Material conditional

The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain conditionals in logic. In propositional logic, it expresses a binary truth function from truth-values to truth-values. In predicate logic, it can be viewed as a subset relation between the extension of (possibly complex) predicates. In symbols, a material conditional is written as one of the following:

1. $X\rightarrow Y$,
2. $X\supset Y$, and sometimes
3. $X\Rightarrow Y$

The material conditional is false when X is true and Y is false - otherwise, it is true. X and Y, also known respectively as the antecedent and consequent, are variables ranging over formulæ of a formal theory. The material conditional is also commonly referred to as material implication with the understanding that the antecedent (X) materially implies the consequent (Y).

Definition

Logical implication and the material conditional are both associated with an 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.

Truth table

The truth table associated with the material conditional not p or q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:

p	q	→
T	T	T
T	F	F
F	T	T
F	F	T

Difference between material and logical implication

Material implication ($\rightarrow$), the logical connective described in this article, is often confused with logical implication ($\Rightarrow$), the statement, that the material implication is always true.

Venn diagram of the material implication $A\rightarrow B$
meanig the same like $\neg A\lor B$
The left circle represents the statement A, the right circle the statement B.
The material implication is false only in the case, represented by the white area: when A is true, but B is false.

Venn diagram of the logical implication $A\Rightarrow B$
It tells that the material implication $A\rightarrow B$ is always true.
The left circle represents the statement A, the right circle the statement B.
The logical implication tells, that A without B is never the case.

In set theory there is the same difference between the operation $A^{c}\cup B$ and the relation $A\subseteq B$ meaning $A\cap B^{c}=\emptyset$.

Example

On the way from $A\subseteq B$ to $A\cap B^{c}=\emptyset$ the difference between logical ($\Rightarrow$) and material implication ($\rightarrow$) can be seen in an easy calculation:

$A\subseteq B$

$\Leftrightarrow (x\in A\Rightarrow x\in B)$

$\Leftrightarrow \forall {x}(x\in A\rightarrow x\in B)$

$\Leftrightarrow \forall {x}(x\notin A\lor x\in B)$

$\Leftrightarrow \neg \exists {x}(x\in A\land x\notin B)$

$\Leftrightarrow \neg \exists {x}(x\in A\cap B^{c})$

$\Leftrightarrow (A\cap B^{c}=\emptyset )$

The operation $\rightarrow$ can be expressed by $\lor$ and $\neg$. The relation $\Rightarrow$ by $\rightarrow$ and the universal quantifier $\forall$.

Formal properties

The material conditional is not to be confused with the entailment relation ⊨ (which is used here as a name for itself). But there is a close relationship between the two in most logics, including classical logic. For example, the following principles hold:

• If $\Gamma \models \psi$ then $\emptyset \models \phi _{1}\land \dots \land \phi _{n}\rightarrow \psi$ for some $\phi _{1},\dots ,\phi _{n}\in \Gamma$. (This is a particular form of the deduction theorem.)

• The converse of the above

• Both $\rightarrow$ and ⊨ are monotonic; i.e., if $\Gamma \models \psi$ then $\Delta \cup \Gamma \models \psi$, and if $\phi \rightarrow \psi$ then $(\phi \land \alpha )\rightarrow \psi$ for any α, Δ. (In terms of structural rules, this is often referred to as weakening or thinning.)

These principles do not hold in all logics, however. Obviously they do not hold in non-monotonic logics, nor do they hold in relevance logics.

Other properties of implication:

• distributivity: $(s\rightarrow (p\rightarrow q))\rightarrow ((s\rightarrow p)\rightarrow (s\rightarrow q))$

• transitivity: ($a\rightarrow b)\rightarrow ((b\rightarrow c)\rightarrow (a\rightarrow c))$

• commutativity: ($a\rightarrow (b\rightarrow c))\equiv (b\rightarrow (a\rightarrow c))$

• idempotency: $a\rightarrow a$

• truth preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of material implication.

Philosophical problems with material conditional

The meaning of the material conditional can sometimes be used in the natural language English "if condition then consequence" construction (a kind of conditional sentence), where condition and consequence are to be filled with English sentences. However, this construction also implies a "reasonable" connection between the condition (protasis) and consequence (apodosis) (see Connexive logic).

So, although a material conditional from a contradiction is always true, in natural language, "If there are three hydrogen atoms in H₂O then the government will lose the next election" is interpreted as false by most speakers, since assertions from chemistry are considered irrelevant conditions for proposing political consequences. "If P then Q", in natural language, appears to mean "P and Q are connected and P→Q". Just what kind of connection is meant by the natural language is not clearly defined.

• The statement "if (B) all bachelors are unmarried then (C) the speed of light in a vacuum is constant" may be considered false, because there is no discernible connection between (B) and (C), even though (B)→(C) is true.
• The statement "if (S) Socrates was a woman then (T) 1+1=3" may be considered false, for the same reason; even though (S)→(T) is true.

When protasis and apodosis are connected, the truth functionality of linguistic and logical conditionals coincide; the distinction is only apparent when the material conditional is true, but its antecedent and consequent are perceived to be unconnected.

The modifier material in material conditional makes the distinction from linguistic conditionals explicit. It isolates the underlying, unambiguous truth functional relationship. Therefore, exact natural language encapsulation of the material conditional X → Y, in isolation, is seen to be "it's false that X be true while Y false" — i.e. in symbols, $\neg (X\land \neg Y)$. Arguably this is more intuitive than its logically equivalent disjunction $\neg X\lor Y$.

The truth function $\rightarrow$ corresponds to 'not ... or ...' and does not correspond to the English 'if...then...' construction. For example, any material conditional statement with a false antecedent is true.

So the statement "if 2 is odd then 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 problems are known as the paradoxes of material implication, though they are not really paradoxes in the strict sense; that is, they do not elicit logical contradictions.

There are various kinds of conditionals in English; e.g., there is the indicative conditional and the subjunctive or counterfactual conditional. The latter do not have the same truth conditions as the material conditional. For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below.

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.
• Stalnaker, Robert. 'Indicative Conditionals'. Philosophia 5 (1975): 269–286.

See also

Conditionals

• Counterfactual conditional
• Indicative conditional
• Corresponding conditional (logic)
• Strict conditional
• Logical implication

Related topics

• Ampheck
• Boolean algebra
• Boolean domain
• Boolean function

• Boolean logic
• Laws of Form
• Logic gate
• Logical graph

• Peirce's law
• Propositional logic
• Sole sufficient operator
• Conditional quantifier