CHALLENGE: Help Decipher Logical Reasoning Q Forum

[logicallogic]

Hello all,

Below is a statement that I would very much appreciate help with in terms of understanding (necessary/sufficient conditions, contrapositive, takeaways, etc.)

Here is the statement:

This transition to a state of jello cannot take place without water or collagen.

Does this mean:

(a) this phrase in if/then format is: not jello --> not water or not collagen
(b) the contrapositive is: if water and collagen --> jello

All tips and explanations for how to address negatively framed statements with multiple necessary conditions would be greatly appreciated!

[tls1885]

The way you wrote it is actually an incorrect reversal. "Cannot" introduces a necessary condition.

The proper form would be:
a) If jello then water and collagen
b) if not water or collagen then not jello

[logicallogic]

Thank you! A couple follow ups and requests to elaborate so I can be certain I understand.

First, what difference exists if the phrase was "This transition to a state of jello cannot take place without water AND collagen." (I would venture: if jello then water or collagen; contra: if not water and not collagen then not jello)

Second, could you explain how you derived (a)? How did you know to render "water" and "collagen" in the positive sense? I assume that the necessary conditions need to be negated which (a) accounts for OR to AND, and (b) the relationship between "cannot" and "without" causes the "not water" and "not collagen" to simply become "water" and "collagen" (two negatives = positive). Correct?

Third, why does "cannot" introduce the necessary conditions as opposed to "without" or does it matter?

Finally, do you have any additional information regarding "incorrect reversals"?

Again, very appreciative for the help.

[Platopus]

.

[logicallogic]

Platopus, looks like your reply got cut off.

Still very much looking forward to receiving a (bit) more of an explanation as to how tls1885 arrived at "a) If jello then water and collagen
b) if not water or collagen then not jello"

[IvoryTowerTP]

Thank you! A couple follow ups and requests to elaborate so I can be certain I understand.

First, what difference exists if the phrase was "This transition to a state of jello cannot take place without water AND collagen." (I would venture: if jello then water or collagen; contra: if not water and not collagen then not jello)

The original seems a little poorly written, and it's kind of unclear to me whether they originally intended to mean:
1) 'This transition to a state of jello cannot take place without both water and collagen.'
or
2) 'This transition to a state of jello cannot take place without at least one of these two things: water or collagen'.

Knowing how jello works, I think it'd be safe to say that they meant 1). But if you're just looking at the or and the without and thinking on a granular word-level with no content knowledge, it's not clear.

1) means that each thing, water and collagen, is needed, and that lacking either will prevent the transition. Consider the equivalent "Unless you take the LSAT and fill out an application, you won't be accepted to a law school". Each is needed, lacking either you're not going to be accepted.

2), on the other hand, means that you can substitute water for collagen or collagen for water, but if you don't have either, your jello won't work. It's like saying "My recipe won't work unless baking powder or baking soda is used as a leavening agent.' It doesn't matter which I use, but my recipe won't work without using one of them.

In formal logic shorthand, 1) is: transition → water AND collagen; contrapositive: ~water OR ~collagen → ~transition.
In formal logic shorthand, 2 is: transition → water OR collagen; contrapositive: ~water AND ~collagen → ~transition.

[logicallogic]

Thank you! A couple follow ups and requests to elaborate so I can be certain I understand.

First, what difference exists if the phrase was "This transition to a state of jello cannot take place without water AND collagen." (I would venture: if jello then water or collagen; contra: if not water and not collagen then not jello)

The original seems a little poorly written, and it's kind of unclear to me whether they originally intended to mean:
1) 'This transition to a state of jello cannot take place without both water and collagen.'
or
2) 'This transition to a state of jello cannot take place without at least one of these two things: water or collagen'.

Knowing how jello works, I think it'd be safe to say that they meant 1). But if you're just looking at the or and the without and thinking on a granular word-level with no content knowledge, it's not clear.

1) means that each thing, water and collagen, is needed, and that lacking either will prevent the transition. Consider the equivalent "Unless you take the LSAT and fill out an application, you won't be accepted to a law school". Each is needed, lacking either you're not going to be accepted.

2), on the other hand, means that you can substitute water for collagen or collagen for water, but if you don't have either, your jello won't work. It's like saying "My recipe won't work unless baking powder or baking soda is used as a leavening agent.' It doesn't matter which I use, but my recipe won't work without using one of them.

In formal logic shorthand, 1) is: transition → water AND collagen; contrapositive: ~water OR ~collagen → ~transition.
In formal logic shorthand, 2 is: transition → water OR collagen; contrapositive: ~water AND ~collagen → ~transition.

Thank you for taking the time to explain. I would agree with you that the statement appears to be confusingly written, but that's what I am dealing with. That said, the truth value of a statement functions independent of its validity from a formal logic standpoint, correct? Although we know, in reality, both water and collagen are necessary components of making jello, this would have no impact on how the phrase is actually written and accepting it as written. At least, that's my understanding. Therefore, 2) seems correct just accepting the words as written. Would you agree?

[IvoryTowerTP]

Thank you for taking the time to explain. I would agree with you that the statement appears to be confusingly written, but that's what I am dealing with. That said, the truth value of a statement functions independent of its validity from a formal logic standpoint, correct? Although we know, in reality, both water and collagen are necessary components of making jello, this would have no impact on how the phrase is actually written and accepting it as written. At least, that's my understanding. Therefore, 2) seems correct just accepting the words as written. Would you agree?

The words as written aren't clear, and I don't think you'd find anything that vaguely written on the LSAT.

