# Interpreting conditional statements

If we have the conditional statement:

`a -> b -> c`

Do we interpret it as "if a, then b, if b, then c" or "if a, then if b then c"?

In other words, which of the arguments would be valid:

1) `a -> b -> c, a ∴ b ^ c`

2) `a -> b -> c, a ∴ b -> c`

Or would both interpretations and both arguments be valid?

You're suggesting as possible interpretations `(a → b) & (b → c)` and `a → (b → c)`, but the two possible interpretations are `(a → b) → c` and `a → (b → c)`. It's never `(a → b) & (b → c)` (at least not in standard logic textbooks). As was pointed out in the other answer, it's a matter of convention which one of the two is intended.

• So, if we have `a->b & b->c`, then we can conclude `b & c` from `a`?
– Matt
Jun 6, 2019 at 15:19
• @MatthewLerner Yes.
– E...
Jun 6, 2019 at 15:21

It depends on the convention adopted by the textbook about omitting parentheses...

Usually, when one connective symbol is used repeatedly, grouping is to the right:

a → b → c is a → (b → c).

If so, the valid argument is the second one.

• And 1) won't be valid under either interpretation. Jun 5, 2019 at 15:39