Diagraming a Formal Logic Sentence: John Coltrane did not play tenor sax unless he also played soprano sax.

Question

Problem

Sentence: John Coltrane did not play tenor sax unless he also played soprano sax.

 T: John Coltrane played tenor sax.
 S: John Coltrane played soprano sax.
~: negation
 ^: conjunction ("and")
 v: disjunction ("or")
 ->: conditional (if-then)
 <->: biconditional (if and only if)

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"Solutions"

The book gives the correct solution as:

 ~T v S

From what I have read, I understand that that keyword unless indicates or but it doesn't feel right to me that this is the only solution. The way I am reading the solution is that either John Coltrane does not play the tenor sax OR he plays the soprano sax. I am not arguing that this is wrong.

But, a clearer solution to me seems to be:

 S -> T

because if John Coltrane plays the soprano sax then he will play the tenor sax.

Or alternatively,

 ~S -> ~T

because if John Coltrane does not play the soprano sax then he will not play the tenor sax.

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Conclusion?

Are my solutions S -> T or ~S -> ~T incorrect, and I am misunderstanding the sentence? Or, are my solutions also relevant (and perhaps equivalencies that I have yet to formally learn)? If so, could you attempt to explain it in my layman terms?

Answer 1

Logically, ~T v S can be written as T -> S.

So, in this condition, if John played the tenor, he played the soprano.

In the original statement, he only played tenor if he played soprano. With this solution, if he played tenor, it's a sufficient condition to satisfy he played soprano, but not necessary (he doesn't need to play tenor to play soprano).

However, playing soprano is a necessary but not sufficient condition. If he didn't play soprano, he didn't play tenor, but John playing soprano is not enough to satisfy he played tenor.

In summary, it follows the logical form of a conditional operator. It's often confusing due to phrasing, but in this case it's in this order.

Answer 2

Sentence: John Coltrane did not play tenor sax unless he also played soprano sax.

Let's call this Proposition P.

T: John Coltrane played tenor sax.
S: John Coltrane played soprano sax.

The book gives the correct solution as:

1. ~T v S
   

I am not arguing that this is wrong. But it doesn't feel right to me that this is the only solution. A clearer solution to me seems to be:

2. S → T
   

because if John Coltrane plays the soprano sax then he will play the tenor sax.

(Note that two sentences with different tenses are not equivalent to each other; please don't frivolously change "played" to "plays" or "will play".)

I think you're trying to say that Proposition P means Solution 1 and Solution 2, jointly. Solution 2 (that is, S → ~(~T)) is merely conversationally implicated by Proposition P. On the other hand, Solution 1 is the literal meaning and correct formalisation of Proposition P.

Generally, the literal meaning and above implicature of ‘Y unless X’ is jointly equivalent to its often-intended (pragmatic) meaning

• Y if and only if (not X)
• equivalently: X exclusive-or Y,

which is thus a logically stronger assertion than its strict literal meaning

• $\color\red{\textbf{if (not X), then Y}}$
• equivalently: $\color\red{\textbf{X inclusive-or Y}}.$

\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}

Or alternatively,

3. ~S → ~T
   

Solution 3 is, as a matter of fact, logically equivalent to Solution 1, so is another correct translation of Proposition P. Yet another correct translation is the following:

4. T → S.
   

Summing up: Solutions 1, 3 and 4—but not Solution 2—are correct translations of Proposition P.