How do you decide how these Conditional Statements written in prose should be symbolised: (A → B) → C or A → (B → C)?

Question

_{Even after rereading this, I still cannot distinguish intuitively (A → B) → C vs A → (B → C). So please allow me to concretise that question with this example (from scrolling 40% down here):}

[1.] If Hyperion and Starbucks are both closed, then if you insist on having coffee, you’re going to go to MacDonald’s but you’ll wish you hadn’t.
[2.] If Hyperion is closed, then Starbucks is too.
[3.] So, if Hyperion is closed, if you insist on having coffee, you’re going to wish you hadn’t.

I abbreviate each proposition as follows:

H = Hyperion is closed, S = Starbucks is closed, M = You go to McDonald's,
W = You wish you had not visited some store (which context determines).

Without relying on probability of occurrence, how do you deduce and decide whether:

4. [1] should be:

1.1.  [(H ● S) ⊃ C] ⊃ (M ● W)    or     1.2.  (H ● S) ⊃ [C ⊃ (M ● W)]?

5. [3] should be:

3.1.  H ⊃ (C ⊃ W)    or    3.2.  (H ⊃ C) ⊃ W?

Answer 1

In both your examples, the right-nested conditional is the correct interpretation, i.e. 1.2 and 3.1. One way to identify a right-nested conditional easily is to employ the import-export rule by seeing whether the sentence makes equal sense when you replace the first conditional with "and". So 1 is the same as "if Hyperion and Starbucks are both closed and you insist on having coffee, you're going to go to MacDonald’s but you’ll wish you hadn’t." And 3 is the same as "if Hyperion is closed and you insist on having coffee, you're going to wish you hadn't."

One can, of course, construct examples where the meaning is genuinely ambiguous, but in practice right-nesting is usually right.

Answer 2

Officially, in well written English, you can follow the dependent clause construction marked by the commas and the succession of the pronouns.

1. If A, [then] if B, [then] C

is always going to mean

2. A → (B → C).

Hereafter, '{Coffee Store}' means '{Coffee Store} is open.' So 'Starbucks' means 'Starbucks is open', and 'not Starbucks' means 'Starbucks is closed.' For example:

3. If you want coffee, then if Starbucks is closed, I suggest we make it ourselves.

Can be encoded only as

4. Coffee ⟹ (not Starbucks ⟹ Make_Coffee)

And never with the other nesting order.

The less likely

5. (A → B) → C

would have to be written

6. If₁  if₂  A,  [then₁] B;  [then₂] C.

or

7. If₁  B  if₂  A,  then C.

In 7, note: if₂ is the '←' Logical Connective, with no comma.

You have to close clauses in order. Put short: like parentheses, references in most Latinate grammars are 'last-in, first-out'.

"Then" has a referent, and officially, references must bind to the closest available potential candidate, that is the rightmost "if" that is not already spoken for. So in 6, then₁ must go with if₂ and then₂ with if₁.

When the "then" is merely implied, the comma that closes the dependent clause must close the clause most recently opened that has not already been closed (by another comma or reference). Same rule, different official reason. So you can just read it as if the implied 'then' were inserted.

When each "if" does not come with a comma or a "then" of its own, the comma or pronoun must split the one that has a clause, not the one that is a connective. Connective conditions like "B whenever A" have to bind most tightly.

Of course, in practice one is seldom going to see the two 'ifs' in a row. One or the other is going to be since/when/while, or some equivalent.

Since when Hyperion is closed, Starbucks is also closed, we will have to make our own coffee.

or the equivalent

If Starbucks is closed whenever Hyperion is, then we will have to make our own coffee.

unwinds as

(not Starbucks ⟹ not Hyperion) ⟹ Make_Coffee

And never the other order. The 'when' clause here must be closed before the preceding 'since' and connectives bind tighter than clauses.

Answer 3

It is quite simple, but I'll use a little different example.

We will "unpack" the following nested conditional :

"if Hyperion and Starbucks are both closed, then (if you want a coffee, you’ll not have it)"

proceeding step-by-step.

We have to start with :

"if Hyperion and Starbucks are both closed, then tomorrow will rain",

that we translate with the formula :

(H ∧ S) ⊃ R.

Now, instead of the "simple" consequent : "tomorrow will rain", we have a "complex" statement, like "if you want a coffee, [then] you’ll not have it"; it is itself a conditional, and thus we have to formalize it with : C ⊃ N.

To get the final result, we have only to replace R with it, using suitable parentheses :

(H ∧ S) ⊃ (C ⊃ N).