How would one write "There are teachers that teach students" using quantifiers?

Question

Regarding the statement "There are teachers that teach students" I'm unsure on how to properly write it as a proposition. The way I wrote it in my notes was:

Let: I(x) : x is a teacher

S(y) : y is a student

T(x, y) : x is teaching y

followed by:

∃x(I(x) ∧ T(x, ∃xS(y))

my logic being as "There are some teachers AND they teach some students", but I am very unsure if I am allowed to use ∃xS(y) in the middle of the proposition. Or is the ∃x quantifier before the S(y) superfluous and it would be correct to only write "∃x(I(x) ∧ T(x, S(y))"? Have I done it correctly or is there a different way one is meant to do it?

I'm not so familiar with notations, but couldn't we write $ \exists x,\exists y(I(x)\land S(y)\land T(x,y))$ ? — zwim, Apr 02 '23 at 18:47
Predicates cannot usually be substituted into variables as you did. A viable option in the spirit of what you've written is $\exists x(I(x)\land \exists y (S(y)\land T(x,y)))$. — Sassatelli Giulio, Apr 02 '23 at 21:09

score 1 · Accepted Answer · answered Apr 03 '23 at 02:47

Regarding the statement "There are teachers that teach students"

Let: I(x) : x is a teacher

S(y) : y is a student

T(x, y) : x is teaching y

$∃x(I(x) ∧ T(x, ∃xS(y))$

Correction: $$∃x\,∃y\,(Ix∧Sy∧Txy).$$

Analogously, there are even numbers that are also multiples of 3: $$∃x\,∃y\,(Ex∧My∧x=y).$$

score 0 · Answer 2 · answered Apr 02 '23 at 20:17

0

I don't think you should use $\exists x$ inside of your expression, as quantifiers are usually put in front of an expression. So I'd make it $\exists x\exists y:I(x)\land T(x,y)$. However this says that there is $\textbf{a}$ teacher teaching a student, whereas your question asks for multiple teachers.

answered Apr 02 '23 at 20:17

Je Mehr Testo Besser

98

Doesnt the existential quantifier simply mean that there is at least one of something but not limited to one? That is at least how my lecturer explained it to me but maybe that is an incomplete explination? – SERO9 Apr 03 '23 at 00:51
No, that's exactly right! But I think his question demands multiple teachers, whereas "exists" only guarantees one (but multiple are of course possible). – Je Mehr Testo Besser Apr 03 '23 at 13:57
@SERO9 Re: the singular 'exists' versus plural 'exist', there's a recent similar discussion here. – ryang Apr 12 '23 at 03:01

score 0 · Answer 3 · answered Apr 03 '23 at 05:29

0

So I think that most people here are almost entirely correct, but when I took discrete, I remember it being important that you specify that the teacher and the student are not the same person:

∃x∃y(I(x)^S(x)^T(x,y)^x!=y)

answered Apr 03 '23 at 05:29

Evan Semet

535

Your additional constraint is unnecessary, as it is conceivable that everyone is self-taught. “3 divides every multiple of 3” does imply that 3 divides itself. – ryang Apr 03 '23 at 07:15

Ryan Pierce Williams · Answer 4 · 2023-04-02T20:59:29.193

-1

Could you just define a set that contains all teachers who are teaching students and assert the size of the set is greater than 1?

$|\{ x | \exists y (I(x)\land S(y) \land T(x, y))\}| > 1$

edited Apr 02 '23 at 20:59

answered Apr 02 '23 at 20:53

Ryan Pierce Williams

533
4
14

@SassatelliGiulio the point is to express that there exist multiple such entities. Not to say you can’t do it without the use of sets and cardinality - but it’s easy to do so with them – Ryan Pierce Williams Apr 02 '23 at 21:35
@SassatelliGiulio as in you think I’ve misused something, or you just don’t like it? – Ryan Pierce Williams Apr 02 '23 at 21:39

How would one write "There are teachers that teach students" using quantifiers?

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