Unless Equation

[scharles]

I have a question regarding a problem I found online, "Diagram M or Q unless Y."

Following the Powerscore Method, my diagram is:

Diagram: M or Q (slash)-->Y
Contrapositive: Y (slash) --> Q and M


However, the answer is:

Diagram: Y (slash) -->M or Q
Contrapositive: Q and M (slash) --> Y

(1) Am I doing something wrong??
(2) Lastly, does the Unless Equation work for all instances of "unless, until, without and except" for conditional statements? Or are there variations for more difficult cases? I haven't yet completed the book..I'm finishing the conditional reasoning chapter now.


Thanks for your time and help!!

S

[Steve Stein]

Hi S,

Thanks for the question. In applying the Unless Equation, remember that whatever follows "unless" simply becomes the necessary condition as-is; the remainder is negated and becomes the sufficient condition. The necessary condition is easy: it's just "Y." The sufficient condition is more challenging, and it is the negation of "M or Q."

The logical opposite (or negation) of "M or Q" is "No M and No Q." Remember, when you negate an "or" statement, the "or" becomes "and," and both the variables are negated. So, using the Unless Equation, the original diagram of the statement "Diagram M or Q unless Y" is:


M
and Y
Q


When we are told that M or Q will be there unless Y is there, that means that the only way that one or the other won't be there is if Y is there. In other words if both are absent, Y must be present.

Next, when you take the contrapositive, you reverse and negate both terms, thus arriving at the result that if Y is not there, then M or Q (or both) must be there:


M
Y or
Q


Also, in answer to your two questions,

• 1. It looks like you didn't negate the sufficient condition when you made you first diagram, and that caused problem when you went to the contrapositive too.
  
  2. Yes, it always works

I hope that's helpful—please let me know whether this is clear—thanks!

Steve

[SGD2021]

Hello, I am also looking at the Unless Equation section of chapter 6. For the example on page 95 ("There can be no peace without justice"), following the Unless Equation and according to the Logical Reasoning bible, the statement becomes: "if peace occurs, there must be justice." (P->J). However, I am confused as to why this works since, wouldn't the new sentence be an example of a "Mistaken Negation," which are always invalid statements?

Are the "mistaken negations" and "mistaken reversals" defined on page 185 always invalid statements?

[Adam Tyson]

Mistaken Reversals and Mistaken Negations are always invalid inferences, SGD2021, but the Unless Equation doesn't lead to one of those mistakes. "There can be no peace without justice" is properly interpreted as meaning "if there is peace, there is justice" and also the contrapositive of that statement, which is "if there is no justice, there is no peace."

The word "without" indicates a Necessary Condition, but that condition is taken as is and is not negated. You might be thinking of "without" as being a negative, and that's why you're seeing this as a Mistaken Negation. But the word "without" in this case doesn't negate anything, it just indicates that what comes after it (justice) is necessary.

An alternative approach to the Unless Equation is what some folks might call the "if not" approach. Using that technique, those four special indicators act like the phrase "if not," and what follows gets negated and is the sufficient condition. The other term is necessary and doesn't get negated, but stays as it was in the original statement. Using that approach, you would set "no justice" as the Sufficient Condition and "no peace" as Necessary, and you would get the contrapositive above. Since the contrapositive of a statement is logically equivalent to the original statement, you've arrived at the exact same place! So perhaps in this case you would find the "if not" approach to be more intuitively correct?

[kbreann01]

Hello - I am wondering if the word "or" applies in this situation as well (the Unless Equation). 7Sage material states that it does. What is your opinion on this? Thank you.

[Luke Haqq]

Hi kbreann01!

If I understand your question correctly, you're asking if the word "or" is an indicator letting you know that there is an "unless" statement to be diagrammed.

If that's the question, then no, "or" isn't a cue for there being "unless" conditional reasoning present.

If I've misunderstood your question, of course feel free to clarify or provide an example.