How to translate "No dolphin sings unless it jumps" into predicate logic?

Question

i have a silly logic question again. How would you translate the following sentence into predicate logic?

No dolphin sings unless it jumps.

My guess is that it is an E-sentence of the form "no A is B", where A stands for "dolphin" and the predicate B stands for "is such that it sings unless it jumps". But how can we analyze B? It seems that the predicate B has the form "p unless q" which can be symbolized as "∼q → p". That's why the predicate has the can be analyzed as "∼Fx → Gx", with "Fx" meaning "x jumps" and "Gx" meaning "x sings". This leads to the symbolization:

∼∃x (Kx ∧ (∼Fx → Gx))

But this is incorrect! The correct solution is the following:

∼∃x (Kx ∧ ∼(Gx → Fx))

What am I doing wrong? I'm a bit confused in figuring out the structure of the predicate. Thanks for your help again!

Answer 1

Because of the double negation (no + unless) I would first try to paraphrase the sentence before trying to translate it into predicate logic. All the following have the same meaning:

No dolphin sings unless it jumps

All dolphins don't sing unless they jump

All dolphins don't sing if they don't jump

All dolphins jump if they sing

The last one is easiest to translate. The translation is:

∀x(Kx → (Gx → Fx))

Which is equivalent to:

∼∃x∼(Kx → (Gx → Fx))

And thus:

∼∃x(Kx ∧ ∼(Gx → Fx))

Answer 2

In any translation process there is a "source" lilly-pad and a "destination" lilly-pad. Like a frog, we leap from lilly-pad to lilly-pad.

Below is a numbered list.
Item 1 is the original/unmodified English, "no dolphin sings unless it jumps"
The last numbered item is a statement written in predicate logic.

1. No dolphin sings unless it jumps
2. There does not exist a dolphin which sings unless it also jumps.
3. There does not exist a dolphin which sings and does not jump.
4. There does not exist d element of the set of dolphins, such that d sings and d does not jump

DIGRESSION: Technically, predicate logic does not contain "sets".
predicate logic is considered to be simpler than set theory.
predicate logic is like a simple machine with a very small buttons.
Set theory is like the same machine with a dozen new additional extra bells, whisles, buttons, levers and gauges added to the original ones.
Instead of "sets" sometimes people will say that predicate logic contains "relations"
So, you might want to write one of the following instead of "d element of the set of all dolphins":

• "d in the dolphin relation"
• "d element of the dolphin relation"
• "d participates in the dolphin relation"

Let us continue with the translation:

5. ∄ d ∈ D such that d sings and d does not jump...... D is the dolphin relation

DIGRESSION: ∈ is short-hand for "element of"

6. ∄ d ∈ D such that d sings and d does not jump...... D is the dolphin relation

7. ∄ d ∈ D ϶ d sings and d does not jump

DIGRESSION: ϶ is short-hand for "such that"
϶ is looks like the symbol for "element of" written backwards.
϶ is old notation. ϶ is no longer in vogue.
For the last decade or more, American mathematicians have started using the colon character : to denote "such that"

8. ∄ d∈D ϶ SING(d) and not JUMP(d)
   We define SING(d) to be notation equivalent to "d sings" We define JUMP(d) to be notation equivalent to "d jumps"

At this point, let us decide to be consistent in notation. Either always write something like A(x) or x∈A, but not mix and match the two notations.

9. ∄ D(d) ϶ SING(d) and not JUMP(d)

9. ∄ d∈D ϶ d ∈ SING and not d∈JUMP

Notice that there are two lines 9s. They are the same in all but notation. Pick whichever notation is closest to what your college professor uses.

If you want, you can define SING as the set (excuse me... "relation") of all things which sing.

Similarly, you can define JUMP as the relation of all objects in the universe which jump.

Answer 3

Yes, "No dolphin sings unless it jumps" is interpreted in predicate logic straightforwardly as:

Is_Dolphin(x) → (Sings(x) → Jumps(x))

Here the character '→' denotes a logical implication, not a material implication.