Paradoxes of material implication
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The paradoxes of material implication are a group of true formulae involving material conditionals 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
Paradox of entailment
As the best known of the paradoxes, and most formally simple, the paradox of
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
Construction
Validity is defined in classical logic as follows:
- An argument (consisting of premises and a conclusion) is valid if and only ifthere 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, the argument is valid.
But one could construct an argument in which the premises are
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,
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 logics were designed to exclude the paradoxes of material implication
- Correlation does not imply causation
- Counterfactuals
- False dilemma
- Import-Export
- List of paradoxes
- Modus ponens
- The Moon is made of green cheese
- Relevance logic arose out of attempts to avoid these paradoxes
- Vacuous truth
References
- ISBN 978-3-11-018523-2.
- 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.
- "Strict implication calculus", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- 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.