# Understanding [(a→b).(c→d)] formula in section 36 of The Logic of Scientific Discovery

Excerpt from footnote 1 in section 36 of Popper's "The Logic of Scientific Discovery":

In the present section, the arrow is used to express a conditional rather than the entailment relation.

We can write ... [(a → b) . (c → d)] → [[(b → c) → (a → d)]

I understand that the dot, in logic notation, means "AND". But I don't quite see how this whole formula works. How can the second block be derived from the first? (block = statement bounded by "[]" brackets).

• What do you mean by "derived"? The expression is explicitly not an entailment, just a statement that can be true or can be false. Nothing needs ot be derived. Commented Jan 12, 2021 at 21:35
• It's the middle arrow that troubles me. Why is it there? Why not write the two blocks as two separate expressions that could be true or false? Commented Jan 12, 2021 at 21:39
• Because he wants to express a conditional from the first to the second part that can be true or false, not two two isolated statements. Commented Jan 13, 2021 at 0:19

The formula you quote is a tautology of the propositional calculus. To see this more clearly, we can use the import-export rule to move the (b → c) from the consequent into the antecedent. The formula is then equivalent to

[(a → b) . (b → c) . (c → d)] → (a → d)

and this is easier to understand, because it just says that if we chain a, b and c together then we can get from a to d.

In the context, Popper is making the point that if a property p is both more universal and more precise than some property q, then everything that is p is q. His discussion is rather laboured.

Also, he makes a mistake. He says,

"p is of greater universality than q if the antecedent statement function of p [...] is tautologically implied by [...], but not equivalent to, the corresponding statement function of q."

However, the formula he writes, "(x)(Φ(sub q)(x) → Φ(sub p)(x))" does not capture the 'not equivalent to' part. His formula would allow that p and q are of equal universality.