/sci/ - Science & Math

[Anonymous No.6016301]

/sci/,
This doesn't make sense (at least from a logical standpoint). Why is the result of the implication operator (if, then) true if the condition is false?

In programming for instance, if the condition isn't met in a conditional statement, it skips over the "then" part of the condition.

[Anonymous No.6016311]

>>6016301
if you >imply something which is wrong you can get a wrong result

[Anonymous No.6016312]

it is "vacuously" true. god i loved logic

[Anonymous No.6016329]

Combinatorically, there 16 functions which take two inputs and give up to two different ouputs, see

https://en.wikipedia.org/wiki/Truth_function#Table_of_binary_truth_functions

The material implication you refer to is one of them.
For practical, as well as for historical reasons, this operation is part of the standard presentation of classical propositional logic - but one doesn't have to adopt it.
Moreover, you're right is feeling uneasy regarding to what extend the material implication really captures the intuitive "if then" relation. Many more complicated logics are formed around extending this, e.g.
https://en.wikipedia.org/wiki/Relevant_logic

That being said, evidently it works for doing formal mathematics and these two questions on math stackexchange ask exactly your question:

http://math.stackexchange.com/questions/38713/help-to-understand-material-implication

http://math.stackexchange.com/questions/232309/how-to-interpret-material-conditional-and-explain-it-to-freshmen

[Anonymous No.6016333]

>>6016301
IF 2=-2
THEN (2)^2 = (-2)^2

[Anonymous No.6016341]

Implication is strongly related to subset, just like conjunction is related to intersection, and disjunction to union.

If we let P = { x | p(x) } and Q = { x | q(x) } (meaning P is the set of x that has property p, and Q the set of x that has property q), we can write:

P ⊂ Q ≡ ( x ∈ P → x ∈ Q)

Now you can think of it visually using the analog of subset, and it should become clear.

[Anonymous No.6016348]

>>6016341
P ⊂ Q ≡ ( x ∈ P → x ∈ Q) ≡ ( p(x) → q(x) ) *