Embedded Conditionals

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nessa.k13.0

Hi @generationhar embedded conditionals usually occur when a conditional statement is inserted as part of a sufficient or necessary condition. An example of that is: If cats like dogs then my terrier will not bite the kitten unless it is angry. This gives rise to so many more worlds than a more simple conditional. (CD)----->(/A----->/B) Yes, there's more room for error if one translates the phrases incorrectly. The best way to learn not to translate incorrectly would be to check out the lesson on embedded conditionals https://7sage.com/lesson/mastery-embedded-conditional/
When I see a conditional phrase in a statement on the LSAT, I translate each statement within each section of the conditional phrase and then apply the conditional phrase indicators (for the larger sentence) to those phrases.

nessa.k13.0

Edited the first term in the conditional statement^

Cant Get Right

The really tricky part is when you start toying around with taking the contrapositive. While negating an entire conditional phrase is functionally the same as negating an isolated condition, it can be difficult to wrap your head around at first.

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Q.E.D

Hm that's interesting.

The contrapositive of your schema:
(~M & ~B) -> ~P

But your vernacular feels more like a biconditional to me:
P->(M<->~B)

'Reason being it seems to assert that, indeed, Mike WILL be there in the abscence of the birds, in addition to the birds guaranteeing Mike's abscence, so the abscence of one is necessary and sufficient for the abscence of the other.

In that case, the contrapositive:
[(M & or (~M & ~B)] -> ~P

- which has a long form of the negated biconditional:
(M<->B) -> ~P

Finally, just want to remark on the potential for scope ambiguity in the vernacular. You could read it to mean:
(P->M) <-> ~B

In that case, the birds being at the park would guarantee your own presence there (without seeing Mike). 'Very influential birds, at any rate.

But that comma really clears it up by placing the "unless" grammatically within the consequent of the leading conditional.

Cant Get Right

@generationhar said:
I will see Mike unless the birds are there.

In English, you can simply negate this by saying "It is not the case that I will see Mike unless the birds are there." In logic, this would look like: ~(~M -> . You're not just negating the terms, you're negating the relationship between them. In a sense, you are negating the arrow. That's how I like to think of these anyway.

nessa.k13.0

The contrapositive of P --->(~M--->B) is ~B---> M or ~P
Why did I get that? I think the best way to start is to first simplify the statement P --->(~M--->B) before figuring out the contrapositive. To do that, take the embedded phrase (the one in the parenthesis) and move it outside of the parenthesis by changing the first arrow to an "and" and keeping the other terms the same. So it changes from P---> (~M--->B) to P and ~M --->B (or P and ~B ---> M). Which means: if I go to the park and do not see Mike, then the birds are there.
How/why did we do that? What's happening here is we now have two sufficient terms that should occur to give us the necessary condition (which is going to the park and not seeing mike) being sufficient to the birds being there. Below I provided the logistic proof for why/how, using the negate sufficient "or" rule with the arrows you can arrive at the simplified answer that I did. Use the "or" arrow rule (which is negating the sufficient term that you remove the arrow for along each step in the proof starting with the original statement) the following is just the proof and a step by step application of the arrow to or translation rule.
P --->(~M--->B)
~P or (~M--->B)
~P or (M or
[~P or M] or B
not[~P or M] --->B *apply De Morgan's law (basically two negations cancel each other out)
P and ~M ---->B

The contrapositive of the last logical statement (our simplified statement) is ~B---> M or ~P
(don't forget that you switch "and" with "or" when getting the contrapositive).

Hmm @Q.E.D Doesn't P--->(~M-->B) mean going to the park is sufficient for not seeing Mike, while not seeing Mike is sufficient for birds being there? Or stated another way P--->(~B--->M) : going to the park is sufficient for the birds not being there, which is sufficient for seeing Mike? The relationship of the two terms B and M are such that they must exist in an oppositely negated relationship to each other. B and M are both not going to be positive or both be negative in the same statement because the term "unless" in the provided sentence requires that we negate one of those terms and make it a sufficient condition for the other.

Q.E.D

Hey @nessa.k13.0

Cool stuff you're putting out there. I wish more people had interests like yours.

'Just gonna go ahead and hit these prompts. First, I couldn't agree more with your thoughts on the biconditional:

B and M are both not going to be positive or both be negative in the same statement because the term "unless" in the provided sentence requires that we negate one of those terms and make it a sufficient condition for the other.

And in response to these comments...

Doesn't
[1] P--->(~M-->B)
mean going to the park is sufficient for not seeing Mike, while not seeing Mike is sufficient for birds being there?

I read (1) to mean P is sufficient for the sufficiency of ~M for B. P guarantees you can't have ~M without B, though possibly M is true - in which case (on this schema), either B or ~B. Here's where the familiar LG-type chains, though fantastic, don't capture critical scope ambiguities in cases like this. In the games, you often see things like P->~M->B, where every item is sufficient for the next individual item down the line, as in the quoted interpretation. In sentence logic, however, you often have to explicitly set the scope of each conditional to apply to a connective (e.g. ->) rather than a propositional atom, as in (1), where the necessary condition doesn't itself guarantee the truth of anything.

Here is some scope syntax totally lost on the straight chain:
P--->(~M-->B)
(P--->~M)-->B

In that vein, I would add about the proof above that the endpoint has a scope ambiguity.

'P and ~M ---->B'
could be

(2) (P and ~M) ---->B or
(3) P and (~M ---->B),

...where I'd advocate for (2).

Awesome proof, btw.

