translating a complex-conditional into lawgic

confusedandenthused

"Without strength or endurance, the fight is all but lost."

This is #5 on the quiz on complex conditional translations to lawgic.
We are to translate this into lawgic and write the contrapositive.

So I figured for statements containing “unless, without, until” we are to negate sufficient and leave the necessary alone.
Following that formula- I got
~L—> S or E (if the fight was not lost, there was either strength or endurance)
~S and ~E—> L (if there was neither strength nor endurance, the fight was lost)

But according to the answer, the correct translations are as follows,
~S or ~E—> L ~S—> L ~E—> L
~L—> S and E

The “or”s and “and”s are switched. What did I miss in translating this?
I understand the logic of A or B —> C
That is, either A or B will get you C, and clearly that’s what’s applied in the correct translation it seems.

Does the formula (the one pertaining to unless,without, until) not apply in this situation?
If so, why does it not? Does “without” have a different meaning here?

confusedandenthused

The following is #21 from the same quiz.
“Unless you understand the homework, which cannot happen without either paying attention to the lecture, or understanding the text, or both, you will find the final on Monday extremely difficult.”

The correct translation for this is
~FED—> UH—> PAL and UT
the contrapositive of which is ~UT or ~PAL—> ~UH—> FED
But could the statement not also be translated as
~FED—> UH—> PAL or UT
( Notice the switch from and to or)

My question is, how do you go about accurately translating something like “which cannot happen without either paying attention to the lecture, or understanding the text, or both”
The “or both” seems to leave open both of the possibilities illustrated here. Any thoughts?

Alex Short

Okay, so "without strength or endurance, the fight is not lost":

Without indicates the need to negate the sufficient condition, and in this sentence we have 2. Either one condition or another condition can be met for the necessary to follow.

Negate strength:- ~S
Negate endurance: ~E

~S or ~E --> F

Where F = fight is all but lost
S = strength
E = endurance

To contrapose, we would flip and negate: ~F --> S and E

If drawn out, we could split the sufficient condition in the first translation and split the necessary in the contrapositive translation. I'm on my phone, otherwise I'd try to write it as such.

It looks as though you may not understand the nature of "without" as it applies to negating certain conditions. Much like "unless", or "either or", you have to negate the sufficient condition (or conditions). The tricky part here is that two ideas exist in the sufficient condition.

Does that clear it up?

Alex Short

I reread your post and think you changed the order of the sentence before applying the negation. In this sentence, the without is your only logical indicator (depending on whether one wishes to use "not" as such in the necessary condition).

Also: "All but lost" cannot be treated as the same as "not lost". There is a significant difference between these two about which the LSAT will test your understanding.

When you read "Without strength or endurance, the fight is not lost", you should be careful before treating that as "the fight is not lost without Stength or endurance". This can lead to errors in translating them correctly if one isn't careful.

Whatever word or phrase "Without" introduces is what will become your (after being negated) sufficient condition. Same as any "unless" you see.

fishtwentyfive

@confusedandenthused

"Without strength or (without) endurance, the fight is all but lost."

S = Strength
E = Endurance
F = the fight is all but lost

original sentence, translated:

(~ S v ~ E ) > F

do a D'Morgan equivalence:

~ ( S & E ) > F

contrapose:

~ F > ( S & E )

...
“Unless you understand the homework, which cannot happen without either paying attention to the lecture, or understanding the text, or both, you will find the final on Monday extremely difficult.”

This will be fun!

H = Understand the homework

P = Paying attention to the lecture

T = Understanding the text

F = Find the final on Monday extremely difficult

translating original sentence:

( ~ H > F ) & ( H > ( P v T ) )

...

And your question:

"My question is, how do you go about accurately translating something like “which cannot happen without either paying attention to the lecture, or understanding the text, or both”
The “or both” seems to leave open both of the possibilities illustrated here. Any thoughts?

I'm going to give you a lesson!

In logic, the word 'or' is a logical operator, just like 'not,' 'and,' 'if... then...' etc.

As a logical operator, 'or' differs from the way we use it regularly in English!

As a logical operator, 'or' is inclusive.

This means that if both sides of a disjunction (a statement containing 'or') are true, then the disjunction itself is also true. A disjunction is not true only when one side of it is true! So for example, take the sentence, "Steve had eggs or ham."
This is easily symbolized as:
E v H
But what if Steve had both eggs and ham ?
That would look like this:
E & H

So, the question arises, if the sentence, "Steve had both eggs and ham" is true, does that mean the sentence "Steve had eggs or ham" is also true? In logic, it does!

It doesn't matter if this matches with your intuition about language, because we are talking about the formal language of logic, which relies on strict norms about truth! Logic doesn't care about your intuitions!

So now take the sentence, "Steve had eggs or ham or both."

It might be your desire to translate it as:

( E v H ) v ( E & H )

But considering what we just learned, the second half of this equation is superfluous!

We only need

( E v H )

to capture the logical force of "Steve had eggs or ham or both."

The sentences ( E v H ) v ( E & H ) is actually equivalent to ( E v H ) ! This means that the sentences will be true at the same times. You can try this out if you want, and discover that indeed they will be true in the same situations!

Thus is the power of the inclusive 'or' !

I hope this helps!