Paradoxes of material implication
Encyclopedia
The paradox
es of material implication are a group of formula
s which are truths of classical logic
, but which are intuitively problematic. One of these paradoxes is the paradox of entailment.
The root of the paradoxes lies in a mismatch between the interpretation of the validity of logical implication
in natural language, and its formal interpretation
in classical logic, dating back to George Boole
's algebraic logic. In classical logic, implication
describes conditional if-then statements using a truth-functional interpretation, ie "p implies q" is defined to be "it is not the case that p is true and q false". Also, "p implies q" is equivalent to "p is false or q is true". For example, "if it is raining, then I will bring an umbrella", is equivalent to "it is not raining, or I will bring an umbrella, or both". This truth-functional interpretation of implication is called material implication or material conditional.
The paradoxes are logical statements which are true but whose truth is intuitively surprising to people who are not familiar with them. If the terms 'p', 'q' and 'r' stand for arbitrary proposition
s then the main paradoxes are given formally as follows:
The paradoxes of material implication arise because of the truth-functional definition of material implication, which is said to be true merely because the antecedent
is false or the consequent
is true. By this criterion, "If the moon is made of green cheese, then the world is coming to an end," is true merely because the moon isn't made of green cheese. By extension, any contradiction implies anything whatsoever, since a contradiction is never true. (All paraconsistent logic
s must, by definition, reject (1) as false.) Also, any tautology
is implied by anything whatsoever, since a tautology is always true.
To sum up, although it is deceptively similar to what we mean by "logically follows" in ordinary usage, material implication does not capture the meaning of "if... then".
makes the best introduction.
In natural language, an instance of the paradox of entailment arises:
And
Therefore
This arises from the principle of explosion
, 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, as it suggests that the above is a valid argument.
is defined in classical logic as follows:
For example an argument might run:
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 inconsistent
. 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:
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.
comes from the fact that the definition of validity in classical logic
does not always agree with the use of the term in ordinary language. In everyday use validity suggests that the premises are consistent. In classical logic, the additional notion of soundness is introduced. A sound argument is a valid argument with all true premises. Hence a valid argument with an inconsistent set of premises can never be sound. A suggested improvement to the notion of logical validity to eliminate this paradox is relevant logic
.
the principle of Simplification, which can be derived from the paradox formulas rather easily (e.g. from (1) by Importation).
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 either true that if John is in London then he is in France, or that if he is in Paris then he is in England." Either John is in London or John is not in London. If John is in London, then John is in England. Thus the proposition "if John is in Paris, then John is in England" holds because we have prior knowledge that the conclusion is true. If John is not in London, then the proposition "if John is in London, then John is in France" is true because we have prior knowledge that the premise is false.
The second 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 if switch B is closed, the light is on." If the two switches are in series, then the premise is true but the conclusion is false. Thus, using classical logic and taking material implication to mean if-then is an unsafe method of reasoning which can give erroneous results.
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
es of material implication are a group of formula
Formula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....
s which are truths of classical logic
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
, but which are intuitively problematic. One of these paradoxes is the paradox of entailment.
The root of the paradoxes lies in a mismatch between the interpretation of the validity of logical implication
Implication
Implication may refer to:In logic:* Logical implication, entailment, or consequence, a relation between statements* Material implication, or conditional implication, a binary truth functionIn linguistics, specifically in pragmatics:...
in natural language, and its formal interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
in classical logic, dating back to George Boole
George Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...
's algebraic logic. In classical logic, implication
Implication
Implication may refer to:In logic:* Logical implication, entailment, or consequence, a relation between statements* Material implication, or conditional implication, a binary truth functionIn linguistics, specifically in pragmatics:...
describes conditional if-then statements using a truth-functional interpretation, ie "p implies q" is defined to be "it is not the case that p is true and q false". Also, "p implies q" is equivalent to "p is false or q is true". For example, "if it is raining, then I will bring an umbrella", is equivalent to "it is not raining, or I will bring an umbrella, or both". This truth-functional interpretation of implication is called material implication or material conditional.
