The difference between "if" and "and" in symbolic logic

Question

Let $M(x, y)$ be “$x$ has sent $y$ an e-mail message” and $T (x, y)$ be “$x$ has telephoned $y,$” where the domain consists of all students in your class. Assume that all e-mail messages that were sent are received. Use quantifiers to express each of these statements.

1. There is a student in your class who has not received an e-mail message from anyone else in the class and who has not been called by any other student in the class.

2. Every student in the class has either received an email message or received a telephone call from another student in the class.

The answer in book for 1 is: $$∃x∀y(x ≠ y → (¬M(y, x) ∧¬T (y, x)))$$ and the answer for 2 is: $$∀x(∃y(x ≠ y ∧ (M(y, x) ∨ T (y, x)))).$$

My problem is with parts $$x ≠ y\: ∧$$ and $$x ≠ y →.$$ When should I use "→" or "∧" after $x ≠ y?$ I don't understand the difference in their literature in the question. When I want to use "if" in any other question, the answer is "and" and vice versa.

Answer 1

1. • “There is a student who is furry and violet and handsome” is a simple statement of three properties of a particular existent (albeit unnamed) student $$\exists s \;\Big(Fs \land \big(Vs \land Hs\big)\Big);$$ because there is nothing hypothetical or conditional about this statement (the student's colour and looks are not dependent on their furriness), replacing the first $\land$ with $\to$ will muck up the translation.
   • “Every furry student is violet and handsome”, on the other hand, states that if a student is furry, then they must also be violet and handsome $$\forall s \;\Big(Fs \to \big(Vs \land Hs\big)\Big);$$ notice that the statement makes no claim about non-furry students, or that furry students even exist. If $\to$ is replaced with $\land,$ the meaning will become “Every student is furry and violet and handsome”.
2. • In your second exercise $$∀x∃y\;\Big(x{≠}y ∧ \big(M(y, x) ∨ T (y, x)\big)\Big),$$ we are merely exhibiting two properties of $y$ (who is claimed to exist), the first property being that $y$ is distinct from $x.$
   • In your first exercise $$∃x∀y\;\Big(x{≠}y → \big(¬M(y, x) ∧¬T (y, x)\big)\Big),$$ we are exhibiting a property of $y\:$ for the case where $\mathbf y$ is distinct from $\mathbf x.$ Changing the → to ∧ will, by additionally asserting that no student ever emails or calls themself, result in a stronger statement than was given.

Answer 2

Suppose we have a property $P$.

• When $P$ to hold somewhere, we translate it as $$(\exists y)P(y)$$ If we want to exclude a group $G$ of people, we can leave $(\exists y)P(y)$ alone…but then the risk is that someone in $G$ will satisfy $P$ and make our statement spuriously true. (You can check that, on the other hand, someone in $G$ failing $P$ is irrelevant.)

  The condition $$(y\notin G)\wedge P(y)$$ does not suffer from that problem: in $G$, it is always false, so the existential will only be satisfied outside of $G$. In (2), we want $M(y,x)\vee T(y,x)$ to hold for some $y$…except our original student $x$. So $G=\{x\}$ and $y\notin G\iff y\neq x$.

  Thus we write $$(y\neq x)\wedge(M(y,x)\vee T(y,x))$$

• When we want $P$ to hold for everyone, we translate it as $$(\forall y)P(y)$$ To exclude a group $G$ of people, here the risk is that someone in $G$ will fail $P$ and force our statement false. The condition $$P(y)\vee(y\in G)$$ does not suffer from that problem: in $G$, it is always true; outside, it still requires $P$.

  In (1),$G$ is the same as (2), but now $P(y)=\neg M(y,x)\wedge\neg T(y,x)$. So we write $$(y=x)\vee(\neg M(y,x)\wedge T(y,x))$$

  Wait! That's not what the book wrote! But $A\vee B$ is equivalent to $\neg A\rightarrow B$. Applying that identity to my solution, the book's one reappears.

Answer 3

A footnote to the other answers.

If you go https://www.logicmatters.net/ifl you can freely download the 2nd edn of my Intro. to Formal Logic. And in the (short-ish) Chs. 27 to 30, I spend a lot of time on translational issues (because getting clear about this kind of thing is the beginning of quantification wisdom!).

Also linked on the same page are very detailed worked answers to relevant end-of-chapter exercises which you might well find helpful too.