Question about modus tollens

Hi,

 

I would like to know if this form of argument is valid or not:

P1) If A is not true, then B is not true

P2) B is true

C1) A is true

 

Thanks!

The argument is valid.  It is an example of the general truth statement that if a statement itself is true, so is its contrapositive.

However, it's made a bit more complex than usual to follow because P1 is stated in the negative.  Just like double negatives in normal conversation can help confuse the discussion, considering the contrapositive of a negative statement takes a bit of mental juggling.

The usual form used to describe statements and contra positives is something like:

          If P, then Q.  ~Q, therefore ~P

The above statement looks more like:

          If ~A, then ~B.  ~ ~ B (or just B), therefore ~ ~ A (or just A)

The contrapositive form is still there.  It just gets a bit messy saying things like "it is true that A is not true".

An equivalent rephrasing of the argument could be (and it might be easier to follow):

If A is false, then B is false

B is true.

Therefore, A is true

Hi, Alex!

      The argument can be quickly evaluated by using a method called indirect proof (or reductio ad absurdism). Now, an argument is valid just in case its formal structure is such that it is impossible for the premises to be true and the conclusion false. An indirect proof works by deriving a contradiction (an impossibility) from the assumption that the premises are true and the conclusion false.

      To begin the indirect proof, assume the two premises of your argument. 

      1. If not A, then Not B.
      2. B

      Now, add a third line that states the negation of the argument’s conclusion.

          3. Not A 

      Having assumed the conclusion of the argument is false, while assuming the premises in the background, the goal now is to derive a contradiction. I will derive a contradiction with just two more steps that logically follow from the previous steps. Step 4 is as follows. 

         4. Not B (1, 3 MP)

      How did I reach step 4? I used the statements in steps 1 and 3 to make a Modus Ponens inference, as noted in the parentheses. Step 5, the final step, is as follows. 

         5. B and Not B (2, 4 CONJ)

      How did I reach step 5? I took the statements in steps 2 and 4 and conjuncted them. In sum, by assuming the premises of the argument (in steps 1 and 2), and then further assuming the negation of the argument’s conclusion (in step 3), I derived a contradiction. Of course, a contradiction is not possibly true. In other words, it is not possible that the premises of your argument are true and the conclusion false. For assuming that the premises are true and the conclusion false leads to a contradiction, as I just proved. Therefore, an argument which has steps 1-2 as premises, and the negation of step 3 as the conclusion, is valid. 

      This is an informal exhibition of the reductio ad absurdum method of proof, of course. But I hope it helps. It is invaluable to invest in learning methods for assessing argument's for validity. Indirect proof is one method, but you can also use a truth-table, a tableau, or natural deduction. Let me recommend a helpful textbook for learning truth-tables and natural deduction, called “Forall x: Calgary remix. An Introduction to Formal Logic” by P.D. Magnus and Tim Button. 

Thank you, Alex

From, Kaiden

The way I see it, the fact B is true does not imply, infer, suggest, mandate or dictate that A is true. It is sophistry and a non sequitur.

I believe that C1 should properly be. Therefore, A is possibly true." This is because the truth or falsity of B could as well be caused by something other than A. 

This appears valid (I say with 86.2% confidence).