“You cannot... unless...” and “You can... only if...”

Question

I am having a hard time translating implications from English to logic. It seems that a lot of the time, $P {\implies}Q$ is the same as $Q {\implies}P.$

“You cannot ride the roller coaster if you are under $4$ feet tall unless you are older than $16$ years old.” Let $X, Y,$ and $Z$ represent “You can ride the roller coaster,” “You are under $4$ feet tall,” and “You are older than $16$ years old,” respectively. Then the sentence can be translated to

$(Y \wedge \neg Z) \implies ¬X$

“You can access the Internet from campus only if you are a computer science major or you are not a freshman.” We let $A, C,$ and $F$ represent “You can access the Internet from campus,” “You are a computer science major,” and “You are a freshman,” respectively. Noting that “only if” is one way a conditional statement can be expressed, this sentence can be represented as

$A \implies (C \vee \neg F )$.

Why can't we say $(C \vee \neg F ) \implies A$? These English sentences both seem to say "you can/cannot access a particular thing if you meet / don't meet certain requirements", yet the goal $A$ is on the left hand side for the first example while the goal $\neg X$ is on the right hand side for the second example. I have looked around for answers but nothing has seemed to clear this up for me. It seems like you could reverse the implications. How do I know that it is $P {\implies}Q$ instead of $Q {\implies}P$?

Answer 1

If "you can access the internet from the campus only if you are a computer science major or a freshman" is a true sentence then also the following sentence is true: "you can access the internet from the campus only if you are a computer science major".

You are dealing with necessary conditions. They do not have to be sufficient.

However, in daily language mostly all necessary conditions are mentioned (or are supposed to be mentioned).

That makes the "bundle of conditions" sufficient after all.

The fact that we tend to expect that is the source of confusion.

Answer 2

Your confusion is actually not about the direction of implication, but due to ‘unless’ and ‘only if’ being particularly tricky to translate, which is in turn because the colloquial meaning of each tends to be loose.

1. Strictly speaking, ‘Y unless X’ says precisely that

   • if X is untrue, then Y must be true
   • $\lnot X\implies Y.$

   As such, all these have the same literal (denotative) meaning:

   • Y unless X
   • Y if not X
   • X or Y
   • not neither X nor Y
     (i.e., at least one of X and Y)

   Note that satisfying X does not preclude Y from holding; the directive ‘don't compliment unless you mean it’ isn't saying that a compliment must be paid whenever it is sincere.

   In everyday English, however, ‘Y unless X’ may, by pragmatic inference, additionally carry the conversational implicature ‘if X, then not Y’ and, therefore, shift in meaning to ‘Y if and only if not X’, that is, ‘either X or Y but not both (i.e., exactly one of X and Y)’.

   Summing up: ‘Y unless X’ literally means X inclusive-or Y, even as it is often (pragmatically) intended to mean the stronger assertion ‘X exclusive-or Y’.

2. In deductive reasoning, ‘X only if Y’ says precisely that

   • if Y is untrue, then X must be untrue
   • $\lnot Y\implies \lnot X$
   • X implies Y.

   Note that X actually needn't hold when condition Y is satisfied.

   ‘X only if Y’ technically does not mean ‘X if Y; otherwise, not X’, that is, ‘X if and only if Y’.

(You cannot ride the roller coaster if you are under $4$ feet tall) unless you are older than $16$ years old.

$$\big(\text{Height}<4 \;→\; \text{Can't Ride}\big)\;∨\; \text{Age}>16\\ \big(\text{Height}\geq4 \;∨\; \text{Can't Ride}\big)\;∨\; \text{Age}>16\\ \big(\text{Height}\geq4 \;∨\; \text{Age}>16\big) \;∨\; \text{Can't Ride}\\ \lnot(\text{Height}<4 \;\land\; \text{Age}\le16 \big) \;∨\; \text{Can't Ride}\\ \big(\text{Height}<4 \;\land\; \text{Age}\le16 \big) \;→\; \text{Can't Ride}.$$

Kanti will not attend unless Jasmine does.

$$\lnot J\to (\lnot K)\\K\to J.$$

You can access the Internet from campus only if (you are a computer science major or you are not a freshman).

$$\text{Can Internet} \;→\; \big(\text{Is CS Major} \;∨\; \text{Isn't Freshman}\big).$$

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Addendum \begin{array}{|c|c|c|c|c|c|} \hline X & Y & \lnot X & \color\red{\lnot X \rightarrow Y} & \color\red{X \lor Y} & \text{Equivalent?} \\ \hline 0 & 0 & 1 & 0 & 0 & \color\red\checkmark \\ 0 & 1 & 1 & 1 & 1 & \color\red\checkmark \\ 1 & 0 & 0 & 1 & 1 & \color\red\checkmark \\ 1 & 1 & 0 & 1 & 1 & \color\red\checkmark \\ \hline \end{array}

$$¬X → Y ≡ ¬Y → X ≡ X ∨ Y ≡¬(¬X ∧ ¬Y).$$

Answer 3

This is, as Element118 has said, due to the difference between "if" and "only if" which is annoying to wrap your head around at first, but makes sense once you get it. The point is that the sentences

if $P$ then $Q$

and

$P$ only if $Q$

are two ways of representing the same thing. They both can be expressed as

$P\implies Q$

so it's really just a subtlety of the language used. If I misunderstood and that's not where your problem lies, just let me know and I'll update this answer