# Is it logically correct to say that if A implies B then not A implies not B?

Here is an argument:

The government has announced that it wants to reduce the level of ill-health due to workplace stress. Ministers could learn a lot from a recent study of 8000 white-collar workers in America which found that men who had a high level of control over the way they carried out their jobs had a low rate of heart disease, a stress-related disease. Clearly, therefore, the most stressful jobs are those in which employees have little control over the pace of their work and how it is organised. If the government is serious about wanting to reduce the level of stress-related disease, then it needs to encourage employers to give their workers greater control over their work.

The part that I want to focus on is the one that says:

1. High control implies low rate of heart disease.
2. THEREFORE, little control implies high stress.

(Please assume true that the low incidence of heart disease was caused by a reduction in stress. Also, if you are someone who understands logic very well, please explain to me, using formal logic, why it's necessary to make this assumption.)

It would be much appreciated if you use formal logic in order to break down the argument and answer my two questions using that. Thank you in advance.

• NO, it is not correct. P and Q implies P but not-(P and Q) does not implies not-P. – Mauro ALLEGRANZA Aug 31 '17 at 17:13
• I fear you are mixing different things here. Implication is not causation. High control does not imply a low rate of heart diase; it causes a low rate of heart disease. The distinction can be counter-intuitive: If Mary stays home, it is raining. Here Mary staying at home implies it is raining. It of course doesn't cause the rain; likely, the converse is true. – Luís Henrique Oct 6 '17 at 14:02

The argument presented looks like this:

1. (P1) `Control -> ~HeartDisease`

2. (P2) `~HeartDisease -> ~Stress`

3. (Modus ponens on 1 and 2) `Control -> ~Stress`

4. (Transposition on 3) `~~Stress -> ~Control`

5. (Double negation on 4) `Stress -> ~Control`

So to keep people from being stressed, we should give them control.

If we didn't have P2, we wouldn't be able to conclude that the people with control and no heart disease did not still have high stress. (They might have high stress and no heart disease because they're physically fit, for instance.) Then we wouldn't be able to say that having control meant not having stress. This is equivalent to not being able to assert that having stress meant not having control.

Is it logically correct to say that if A implies B then not A implies not B?

“If not-A, then not-B” is the converse of “If A then B”. The truth of a statement does not imply the truth of its converse. Result B could have many causes besides Cause A, so the negation of Cause A can not imply anything. The fallacy here is Denying the Antecedent.

The part that I want to focus on is the one that says: 1. High control implies low rate of heart disease. 2. THEREFORE, little control implies high stress. (Please assume true that the low incidence of heart disease was caused by a reduction in stress….)

The argument is as follows:

-If high control, then low rate of heart disease. Premise 1.

-If low stress, then low rate of heart disease. Additional Assumption.

-If low control, then high stress. Conclusion.

“low rate of heart disease” is the middle term, and is undistributed in both premises. The fallacy is Undistributed Middle. Here, the middle term fails to connect Premise 1 to the additional assumption.

The form of argument, if this inference were valid, would be :

If A then B

## Not-A

Not-B

However :

If A then B (t) Not-A (t)

could be true as indicated (t) yet Not-B false because :

If C then B (t)

## C (t)

B (t)

is true consistently with Not-A.

Take an example. If (A) it is raining (B) then the pavements are wet, it does not follow that if not-A (it is not raining) then not-B (the pavements are not wet). The pavements could be wet because of a broken drain (C).