Logical implication
The concept of logical implication encompasses, depending on the context of use, a specific logical function, a specific logical relation, and the various symbols that are used to denote this function and this relation. In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them.
A close approximation to the concept of logical implication or material conditional is expressed in ordinary language by means of the following conditional form:
- If p then q.
Here p and q are propositional variables that stand for any propositions in a given language. In a statement of the form "if p then q", the first term, p, is called the antecedent and the second term, q, is called the consequent, while the statement as a whole is called either the conditional or the consequence. Assuming that the conditional is true, then 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.
Definition
The material conditional and logical implication are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if the first operand is true and the second operand is false.
The truth table associated with the material conditional if p then q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:
| p | q | p ⇒ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
Discussion
The usage of the terms logical implication and material conditional varies from field to field and even across different contexts of discussion. One way to minimize the potential confusion is to begin with a focus on the various types of formal objects that are being discussed, of which there are only a few, taking up the variations in language as a secondary matter.
The main formal object under discussion is a logical operation on two logical values, typically the values of two propositions, that produces a value of false just in case the first operand is true and the second operand is false. By way of a temporary name, the logical operation in question may be written as Cond (p, q), where p and q are logical values. The truth table associated with this operation is as follows:
| p | q | Cond (p, q) |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
Some logicians draw a firm distinction between the conditional connective (the syntactic sign "\(\rightarrow\)"), and the implication relation (the formal object denoted by the sign "\(\Rightarrow\)"). These logicians use the phrase if–then for the conditional connective and the term implies for the implication relation. Some explain the difference by saying that the conditional is the contemplated relation while the implication is the asserted relation. In most fields of mathematics, it is treated as a variation in the usage of the single sign "\(\Rightarrow\)", not requiring two separate signs. Not all of those who use the sign "\(\rightarrow\)" for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a so-called syncategorematic sign, that is, a sign with a purely syntactic function. For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign "\(\rightarrow\)" to denote the boolean function that is associated with the truth table of the material conditional. These considerations result in the following scheme of notation.
\[\begin{matrix}
p \rightarrow q & \quad & \quad & p \Rightarrow q \\
\mbox{if}\ p \ \mbox{then}\ q & \quad & \quad & p \ \mbox{implies}\ q
\end{matrix}\]
Let \(\mathbb{B} = \{\mathbf{F},\ \mathbf{T}\}\) be the boolean domain of two logical values. The truth table shows the ordered triples of a triadic relation \(L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\) that is defined as follows:
\[L = \{(p,\ q,\ r) \in \mathbb{B} \times \mathbb{B} \times \mathbb{B}\ :\ Cond (p,\ q)\ = r \}\,.\]
Regarded as a set, this triadic relation is the same thing as the binary operation:
\[Cond : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\]
The relationship between \(Cond\!\) and \(L\!\) exemplifies the standard association that exists between any binary operation and its corresponding triadic relation.
The conditional sign "\(\rightarrow\)" denotes the same formal object as the function name "\(Cond\mbox{ }\)", the only difference being that the first is written infix while the second is written prefix. Thus we have the following equation:
\[(p \rightarrow q) = Cond (p,\ q)\,.\]
Consider once again the triadic relation \(L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\) that is defined in the following equivalent fashion:
\[L = \{(p,\ q,\ Cond (p,\ q))\ :\ (p,\ q) \in \mathbb{B} \times \mathbb{B} \}\,.\]
Associated with the triadic relation \(L\!\) is a binary relation \(L_{..T} \subseteq \mathbb{B} \times \mathbb{B}\) that is called the fiber of \(L\!\) with \(T\!\) in the third place. This object is defined as follows:
\[L_{..T} = \{ (p,\ q) \in \mathbb{B} \times \mathbb{B}\ :\ (p,\ q,\ T) \in L \}\,.\]
The same object is achieved in the following way. Begin with the binary operation:
\[Cond : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\]
Form the binary relation that is called the fiber of \(Cond\!\) at \(T\!\), notated as follows:
\[Cond^{-1}(T) \subseteq \mathbb{B} \times \mathbb{B}\,.\]
This object is defined as follows:
\[Cond^{-1}(T) = \{ (p,\ q) \in \mathbb{B} \times \mathbb{B}\ :\ Cond (p,\ q) = T \}\,.\]
The implication sign "\(\Rightarrow\)" denotes the same formal object as the relation names "\(L_{..T}\mbox{ }\)" and "\(Cond^{-1}(T)\mbox{ }\)", the only differences being purely syntactic. Thus we have the following logical equivalence:
\[(p \Rightarrow q) \iff (p,\ q) \in L_{..T} \iff (p,\ q) \in Cond^{-1}(T)\,.\]
This completes the derivation of the mathematical objects that are denoted by the signs "\(\rightarrow\)" and "\(\Rightarrow\)" in this discussion. It needs to be remembered, though, that not all writers observe this distinction in every context. Especially in mathematics, where the single arrow sign "\(\rightarrow\)" is reserved for function notation, it is common to see the double arrow sign "\(\Rightarrow\)" being used for both concepts.
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.
Using the horseshoe "⊃" symbol for implication is falling out favor due to its conflict with the superset symbol \(\supset\) used by the Algebra of sets. A set interpretation of "\( A \to B\)" is "{x| A(x) is true} \(\subseteq\) {x| B(x) is true}".
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 \(\neg A \vee B\), using negation and disjunction.)
References
- Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
- Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell.
- Edgington, Dorothy (2006), "Conditionals", in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Eprint.
- Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.
See also
Logical operators
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Aficionados
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