Material conditional
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In propositional calculus, or logical calculus in mathematics, the material conditional or the implies operator is a binary truth-functional logical operator yielding the form
If a then c,
where a and c are statement variables (to be replaced by any meaningful indicative sentence of the language). In a statement of this form, the first term, in this case a, is called the antecedent and the second term, in this case c, is called the consequent. The truth of the antecedent is a sufficient condition for the truth of the consequent, while the truth of the consequent is a necessary condition for the truth of the antecedent.
The operator is symbolized using a right-arrow "→" (or sometimes a horseshoe "⊃"). "If A then B" is written like this:
<math> A \to B</math>
Symbolization
A common exercise for an introductory logic text to include is symbolizations. These exercises give a student a sentence or paragraph of text in ordinary language which the student has to translate into the symbolic language. This is done by recognizing the ordinary language equivalents of the logical terms, which usually include the material conditional, disjunction, conjunction, negation, and (frequently) biconditional. More advanced logic books and later chapters of introductory volumes often add identity, Existential quantification, and Universal quantification.
Different phrases used to identify the material conditional in ordinary language include if, only if, given that, provided that, supposing that, implies, even if, and in case. Many of these phrases are indicators of the antecedent, but others indicate the consequent. It is important to identify the "direction of implication" correctly. For example, "A only if B" is captured by the statement
A → B,
but "A, if B" is correctly captured by the statement
B → A
When doing symbolization exercises, it is often required that the student give a scheme of abbreviation that shows which sentences are replaced by which statement letters. For example, an exercise reading "Kermit is a frog only if muppets are animals" yields the solution:
A → B, A - Kermit is a frog. B - Muppets are animals.
Truth table
The truth value of expressions involving the material conditional is defined by the following truth table:
| p | q | p → q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
Comparison with other conditional statements
The use of the operator is stipulated by logicians, and, as a result, can yield some unexpected truths. For example, any material conditional statement with a false antecedent is true. So the statement "2 is odd implies 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement, "If pigs fly, then Paris is in France" is true.
These unexpected truths arise because speakers of English (and other natural languages) are tempted to equivocate between the material conditional and the indicative conditional, or other conditional statements, like the counterfactual conditional and the material biconditional. This temptation can be lessened by reading conditional statements without using the words "if" and "then". The most common way to do this is to read A → B as "it is not the case that A and/or it is the case that B" or, more simply, "A is false and/or B is true". (This equivalent statement is captured in logical notation by <math>\neg A \vee B</math>, using negation and disjunction.)mk:Материјална импликација zh:实质条件