Why do we qualify "implication" with "material"? This seems to imply that there are other kinds of implication.
There is no "immaterial" implication.
- Two kinds of implication, the material and the formal.
See in W&R, Principia Mathematica the notation for implication (the "horseshoe") ⊃; in the "material" usage, it is a connective between propositions :
*1.2 ⊢ : p v p . ⊃ . p,
while in the formal usage it is a relation between "classes" :
*10·02 φx ⊃x ψx . = . (x). φx ⊃ ψx.
- Alfred North Whitehead & Bertrand Russell, Principia Mathematica to *56 (2nd ed - 1927), page 7.
Today, the material conditional has to be compared with other conditionals : the subjunctive and the counterfactual conditionals; see :
- Graham Priest, An Introduction to Non-Classical Logic : From If to Is (2nd ed - 2008), page 13.
Material implication -- the usual form implication in mathematics -- defines P => Q equivalent to ~[P and ~Q]. Some are uncomfortable with the notion that if P is false and Q is false, then P => Q true. Perhaps that is because the notation suggests some come causal relationship -- e.g. that maybe P causes Q, or Q causes P. Their thinking may be: How can P cause Q if P is false? Or something along those lines.
I like to use following example. Consider the statement: "If it is raining, then it is cloudy."
Raining => Cloudy
This does not mean that rain causes cloudiness. Or that cloudiness causes rain. Neither is the case. The statement means simply that it cannot be simultaneously both raining and not cloudy.
~[Raining & ~Cloudy]
This will be true if it is not raining and not cloudy (i.e. if both antecdent and consequent of the conditional statement are false). Nothing weird or counter-intuitive about that.
As a math person, I don't see the need for any other kind of conditionals, but here is a list of some alternative formulations from Wiki:
UPDATE 3 YEARS LATER:
For my latest thinking on material implication, see my blog posting Material Implication: If Pigs Could Fly. There I derive the truth table using notions of implication in common usage.
Not quite an answer, but it might be found useful:
In Aristotles Physics he distinguishes between causes; there are four types, amongst them is the formal and material.
The formal cause is what causes the shape or form of something: the formal cause of the pot is the potter. The material cause is what the pot is made of - the clay.
Of course there can be no pot if the potter is not there to shape the clay; nor can there be any pot, if the potter has no clay. Both causes are necessary in this situation.
Causes are consequences; and what is a consequence can be understood as an implication. This links up with two types of logic in Aristotle, the formal and material. The formal logic are propositions true by form - the syllogism - for instance; the first proposition of the famous example being:
Socrates is mortal
The material logic is truth by correspondence. Is there, in fact, a Socrates?
Again both are neccessary to establish objective knowledge.
One might say then the immaterial implication is the formal one.