[logicallogic]

Thank you for taking the time to explain. I would agree with you that the statement appears to be confusingly written, but that's what I am dealing with. That said, the truth value of a statement functions independent of its validity from a formal logic standpoint, correct? Although we know, in reality, both water and collagen are necessary components of making jello, this would have no impact on how the phrase is actually written and accepting it as written. At least, that's my understanding. Therefore, 2) seems correct just accepting the words as written. Would you agree?

The words as written aren't clear, and I don't think you'd find anything that vaguely written on the LSAT.

That does not answer the question of whether the validity functions independent of the truth value. If I'm not mistaken (and correct me if I am wrong), your issue appears to be the fact you know (beforehand) that jello does require both. Not that the actual statement is vague. Correct?

Here is another statement, similarly worded, that I'd like your input on:

The shower cannot take place without water or a desire for it.

How would you render this in terms of formal logic? Note: a shower could theoretically take place if someone does not desire it (e.g., handicapped, unconscious, etc.).

I do appreciate your insights thus far.

[IvoryTowerTP]

I'm not quite sure the point you're aiming at. Truth value and ambiguity are separate statements. Take the specifics completely out. If I say:

no X without Y or Z,

that doesn't provide the context necessary to know if it's

no X without both Y and Z

or

no X without either Y or Z

In the shower example, we'd say the context clearly makes that or into the first one. Same with Jello. Without that context, a single or is not clearly one of the two options provided.

[Mikey]

Here is another statement, similarly worded, that I'd like your input on:

The shower cannot take place without water or a desire for it.

How would you render this in terms of formal logic? Note: a shower could theoretically take place if someone does not desire it (e.g., handicapped, unconscious, etc.).

So water or a desire for it are the necessary conditions. the way you worded it, namely 'or' is confusing, as it did for others above. is or what you meant? or did you mean for both to be used as the nec condition?

if it said 'the shower cannot take place without water AND a desire for it' then you would NEED both in order for the sufficient condition to happen.

Shower --> Water & desire for it
water --> shower
desire for it --> shower

if it said 'or' like in your original statement then I believe you just need 1 of them to fulfill the necessary condition. If you have water but no desire (or vice versa), then you are good for the nec condition.

[logicallogic]

I'm not quite sure the point you're aiming at. Truth value and ambiguity are separate statements. Take the specifics completely out. If I say:

no X without Y or Z,

that doesn't provide the context necessary to know if it's

no X without both Y and Z

or

no X without either Y or Z

In the shower example, we'd say the context clearly makes that or into the first one. Same with Jello. Without that context, a single or is not clearly one of the two options provided.

Why would the context be necessary from the pure logic standpoint? If you know only "no X without Y or Z" are you saying that "without" is always going to be vague" I don't understand that. Why not just apply the normal rules, accept the words as literally written, and continue? Hence "if not Y and Z, then not X / If X, then Y or Z."

And, also, the shower example is not clear because, as mentioned, someone could receive a shower without a desire for a shower (e.g., unconscious, handicapped, in prison, etc.).

[logicallogic]

Thank you! A couple follow ups and requests to elaborate so I can be certain I understand.

First, what difference exists if the phrase was "This transition to a state of jello cannot take place without water AND collagen." (I would venture: if jello then water or collagen; contra: if not water and not collagen then not jello)

The original seems a little poorly written, and it's kind of unclear to me whether they originally intended to mean:
1) 'This transition to a state of jello cannot take place without both water and collagen.'
or
2) 'This transition to a state of jello cannot take place without at least one of these two things: water or collagen'.

Knowing how jello works, I think it'd be safe to say that they meant 1). But if you're just looking at the or and the without and thinking on a granular word-level with no content knowledge, it's not clear.

1) means that each thing, water and collagen, is needed, and that lacking either will prevent the transition. Consider the equivalent "Unless you take the LSAT and fill out an application, you won't be accepted to a law school". Each is needed, lacking either you're not going to be accepted.

2), on the other hand, means that you can substitute water for collagen or collagen for water, but if you don't have either, your jello won't work. It's like saying "My recipe won't work unless baking powder or baking soda is used as a leavening agent.' It doesn't matter which I use, but my recipe won't work without using one of them.

In formal logic shorthand, 1) is: transition → water AND collagen; contrapositive: ~water OR ~collagen → ~transition.
In formal logic shorthand, 2 is: transition → water OR collagen; contrapositive: ~water AND ~collagen → ~transition.

I understand your concerns about vagueness. However, if "OR" is being used, then it seems that such a word should be accepted in the "either/or" sense when an "unless" or "without" are present. Disagree? I cannot conceive of an IF/THEN statement using without/unless that uses the OR in an inclusive sense. The only time such statements (at least) appear to make the OR inclusive is when we possess background content knowledge (like in the case of jello).

If I said: No widgets without bumpers or boppers -- how you would render that in terms of formal logic? Or is your general position that "without/unless" must be supplemented with an "either" or "both" or some background knowledge to make any such statement decipherable?

I do appreciate these insights and thoughts. Thanks again.

[IvoryTowerTP]

Why would the context be necessary from the pure logic standpoint? If you know only "no X without Y or Z" are you saying that "without" is always going to be vague" I don't understand that. Why not just apply the normal rules, accept the words as literally written, and continue? Hence "if not Y and Z, then not X / If X, then Y or Z."

There is no such thing as a 'pure logic standpoint'. Systems of notation in logic have their own rules for how statements should be interpreted. While you could design your own set of notation in which 'No X without Y or Z' is always intended to mean 'Y and Z are not jointly required, though the absence of both would be sufficient to indicate the condition X does not hold' that'd be your rule set, not an absolute standard.

I don't know where your examples are coming from, but they're too ambiguously phrased for the LSAT to make a right answer turn on one interpretation or the other.