# If F is a sufficient condition for G, is lacking G a sufficient condition for lacking F?

If having feature F is a sufficient condition for having feature G, then lacking G is a sufficient condition for lacking F.

I think this statement should be "If having feature F is a sufficient condition for having feature G, then lacking F is not a sufficient condition for G." But I'm not sure. I know this probably is such an easy question but I'm having trouble. I just started using the tables to test and eliminate features.

## 5 Answers

The statement is correct.

We can rephrase it as: (F → G) → (¬G → ¬ F). That this holds is the idea of modus tollens.

About your proposal of rewriting:
If F is sufficient for G (so F → G), then it doesn't follow that lacking F is not sufficient for G (¬(¬F → G)). For example, if G is always true, anything is a sufficient condition for G: both F and ¬F.

There are several tools online with which you can make truth tables yourself and check the truth value of statements. This is a truth table for the correct statement: And this is a truth table for your proposal: 1) I recommend making a truth table with the 6 columns

F, G, F => G, non F, non F => G, non (non F => G).

When setting F true and G true you get a row with

"F => G" true, but "non(non F => G)" false. This counterexample shows

"(F => G) => non(non F => G)" is not valid.

2) A second possibility is to use that "A => B" is equivalent to "non A or B" and to make an analogous truth table.

3) A third possibility is using again the equivalence of "A => B" to "non A or B" and to transform

"(F => G) => non(non F => G)"

by successive application of that equivalence and the tautology "F or non F" to the equivalent formula

"non G".

The latter is false in case "G" is true. Hence the original formula

"(F => G) => non(non F => G)" is not valid.

If having feature F is a sufficient condition for having feature G, then lacking G is a sufficient condition for lacking F.

Lets replace F with "father", G with "son" and rewrite.

If being father is a sufficient condition for having son, then lacking son is a sufficient condition for not being father.

Wikipedia provides a valuable summary of necessary and sufficient conditions and their use in natural language. Here is an overview:

In logic, necessity and sufficiency are terms used to describe a conditional or implicational relationship between statements. For example, in the conditional statement: "If P then Q", Q is necessary for P, because P cannot be true unless Q is true. Similarly, "P is sufficient for Q" because P being true always implies that Q is true, but P not being true does not always imply that Q is not true.

That article also provides a diagram mapping the relationship. One can simplify that diagram as follows: If F is a sufficient condition for G, then the region associated with F is completely contained within the region associated with G as in the diagram. Note there may be other conditions, that do not include F or overlap with F, that are also sufficient for G. The diagram also shows that G is a necessary condition for F. Any other necessary conditions for F would have to overlap with G.

Similarly, the diagram shows the relationship between ~G and ~F. Note that the region associated with ~G is contained within the region associated with ~F. So ~G is a sufficient condition for ~F.

Now consider the statement in question:

"If having feature F is a sufficient condition for having feature G, then lacking F is not a sufficient condition for G."

If lacking F, or ~F, is a sufficient condition for G, then the region associated with ~F in the diagram would have to lie entirely within the region associated with G. That might be the case if G contained everything, that is, if G were a necessary condition for both F and ~F, a tautology, but in general it is not. We would also need to take precautions if G were a contradiction, always false. However, there is usually a region outside G, or ~G. Therefore, ~F, or lacking F, is not a sufficient condition for G, as the statement claims.

Wikipedia contributors. (2019, October 8). Necessity and sufficiency. In Wikipedia, The Free Encyclopedia. Retrieved 16:08, October 9, 2019, from https://en.wikipedia.org/w/index.php?title=Necessity_and_sufficiency&oldid=920256263

The answer is no, and the following example might help.

The existence of the universe (F) is sufficient for the existence of humankind (G). The lack of humankind (G) is not sufficient to cause the lack of the universe (F).

• The example is wrong because the mere existence of the universe is not sufficient for the existence of human kind. – user17695 Nov 5 '15 at 5:20