bi-conditional with third element in diagram

dcdcdcdcdc

Looking to confirm my thinking on the below. Thanks!

https://7sage.com/lesson/or-but-not-both/

From this lesson:

Alan or Chris go to the park. (/A-->C)

And

Alan and Chris cannot both go to the park. (A-->/C)

I'm interested in diagramming these statements in relation to the third idea in the sentence, in this case "go to the park," as (P).

With "A or C go to the park" I would diagram as follows:

A-->P
C-->P

With "A and C cannot both go to the park" I would diagram as follows:

P --> /A or /C which can be diagrammed as P-->(A-->/C)

Now to link up the two statements:

A-->P-->(A-->/C)

I'm getting "If Alan goes to the park, then Chris does not go to the park."

Alternatively:

C-->P-->(C-->/A)

I'm getting "If Chris goes to the park, then Alan does not go to to the park."

tamzidamila

i am not sure. from lesson i understand that, in not both relation , max 1 can go to the park. it might happen that no one goes to the park. whatever you want to show or think, you need to keep in your mind that in not both relation two person might be absent .

Cant Get Right

@dcdcdcdcdc said:
With "A or C go to the park" I would diagram as follows:

A-->P
C-->P

So, I’m not sure what this means. Can you clarify? How does this translate back into English?

dcdcdcdcdc

@tamzidamila I think that is why the OR statement is also supplied, which would be at least one. So combination of "at least one" and "at most one" results in the bi-conditional.

@"Cant Get Right" "A or C got to the park." That is my diagram showing a split OR in the sufficient condition.



I see that as "If Alan then he goes to the park, therefore alan does not go or chris does not go." Obviously Alan did go, so we can't say that alan did not go and we are left saying that Chris did not go. This gives us our bi-conditional relationship of A<-->/C

Cant Get Right

@dcdcdcdcdc said:
I see that as "If Alan then he goes to the park, therefore alan does not go or chris does not go." Obviously Alan did go, so we can't say that alan did not go and we are left saying that Chris did not go. This gives us our bi-conditional relationship of A<-->/C

Okay, so this get’s into some existential issues. So if Alan, then Alan goes to the park. Simultaneously, If Chris, then Chris goes to the park. One of them doesn’t go to the park. So if you run the contrapositive back, then whoever isn’t going to the park gets erased from existence. If Alan does not go to the park, then not Alan.

MrSamIam

Diagramming the third idea would be redundant, as "go to the park" is implicitly diagrammed. You're esentially keeping the idea in the back of your mind. "If not Alan, then Chris" is what you would put on paper. But, in your mind you translate that to, "If not Alan going to park, then Chris goes to park."
If you absolutely have to diagram "going to park" I would do so as a subscript. So, for example:
A = Alan C = Chris P = Going to park ( ) = Subscript

"Alan or Chris go to the park"

/A(P) --> C(P)

dcdcdcdcdc

I agree that P is redundant because it is really the third element which we are using to determine the state of Alan and/or Chris.

I was interested in diagramming each statement (or, but not both) and combining them after re-watching the "neither, nor" lesson. In that lesson we have "Neither Koalas, nor Pandas are cute enough to enter the zoo." The statement is diagrammed as follows:
K --> /CE
P --> /CE

Could we also treat "cute enough" as the third element by which we are grouping the pandas and koalas, so that we arrive at an always together/never apart bi-conditional?

/K <----> /P in regards to "cute enough to enter"

J.Y. Ping

I should have been clearer in the original lesson from which this question arose.

The reason why it's confusing is because in English, you get to use the same predicate for more than one subject. "Alan goes to the park." That's one subject (Alan) matched with one predicate (goes to the park). Now I say "Alan and Chris go to the park." Here we have two subjects (Alan and Chris) yet still the same one predicate (go to the park). That predicate gets matched with both subjects. What's really happening is a grammatical shorthand. If you remove the shorthand and expand the sentence out, you'll see it's actually two sentences. "Alan goes to the park" and "Chris goes to the park". You can symbolize that to A and B.

Once you can derive the fully fleshed sentence from the short hand, I think you can see that the logical relationship (whether it's "and" or "or" or "not both") is operating over the whole sentences.

dcdcdcdcdc

Thanks. I think I was arriving (slowly, but surely) at that understanding. The A / B comparison is operating on Alan going to the park and Chris going to the park. We move the shared predicate into each of the two elements. So it is deceptive for me to label it a third element.

I think the neither nor example is also not a bi-conditional as I earlier stated because the statement isn't about reasoning the position of one element from that of the other. Instead, it is really one of two outcomes: a split sufficient or | a split negated and in the necessary. That is, neither..nor is just one statement: (sufficient) a or b | (necessary) not a and not b
There isn't another "not both" component there, so we can't state a bi-conditional.

Koalas and Pandas will both not ever be in the "cute enough to enter" category. It's a stated fact and we aren't concerned with determining, say, the koalas status of ability to enter based on the panda.

quinnxzhang

@dcdcdcdcdc, the antecedent and consequent of a conditional must truth-apt (or truth-evaluable, if you prefer that terminology). Similarly, the disjuncts or conjuncts of a disjunction or conjunction or whatever must also be truth-apt.

So when you have a sentence like "Alan and Chris cannot both go to the park", what you're really saying is equivalent to "If Alan goes to the park, then Chris does not go to the park". When you split up the conditional into antecedent and consequent, you see that both of them are truth-apt. The sentence "Alan goes to the park" can be true or false, just as "Chris does not go to the park" can be true or false. This is why the symbolization "A --> ~C" makes sense.

Now consider what you were trying to do with your "A --> P" symbolization. Your antecedent is simply "Alan" and your consequent is "go to the park". But neither of these things are truth-apt. "Alan" cannot be true or false -- it's just a name! Similarly, "go to the park" cannot be true or false either -- it's just a predicate! That is to say, it's a *category mistake* to try to put terms and predicates into the antecedent and consequent positions of a conditional formula. This is true for "A or B" and "A and B" and "A <--> B" as well, where "A" and "B" symbolize truth-apt sentences.

This is a very common mistake to make when you start learning symbolic logic. It's just unfortunate that no LSAT course teaches predicate logic, because there are certainly ways to symbolize "Alan goes to the park" in a more sophisticated way. I think you came onto this realization yourself. If you're interested, I recommend Barwise & Etchemendy's Language, Proof & Logic, which will cover both propositional and predicate logic, so you can see the difference yourself.