The paradoxes are logical statements which are true but whose truth is intuitively surprising to people who are not familiar with them. If the terms 'p', 'q' and 'r' stand for arbitrary proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
s then the main paradoxes are given formally as follows:
-
, p and its negationNegationIn logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
imply q. This is the paradox of entailment. -
, if p is true then it is implied by every q. -
, if p is false then it implies every q. This is referred to as 'explosion'. -
, either q or its negation is true, so their disjunction is implied by every p. -
, if p, q and r are three arbitrary propositions, then either p implies q or q implies r. This is because if q is true then p implies it, and if it is false then q implies any other statement. Since r can be p, it follows that given two arbitrary propositions, one must imply the other, even if they are mutually contradictory. For instance, "Nadia is in Barcelona implies Nadia is in Madrid, or, Nadia is in Madrid implies Nadia is in Barcelona." This truism sounds like nonsense in ordinary discourse. -
, if p does not imply q then p is true and q is false. NB if p were false then it would imply q, so p is true. If q were also true then p would imply q, hence q is false. This paradox is particularly surprising because it tells us that if one proposition does not imply another then the first is true and the second false.
The paradoxes of material implication arise because of the truth-functional definition of material implication, which is said to be true merely because the 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 false or the 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 true. By this criterion, "If the moon is made of green cheese, then the world is coming to an end," is true merely because the moon isn't made of green cheese. By extension, any contradiction implies anything whatsoever, since a contradiction is never true. (All paraconsistent logic
Paraconsistent logic
A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...
s must, by definition, reject (1) as false.) Also, any tautology
Tautology (logic)
In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...
is implied by anything whatsoever, since a tautology is always true.
To sum up, although it is deceptively similar to what we mean by "logically follows" in ordinary usage, material implication does not capture the meaning of "if... then".
Paradox of entailment
As the most well known of the paradoxes, and most formally simple, the paradox of entailmentEntailment
In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
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 plastic.
This arises from the principle of explosion
Principle of explosion
The principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...
, a law of classical logic
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. This seems paradoxical, as it suggests that the above is a valid argument.
Understanding the paradox of entailment
ValidityValidity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
is defined in classical logic as follows:
- An argument (consisting of premisePremisePremise 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 a conclusion) is valid if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
there is no possible situation in which all the premises are true and the conclusion is false.
For example an 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
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
, the argument is valid.
But one could construct an argument in which the premises are inconsistent
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
. 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:
- Matter has mass (1st premise; true)
- Matter does not have mass (2nd premise; false)
- All numbers are equal to 12 (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.
Explaining the paradox
The strangeness of the paradox of entailmentEntailment
In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
comes from the fact that the definition of validity in classical logic
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
does not always agree with the use of the term in ordinary language. In everyday use validity suggests that the premises are consistent. In classical logic, the additional notion of soundness is introduced. A sound argument is a valid argument with all true premises. Hence a valid argument with an inconsistent set of premises can never be sound. A suggested improvement to the notion of logical validity to eliminate this paradox is relevant logic
Relevance logic
Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...
.
Simplification
The classical paradox formulas are closely tied to the formula,
the principle of Simplification, which can be derived from the paradox formulas rather easily (e.g. from (1) by Importation).
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 either true that if John is in London then he is in France, or that if he is in Paris then he is in England." Either John is in London or John is not in London. If John is in London, then John is in England. Thus the proposition "if John is in Paris, then John is in England" holds because we have prior knowledge that the conclusion is true. If John is not in London, then the proposition "if John is in London, then John is in France" is true because we have prior knowledge that the premise is false.
The second 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 if switch B is closed, the light is on." If the two switches are in series, then the premise is true but the conclusion is false. Thus, using classical logic and taking material implication to mean if-then is an unsafe method of reasoning which can give erroneous results.