Corresponding conditional (logic)
Encyclopedia
In logic
, the corresponding conditional of an argument
(or derivation) is a material conditional
whose antecedent
is the conjunction
of the argument's (or derivation's) premise
s and whose consequent
is the argument's conclusion. An argument is valid
if and only if
its corresponding conditional is a logical truth
. It follows that an argument is valid if and only if the negation of its corresponding conditional is a contradiction
. The construction of a corresponding conditional therefore provides a useful technique for determining the validity of argument
This argument is of the form:
The corresponding conditional C is:
and the argument A is valid just in case the corresponding conditional C is a necessary truth.
If C is a necessary 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
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:
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.
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 difficulty trying to derive a contradiction proceed as follows. From the negation of the corresponding conditional derive a theorem in conjunctive normal form
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.
By Leigh S. Cauman
Published by Walter de Gruyter, 1998
ISBN 3110157667, 9783110157666, Page 19
The Cambridge Companion to Mill
By John Skorupski
Published by Cambridge University Press, 1998
ISBN 0521422116, 9780521422116, PAge 40
The Languages of Logic: An Introduction to Formal Logic
By Samuel D. Guttenplan
Published by Blackwell Publishing, 1997
ISBN 155786988X, 9781557869883, page 90.
The Value of Knowledge and the Pursuit of Understanding
By Jonathan L. Kvanvig
Published by Cambridge University Press, 2003
ISBN 0521827132, 9780521827133, page 175
Logic
By Paul Tomassi
Published by Routledge, 1999
ISBN 0415166969, 9780415166966, page 153
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, the corresponding conditional of an argument
Argument
In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...
(or derivation) is a material conditional
Material conditional
The material conditional, also known as material implication, is a binary truth function, such that the compound sentence p→q is logically equivalent to the negative compound: not . A material conditional compound itself is often simply called a conditional...
whose antecedent
Antecedent (logic)
An antecedent is the first half of a hypothetical proposition.Examples:* If P, then Q.This is a nonlogical formulation of a hypothetical proposition...
is the conjunction
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....
of the argument's (or derivation's) premise
Premise
Premise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument...
s and whose consequent
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".Examples:* If P, then Q.Q is the consequent of this hypothetical proposition....
is the argument's conclusion. An argument is valid
Valid
Valid is a Brazilian engraving company headquartered in Rio de Janeiro that provides security printing services to financial institutions, telecommunication companies, state governments, and public agencies in Brazil, Argentina, and Spain....
if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
its corresponding conditional is a logical truth
Logical truth
Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement.Logical...
. It follows that an argument is valid if and only if the negation of its corresponding conditional is a contradiction
Contradiction
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
. The construction of a corresponding conditional therefore provides a useful technique for determining the validity of 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 the propositional calculusPropositional calculusIn mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
):
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 necessary truth.
If C is a necessary 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
Truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their...
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
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and 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 difficulty trying to derive a contradiction proceed as follows. From the negation of the corresponding conditional derive a theorem in conjunctive normal form
Conjunctive normal form
In Boolean logic, a formula is in conjunctive normal form if it is a conjunction of clauses, where a clause is a disjunction of literals.As a normal form, it is useful in automated theorem proving...
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
- http://books.google.co.uk/books?id=TQlvJJgUiVoC&pg=PA19&lpg=PA19&dq=Corresponding+conditional&source=web&ots=V0GmWFcKsg&sig=JXjvWnQJpOKjU_-Nr-e3vE6s8PE&hl=en&sa=X&oi=book_result&resnum=3&ct=result
- http://books.google.co.uk/books?id=BVHwg_qNxosC&pg=PA40&lpg=PA40&dq=Corresponding+conditional&source=web&ots=MHRGHboBUd&sig=ha4gxQrKdKsINVcSOWBfrpvNQ00&hl=en&sa=X&oi=book_result&resnum=6&ct=result
- http://www.earlham.edu/~peters/courses/log/terms2.htm
- http://www.csus.edu/indiv/n/nogalesp/SymbolicLogicGustason/SymbolicLogicOverheads/Phil60GusCh2TruthTablesSemanticMethods/TTValidityCorrespondingConditional.doc
- http://books.google.co.uk/books?id=xfOdpyj1bSIC&pg=PA90&lpg=PA90&dq=Corresponding+conditional&source=web&ots=PNBSh6fukg&sig=7BEBKbCD5Qhq9TOIBri9Oa5Zah4&hl=en&sa=X&oi=book_result&resnum=6&ct=result
- http://books.google.co.uk/books?id=OxXopc5AjQ0C&pg=PA175&lpg=PA175&dq=Corresponding+conditional&source=web&ots=FCFY5L4_HB&sig=7pkTUrJ87AtojCVRzeej5eHgqnA&hl=en&sa=X&oi=book_result&resnum=2&ct=result
- http://books.google.co.uk/books?id=tb6bxjyrFJ4C&pg=PA153&dq=Corresponding+conditional+logic
Literature
First-order Logic: An IntroductionBy Leigh S. Cauman
Published by Walter de Gruyter, 1998
ISBN 3110157667, 9783110157666, Page 19
The Cambridge Companion to Mill
By John Skorupski
Published by Cambridge University Press, 1998
ISBN 0521422116, 9780521422116, PAge 40
The Languages of Logic: An Introduction to Formal Logic
By Samuel D. Guttenplan
Published by Blackwell Publishing, 1997
ISBN 155786988X, 9781557869883, page 90.
The Value of Knowledge and the Pursuit of Understanding
By Jonathan L. Kvanvig
Published by Cambridge University Press, 2003
ISBN 0521827132, 9780521827133, page 175
Logic
By Paul Tomassi
Published by Routledge, 1999
ISBN 0415166969, 9780415166966, page 153