The paradoxes of material implication are a group of
true formulae involving
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 ...
s whose translations into natural language are intuitively false when the conditional is translated as "if ... then ...". A material conditional formula
is true unless
is true and
is false. If
natural language conditionals were understood in the same way, that would mean that the sentence "If the Nazis won World War Two, everybody would be happy" is
vacuously true
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she d ...
. Given that such problematic consequences follow from a seemingly correct assumption about logic, they are called ''
paradoxes
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
''. They demonstrate a mismatch between classical logic and robust intuitions about
meaning and
reasoning.
Paradox of entailment
As the best known of the paradoxes, and most formally simple, the paradox of
entailment
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...
makes the best introduction.
In natural language, an instance of the paradox of entailment arises:
:''It is raining''
And
:''It is not raining''
Therefore
:''George Washington is made of rakes.''
This arises from the
principle of explosion
In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (, 'from falsehood, anything ollows; or ), or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a ...
, a law of
classical logic stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any
conclusion at all. This seems paradoxical because although the above is a logically valid argument, it is not
sound
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' b ...
(not all of its premises are true).
Construction
Validity is defined in classical logic as follows:
:''An
argument (consisting of
premise
A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agre ...
s and a conclusion) 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 b ...
there is no possible situation in which all the premises are true and the conclusion is false. ''
For example a valid argument might run:
:''If it is raining, water exists'' (1st premise)
:''It is raining'' (2nd premise)
:''Water exists'' (Conclusion)
In this example there is no possible situation in which the premises are true while the conclusion is false. Since there is no
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
, the argument is valid.
But one could construct an argument in which the premises are
inconsistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
. This would satisfy the test for a valid argument since there would be ''no possible situation in which all the premises are true'' and therefore ''no possible situation in which all the premises are true and the conclusion is false''.
For example an argument with inconsistent premises might run:
:''It is definitely raining'' (1st premise; true)
:''It is not raining'' (2nd premise; false)
:''George Washington is made of rakes'' (Conclusion)
As there is no possible situation where both premises could be true, then there is certainly no possible situation in which the premises could be true while the conclusion was false. So the argument is valid whatever the conclusion is; inconsistent premises imply all conclusions.
Simplification
The classical paradox formulae are closely tied to
conjunction elimination
In propositional logic, conjunction elimination (also called ''and'' elimination, ∧ elimination, or simplification)Hurley is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction ' ...
,
*
which can be derived from the paradox formulae, for example from (1) by
importation
An import is the receiving country in an export from the sending country. Importation and exportation are the defining financial transactions of international trade.
In international trade, the importation and exportation of goods are limited ...
.
In addition, there are serious problems with trying to use material implication as representing the English "if ... then ...". For example, the following are valid inferences:
#
#
but mapping these back to English sentences using "if" gives paradoxes.
The first might be read "If John is in London then he is in England, and if he is in Paris then he is in France. Therefore, it is true that either (a) if John is in London then he is in France, or (b) if he is in Paris then he is in England." Using material implication, if John is ''not'' in London then (a) is true; whereas if he ''is'' in London then, because he is not in Paris, (b) is true. Either way, the conclusion that at least one of (a) or (b) is true is valid.
But this does not match how "if ... then ..." is used in natural language: the most likely scenario in which one would say "If John is in London then he is in England" is if one ''does not know'' where John is, but nonetheless knows that ''if'' he is in London, he is in England. Under this interpretation, both premises are true, but both clauses of the conclusion are false.
The second example can be read "If both switch A and switch B are closed, then the light is on. Therefore, it is either true that if switch A is closed, the light is on, or that if switch B is closed, the light is on." Here, the most likely natural-language interpretation of the "if ... then ..." statements would be "''whenever'' switch A is closed, the light is on," and "''whenever'' switch B is closed, the light is on." Again, under this interpretation both clauses of the conclusion may be false (for instance in a series circuit, with a light that comes on only when ''both'' switches are closed).
See also
*
Connexive logic
Connexive logic names one class of alternative, or 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 t ...
s were designed to exclude the paradoxes of material implication
*
Correlation does not imply causation
*
Counterfactuals
*
False dilemma
*
Import-Export
*
List of paradoxes
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their ...
*
Modus ponens
*
The Moon is made of green cheese
"The Moon is made of green cheese" is a statement referring to a fanciful belief that the Moon is composed of cheese. In its original formulation as a proverb and metaphor for credulity with roots in fable, this refers to the perception of a s ...
*
Relevance logic Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, but ...
arose out of attempts to avoid these paradoxes
*
Vacuous truth
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she d ...
References
*
Bennett, J. ''A Philosophical Guide to Conditionals''. Oxford: Clarendon Press. 2003.
*''Conditionals'', ed.
Frank Jackson. Oxford: Oxford University Press. 1991.
*
Etchemendy, J. ''The Concept of Logical Consequence''. Cambridge: Harvard University Press. 1990.
*
*
Sanford, D. ''If P, Then Q: Conditionals and the Foundations of Reasoning''. New York: Routledge. 1989.
*
Priest, G. ''An Introduction to Non-Classical Logic'', Cambridge University Press. 2001.
{{Paradoxes
Paradoxes
Semantics
Logical consequence