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."
Comments
@"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
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)
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"
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.
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.
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.