (and of course if we construct a truth table for the negation of C it will come out F (false) in every row. These results confirm the validity of the argument A Some arguments need

first-order predicate logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...

to reveal their forms and they cannot be tested properly by truth tables forms. Consider the argument A1:

Some mortals are not Greeks
Some Greeks are not men
Not every man is a logician
Therefore Some mortals are not logicians

To test this argument for validity, construct the corresponding conditional C1 (you will need first-order predicate logic), negate it, and see if you can derive a contradiction from it. If you succeed, then the argument is valid.

Application

Instead of attempting to derive the conclusion from the premises proceed as follows. To test the validity of an argument (a) translate, as necessary, each premise and the conclusion into sentential or predicate logic sentences (b) construct from these the negation of the corresponding conditional (c) see if from it a contradiction can be derived (or if feasible construct a truth table for it and see if it comes out false on every row.) Alternatively construct a truth tree and see if every branch is closed. Success proves the validity of the original argument. In case of the difficulty in trying to derive a contradiction, one should proceed as follows. From the negation of the corresponding conditional derive a theorem in

conjunctive normal form In Boolean algebra, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs. In au ...

in the methodical fashions described in text books. If, and only if, the original argument was valid will the theorem in conjunctive normal form be a contradiction, and if it is, then that it is will be apparent.

External links

Corresponding conditional from the Free On-line Dictionary of Computing
* https://books.google.com/books?id=TQlvJJgUiVoC&pg=PA19 * https://books.google.com/books?id=BVHwg_qNxosC&pg=PA40 * http://www.earlham.edu/~peters/courses/log/terms2.htm * http://www.csus.edu/indiv/n/nogalesp/SymbolicLogicGustason/SymbolicLogicOverheads/Phil60GusCh2TruthTablesSemanticMethods/TTValidityCorrespondingConditional.doc * https://books.google.com/books?id=xfOdpyj1bSIC&pg=PA90 * https://books.google.com/books?id=OxXopc5AjQ0C&pg=PA175 * https://books.google.com/books?id=tb6bxjyrFJ4C&pg=PA153 Conditionals Statements

Example

Application

Further reading

External links