Paradoxes of Material Implication

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The paradoxes of material implication or subjunction are a group of propositional logic formulas that are tautologies but are intuitively problematic. The cause of the paradoxes is that the interpretation of the truth of an implication in natural language does not correspond to its formal interpretation in classical logic by truth tables.

example

The statement "If it rains now, I will take an umbrella with me" is formalized in classic propositional logic . According to the definition of the subjunction, this statement is false if it is true and false, otherwise true (- if false and true, if and both true, if and both false). This follows from the interpretation of the subjunction as a truth value function by the truth table of the seq function. So if it's not raining, the statement “If it rains now, I'll take an umbrella with me” is the same in both cases: it doesn't matter whether I take an umbrella with me or not.

w w w
w f f
f w w
f f w

The statement “If it rains tomorrow, then it will be ” is logically correct, because “ ” is always correct - regardless of whether it will rain tomorrow or not. This example already points to the problematic point of implication: can be true without there being any contextual connection between and - because the truth value of the subjunction depends only on the truth values ​​of and .

Paradoxes of Material Implication

The following list gives an overview of the most important paradoxes of material implication:

That all of these formulas are tautologies can be checked using the truth table method. But you can also see it more quickly if you have the relationship

used: for example in the case of the 6th formula above, the first part of the disjunction is not true only if is true but false In this case the second part of the disjunction is true.

The philosopher Charles Sanders Peirce once illustrated the sixth variant listed above as follows: If you cut up a newspaper sentence by sentence, pour all sentences into a hat and take out any two at random, then the first of these sentences is a consequence of the second or vice versa . This example also shows that the material implication has nothing to do with the content of the statements involved (only with the truth values).

Avoiding the paradoxes

Attempts have long been made to modify classical logic in such a way that the paradoxes of material implication no longer occur. One approach is that of relevance logic . The idea is that for a true implication, one demands that there is a "substantive connection" between the antecedent and the consequent, or that the antecedent is relevant to the consequent .

Another non-classical logic that avoids the paradoxes of implication is Connexive Logic, which is characterized by being Aristotle's thesis , i.e. H. the formula

accepted as logical truth (tautology). Aristotle's thesis is that no statement can follow from its own negation. The formula ~ (~ p → p) is not a tautology in classical propositional logic (see the truth table below), although it appears intuitively correct.

w f w f
f w f w

literature

  • Alan Ross Anderson, Nuel Belnap: Entailment: the logic of relevance and necessity, vol. I . Princeton University Press 1975. ISBN 978-0691071923

Individual evidence

  1. Anderson / Belnap, 1975
  2. ^ S. McCall: Connexive Implication . In: The Journal of Symbolic Logic, Vol. 31, No. 3 (1966), pp. 415-433.
  3. To. Pr. Ii 4.57b3