Quantifier scope

Question

I'm having trouble understanding predicate logic. I get very confused in interpreting the parentheses, to be able to correctly demarcate the scope of quantifiers, particularly in the following (from our unit book):

(x)(Bx → (Cx & ~Dx)) & ((∃y)(Ey & Fy) v (∃z)((~Jz & ~Kz) → (w) Mwz))

Is this correct? Can you please explain where I'm wrong? Thank you!

(x) : Scope is (Bx → (Cx & ~Dx). The x in Bx, Cx, and ~Dx, are bound to (x).

(∃y) : Scope is (Ey & Fy). The y in Ey and Fy are bound to (∃y).

(∃z) : Scope is (~Jz & ~Kz) → (w) Mwz). The z in ~Jz, ~Kz, and Mwz, are bound to (∃z).

Yes, the scope of the leftmost universal quantifier $(x)$ is (Bx → (Cx & ~ Dx)) — Mauro ALLEGRANZA, Dec 07 '19 at 12:09
And Yes and Yes (but the parentheses in your sub-formulas must match. — Mauro ALLEGRANZA, Dec 07 '19 at 12:10
@MauroALLEGRANZA Thank you! Are you able to elaborate further about the parentheses, because I'm still new to this. So, in another example: (x)((Bx → (Cx & ~Dx)) & (y)(Cx & Ly)), does (x) govern ((Bx → (Cx & ~Dx)) & (y)(Cx & Ly))? Or does it only govern ((Bx → (Cx & ~Dx)), because of the double brackets? — explorer, Dec 09 '19 at 06:35
If you have a formula $\forall x ( Px \to Qxy)$ the scope of the quantifier is explicitly stated; it is : $Px \to Qxy$. If instead you have $\forall x Px \to Qxy$ the scope is "narrow : only $Px$. — Mauro ALLEGRANZA, Dec 09 '19 at 08:19

score 1 · Accepted Answer · answered Dec 08 '19 at 21:22

1

Yes, all correct. Also, the scope of the $(w)$ is $Mwz$

answered Dec 08 '19 at 21:22

Bram28

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Thanks for pointing that out. If you have a formula (w)Mww, is it only the first 'w' that is bound (w), or are the two w's both bound to (w), since they're the same? – explorer Dec 11 '19 at 04:22
@explorer Both $w$’s would be bound to $(w)$ since both occur within the scope of $(w)$. Usually, parentheses are used to indicate the scope, as in $(w)(P(w)\to Q(w))$, but if there are no parentheses, then the scope is limited to the first term that follows it. So, with $(w)mww$ the scope is limited to $Mww$, but that would still cover both $w$’s – Bram28 Dec 11 '19 at 14:07
That's really making sense to me now. That leads me to wonder about a scenario (I hope you don't mind answering). How would you translate (x)(Mxx), if x is, say, a mother? Would it be that x is a mother herself? @Bram28 – explorer Dec 13 '19 at 05:19
@explorer it would mean that everyone is a mother of themselves. – Bram28 Dec 13 '19 at 13:46

Quantifier scope

(x)(Bx → (Cx & ~Dx)) & ((∃y)(Ey & Fy) v (∃z)((~Jz & ~Kz) → (w) Mwz))

1 Answers1