# How to translate 'azaleas bloom if and only if they are fertilized' into symbolic logic?

Source: p 460, A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley

Statement: [S1.] Azaleas [(Wiktionary)] bloom if and only if they are fertilized.
Symbolic translation: [S2.] (x) [ Ax ⊃ (Bx ≡ Fx) ]

1. I translate the statement into a Standard Form Categorical Proposition (which necessitates a Quantifier, Subject, Copula, and Predicate):

S3. All azaleas are azaleas that bloom, iff, all azaleas are azaleas that are fertilised.

1. I abbreviate each term as follows: Ax = azaleas; Bx = Azaleas that bloom, Fx = Azaleas that are fertilised. Then S3 becomes:

S4. All Ax are Bx, iff, all Ax are Fx.

S5. (x) [ (Ax ⊃ Bx) ≡ (Ax ⊃ Fx) ]

Why does my S5 differ from S2? Where did I err?

• Translate your S5 back into English (or French) piece by piece. ça dit quoi? Does it come out as being like the original sentence? Jan 3, 2016 at 6:15
• Note that Fx should be "X is fertilized," not anything about azaleas Jan 3, 2016 at 13:28
• @ChristopherE So should S3 be: All azaleas are THINGS that bloom, iff, all azaleas are THINGS that are fertilised.?
– user8572
Jan 3, 2016 at 20:22
• S5 does equal S3, that is not the point, compare S3 to S1.
– user9166
Jan 3, 2016 at 20:55

Their S1.

[All] azaleas bloom if and only if they are fertilized

lets me deduce that if I have an azalea, and I fertilize it, it will bloom.

[All] azaleas [are azaleas that] bloom if and only if all azaleas are [azaleas that are] fertilized

does not imply this. S3 only implies that if all the azaleas in the world are fertilized, then the one I have chosen will bloom, because all of them will bloom. It does not assure me that if my neighbor neglects his azaleas, mine will not be affected.

So these do not mean the same thing from the very first step.

You have introduced two separate 'Alls' where there is only one.

In S1: The nature of the pronoun (they) is to protect its antecedent 'All azaleas' (to the left of the 'iff'), not to choose another instance of new azaleas (separate from 'All azaleas' introduced to the left of the 'iff') at random. So "when they are fertilized" means "when those same azaleas are fertilized".

In your S3: By writing 'all azaleas' again instead of using a pronoun to the right of 'iff', you are wrongly referring to the whole class (of azaleas) named, because you should refer to only the azaleas already introduced to the left of 'iff'.

So

All azaleas bloom, when they (those same azaleas) are fertilized

Is not grammatically similar to

All azaleas bloom, when all azaleas are fertilized.

The former means something like

(x)(Azalea(x) -> (Blooms(x) <- Fertilized(x)))

Naturally, without any noted link via a name or pronoun, each 'all' would get its own free variable.

So this latter would become:

(x)(Azalea(x) -> Blooms(x)) <= (y)(Azalea(y) -> Fertilized(y))

which kind of imagines some magical communion of all azaleas that can determine whether azaleas in far away lands got fertilized or not...

• Thanks. 1. Will you please explain what the pronoun 'they' in S1 means? I think that my failure (to identify its antecedent) initiated all the other problems. 2. Will you please explain how to rewrite S1 correctly into Standard Form? How SHOULD S3 be stated?
– user8572
Jan 3, 2016 at 22:01