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Should

a→b→c

be read as

(a→b)→c

or

a→(b→c)?

I used a online truth table generator (http://logic.stanford.edu/intrologic/secondary/applications/babbage.html) to test and got a→(b→c) is the correct one.

But on this article it says logician use (a→b)→c See:Boolean algebra operation precedence?

So I wondered in the field of logics, which would be the norm to read sentence with multiple implication operators such as a→b→c .

Community
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pikaisalone
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    There are conventions, like: precedence between conncetives: $\lnot$ applies to as little as possible, then $\land$ and $\lor$; where one connective symbol is used repeatedly, grouping is to the right. Thus $a \to b \to c$ must be read as $a \to (b \to c)$. But they are not "universal"... – Mauro ALLEGRANZA Jul 05 '16 at 06:09
  • See if the author tells you enough to imply which they mean. If not, neither is correct. – Doug Spoonwood Jul 05 '16 at 09:49

2 Answers2

3

In almost every context you will encounter $a \implies b \implies c$ it means

$$a \implies (b \implies c)$$

This is so common in constructive logic that it is effectively a universal standard. The fake reason for it is that

$$a_0 \implies (a_1 \implies (a_2 \implies (\dots \implies b)))$$

is propositionally equivalent to

$$(a_0 \land a_1 \land a_2 \dots) \implies b$$

which makes it a very easy convention to work with, since most theorems have a list of conditions and 1 conclusion. But the real reason for the convention comes from typed lambda calculus, which is the basis of constructive logic. Suppose you have a lambda expression

$$\lambda y. \lambda x. \lambda w. V$$

and you have a predicate $T(n)$ that represents "$n$ is of the appropriate type". Let

  • $A$ be $T(y)$
  • $B$ be $T(x)$
  • $C$ be $T(w)$
  • $V$ be $T(V)$

Then the statement that "$\lambda y. \lambda x. \lambda w. V$ is appropriately typed" is propositionally:

$$T(\lambda y. \lambda x. \lambda w. V) = (A \implies B \implies C \implies D)$$

if you associate the implication to the right as above. The similarity between that notation and functions from $A$ to $B$ as $F: A \to B$ is apparent. Since constructive logic is built on top of typed lambda calculus, the convention is preserved. You will probably never encounter anywhere in modern logical publication that doesn't use this convention.

DanielV
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1

This depends on convention, just like the precedence between different operators like $\neg, \land, \lor$ etc.

I think that usually, the connective is interpreted right-associative, i.e.

$a \to b \to c \Leftrightarrow a \to (b \to c)$

but that depends on what the author specified.

Natalie Clarius
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