Best,
QED

nessa.k13.0

Damn y'all are intense haha. I wasn't ready to go that far in the other day, but it was a great exercise to shore up my understanding. @Q.E.D cool stuff, I concur

quinnxzhang

There's a lot of confusion in this thread. The contrapositive of 'P -> (~M -> ' is straightforwardly '~(~M -> -> ~P'. With one application of De Morgan's, this is equivalent to '(~M & ~B) -> ~P', as @Q.E.D said. I don't see the point of going through more convoluted steps and ending up with an unnatural expression.

I think a lot of the confusion here stems from the fact that it's unclear whether "unless" expresses a conditional or a biconditional. For LSAT purposes, these are treated as straightforward conditionals, but in ordinary language, they can behave like biconditionals. This isn't a settled issue among logicians, but I think the trend is towards interpreting "unless" as a conditional and explaining instances that appear to be biconditionals as implicatures.

nessa.k13.0

Hey @quinnxzhang ! I don't think that's how you apply De Morgan's law though. When you negate a negative term you get a positive term in the sense that it is either positive or not positive--negative. Your final M should be positive. The point of going through more steps, at least for me, is to communicate what I'm doing in an attempt to understand why I do it. This is opposed to just doing something and not understanding why.

Q.E.D

Jeece, busy thread.

@nessa.k13.0
Remember that '(A -> ' is equivalent to '(~A or ', so the '~(A -> ' is the same as '~(~A or ', which begets '(A and ~B)'. Plugging '~M' and 'B' in for those schematics gets us from '~(~M -> ' to '~(~~M or ', hence '~(M or ' and '(~M and ~B)'.

It also makes sense to remember that a conditional is only false when the antecedent is true and the consequent is false, as in '(A and ~B)' above. Incidentally, you'll never see a bald negation of a conditional connective on the LSAT, e.g. one that translates as '~(A -> ', because the material conditional we're negating in this thread doesn't quite capture natural language conditionals. That is, an LSAT prob might set us up to infer '~(A -> ' from '(A and ~B)', but never the other way around. Consider this example:

(1) It's not true that if Jones is arrested then he'll confess.

You would schematize it as '~(A -> ', which translates to '(A and ~B)'. Tense aside, classical logic would have us conclude that John was in fact arrested and did not confess, whereas we don't naturally see any commitment to these apparent facts in English. Turns out only non-classical logics have the expressive power to approximate most natural language conditionals. From a formal point of view, it seems the LSAC test writers have some uncomfortable limitations in how they can write questions.

Excuse me. I get excited about logic. On that note, I need to know how quinnxzhang knows that

This isn't a settled issue among logicians...

Do you follow the literature on symbolic logic, @quinnxzhang? Your awareness of correct use-mention notation, as in

straightforwardly '~(~M -> -> ~P'

has analytic philosophy written all over it.

Anyway, good luck on the LSAT to all. 'Fun thread.

nessa.k13.0

Wait nevermind! After reading that section with the conditional statement again I see that @quinnxzhang applied De Morgans and then went to the end result with "&" not "or". Gotcha!

quinnxzhang

@Q.E.D said:
Do you follow the literature on symbolic logic, @quinnxzhang?

I focus more on philosophy of language, but I've TA'd both first-order logic and mathematical logic courses. The logic literature I read tends to bend towards the philosophical side. For example, I'm a big fan of John MacFarlane -- I think his paper on formality is great, and I think his paper on modus ponens is way better than Vann McGee's.

@Q.E.D said:Your awareness of correct use-mention notation

I do care about use-mention, but this is less about use-mention than it is about separating object language and metalanguage.

Q.E.D

@quinnxzhang

MacFarlane -- I think his paper on formality is great

Is this the paper you're referring to?
http://fitelson.org/probability/macfarlane.pdf

It looks solid, man. I haven't seen a lot of literature on the demarcation problem in logic. I think I'll dig into this.

Looks like it covers 2nd order logic. I'm aware that Quine considered it "set theory is sheep's clothing" and bumped it across that demarcation line out of logic, but I never researched the controversy.

Nice crossing paths.

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Q.E.D

Hey @Thoughtful, I assume you're talking about '~(~M -> '. If so, notice that the negation is outside the parentheses, acting on the truth value of the whole sentence '(~M -> '. The truth value of that sentence is determined by the conditional truth function '-->'. The conditional takes in the truth values of the antecedent '~M' and consequent 'B', and then it spits out either true or false.

Remember the definition
of the material conditional (A --> C):

True when A is false and C is true;
True when A is false and C is false;
True when A is true and C is true;
False when A is true and C is false.

So plug in '~M' and 'B' for A and C in the definition. To negate
'(~M -> ' is to say it's false, and there's only one condition where the '-->' is false, namely when '~M' is true and 'B' is false, hence '~(~M --> ' is (~M and ~B)'.

A bit painstaking but very clear, I hope.

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Q.E.D

Hey @Thoughtful, sorry that didn't help. The main point was that the negation you referred to had a larger scope than just the '~M'.

As you said, '~(~M)' is just 'M', but notice the scope of the negation only encompasses '~M'. In the example you cited, '~(~M -> ', the negation outside those parentheses operates on everything inside the parentheses. So it's actually the conditional relationship '-->' between '~M' and 'B' that's being negated, not necessarily '~M'. Does that make sense?

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Q.E.D

Hey @Thoughtful, I'm actually not familiar with that part of the curriculum, sorry to say. It's a unique connective in that it's broken into two pieces, whereas the other truth functions (not, and, or) are solitary. That's why I can't say whether "negating then 'then' part" is negating the whole conditional or just the consequent. Incidentally, you would negate the consequent when negating the conditional, but you would also affirm the antecedent. That is, you negate 'A->B' by saying 'A & ~B', precisely the move we made in that example with M and B. So it sounds like you've got it, but I can't confirm the parallel you saw in the curriculum.