logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...

, the corresponding conditional of an

argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...

(or derivation) is a

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

whose antecedent is the

conjunction Conjunction may refer to: * Conjunction (grammar), a part of speech * Logical conjunction, a mathematical operator ** Conjunction introduction, a rule of inference of propositional logic * Conjunction (astronomy), in which two astronomical bodies ...

of the argument's (or derivation's) premises 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 ...

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" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...

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. 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 Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
):

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 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 logic, 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. As a cano ...

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