Conversion of Conditional Statement to Disjunctions

Let us say that we have the conditional statement A→B. What would be the best way to represent that in the form of a disjunction (A∨B)? Help is much appreciated.

310 4 4 silver badges 14 14 bronze badges asked Mar 11, 2015 at 3:21 45 1 1 silver badge 4 4 bronze badges Commented Mar 11, 2015 at 3:33

So, how would I create a disjunctive syllogism out of the conditional If it is a difficulty, it strengthens the mind? The disjunction is ¬D∨SM. Correct?

Commented Mar 11, 2015 at 3:39

Normally you use only one symbol for each statement so ¬D∨M would be my choice . It's entirely the equivalent to D→M

Commented Mar 11, 2015 at 4:29 This is called Material Implication. Commented Mar 11, 2015 at 20:54

1 Answer 1

Assuming you're looking for some intellectual understanding of this:

What the statement "If A then B" means in logic is that there is never a case where A is true and B is false. If A is true, then B will be true.

Another way of saying that is that either B will be true or A will be false. Therefore

are both alternate ways of expressing A -> B

You can confirm by assigning A and B values and confirming that the overall value of those three expressions always matches.

answered Mar 11, 2015 at 17:19 Chris Sunami Chris Sunami 30.4k 2 2 gold badges 51 51 silver badges 104 104 bronze badges

Some folks get picky about keeping symbols on the same sides of operators when changing the operator being used. If someone reading this has a logic professor who's a stickler about these sorts of things but they want to convert A->B to B V !A in preparation for a further operation, I recommend first converting to B V !A via Material Implication, then swapping sides via Commutation. My inner pedant satisfied, this was an excellent explanation that did a good job of explaining the why in addition to the how, and I thoroughly appreciate it.