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? – Matthew Lerner Jun 6 at 15:19
  • @MatthewLerner Yes. – Eliran Jun 6 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. – Jishin Noben Jun 5 at 15:39

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