The
material conditional (also known as
material implication,
material consequence, or simply
implication,
implies, or
conditional) is a
logical connective (or a
binary operator) that is often symbolized by a forward arrow "โ". The material conditional is used to form
statements of the form ๐โฏโโฏ๐ (termed a
conditional statement) which is read as "if ๐ then ๐" or "๐ only if ๐". It is conventionally compared to the English construction "If...then...". However, unlike the English construction, the material conditional statement ๐โฏโโฏ๐ does not specify a causal relationship between ๐ and ๐. It is merely to be understood to mean "if ๐ is true, then ๐ is also true" such that the statement ๐โฏโโฏ๐ is false only when ๐ is true and ๐ is false. Intuitively, consider that the statement "if ๐ is true, ๐ is always also true" is false when ๐ is true and ๐ is falseโโโeven when "if ๐ then ๐" does not represent a causal relationship between ๐ and ๐. Instead, the statement describes ๐ and ๐ as each only being true when the other is true and makes no claim that ๐ causes ๐. However, note that such a general and informal way of thinking about the material conditional is not always acceptable, as will be discussed. As such, the material conditional is also to be distinguished from
logical consequence.