I'm teaching a discrete math class at the high school level and realize that I'm fuzzier on a topic than I should be.


In their last problem set, I asked my students to translate "There is a triangle that is above every square." into formal notation.


My answer was $\exists t\forall s(\text{triangle}(t)\wedge\text{square}(s)\rightarrow\text{above}(t, s))$.


Many students made a mistake of replacing the implication with conjunction, but once I explained they were saying that all members of the domain were squares, they understood why they needed the implication.


A few students side-stepped the problem by writing $\exists t\ \text{triangle}(t)(\forall s\ \text{square}(s)(\text{above}(t, s)))$. This was inspired by a quasi-formal notation in the textbook that would have written the sentence as $\exists\ \text{triangle}\ t,\ \forall\ \text{squares}\ s,\ \text{above}(t, s)$.


What I'm not sure about is whether this is okay. Is there some notational convention that
$$ \forall xP(x)(Q(x))\equiv\forall x(P(x)\rightarrow Q(x))$$ or am I just making that up?


What confuses this somewhat is that I'm pretty sure that $$\exists xP(x)(Q(x))\equiv\exists x(P(x)\wedge Q(x))$$ under the normal understanding of the notation without any convention, so I'm afraid that I let my students get away with playing fast and loose with the notation when I shouldn't have.


Thanks!
Todd



How do quantifiers limit scope?
I'm afraid your suggested answer is wrong. Suppose $a$ is in the domain and a non-triangle. Then $$\forall s(\text{triangle}(a)\wedge\text{square}(s)\rightarrow\text{above}(a, s))$$ is vacuously true as it always has a false antecedent (recall $\land$ binds tighter than $\to$). Hence $$\exists t\forall s(\text{triangle}(t)\wedge\text{square}(s)\rightarrow\text{above}(t, s))$$ is true even if no triangle is above a square. 


What you need is 
$$\exists t(\text{triangle}(t)\wedge \forall s(\text{square}(s)\rightarrow\text{above}(t, s)))$$
or equivalently
$$\exists t\forall s(\text{triangle}(t)\wedge(\text{square}(s)\rightarrow\text{above}(t, s)))$$
Bracketing matters!


As for
$$\forall xP(x)(Q(x))\equiv\forall x(P(x)\rightarrow Q(x))$$
what's on the left is horribly ill-formed by normal standards and so is to be deprecated. But the thought that restricted universals are to be rendered using conditionals, and restricted existentials rendered using conjunctions is of course right. Your students might find the (freely available) chapters on transcription in Paul Teller's *Logic Primer* very helpful: see the first four chapters of Part 2 at <http://tellerprimer.ucdavis.edu/pdf/> 


