## View Poll Results: Does (P and Q) follow from P1,...,Pn,Q1,...Qm?

Voters
3. You may not vote on this poll
• Yes, (P and Q) logically follow from P1,...,Pn,Q1,...,Qm.

2 66.67%
• No, it is not the case that (P and Q) logically follow from P1,...,Pn,Q1,...,Qm

1 33.33%

0 0%
Multiple Choice Poll.

# Thread: Yet another logic question.

1. ## Yet another logic question.

Suppose that n and m are positive integers, and P, Q, P1,...,,Pn, Q1,...Qm are statements such that:

i. P logically follows from P1,...,Pn. In other words, if we have premises P1,...,Pn, then P follows as a conclusion.
ii. Similarly, Q logically follows from Q1,...,Qm.

Is it true that the conjunction (P and Q) follows from P1,...,Pn, Q1,...,Qm?
In other words, if we have premises P1,...,Pn, Q1,...,Qm, does (P and Q) follow as a conclusion?

2. Originally Posted by Angra Mainyu
Suppose that n and m are positive integers, and P, Q, P1,...,,Pn, Q1,...Qm are statements such that:

i. P logically follows from P1,...,Pn. In other words, if we have premises P1,...,Pn, then P follows as a conclusion.
ii. Similarly, Q logically follows from Q1,...,Qm.

Is it true that the conjunction (P and Q) follows from P1,...,Pn, Q1,...,Qm?
In other words, if we have premises P1,...,Pn, Q1,...,Qm, does (P and Q) follow as a conclusion?
I voted yes. But, I’d rather it didn’t.

3. Originally Posted by fast
Originally Posted by Angra Mainyu
Suppose that n and m are positive integers, and P, Q, P1,...,,Pn, Q1,...Qm are statements such that:

i. P logically follows from P1,...,Pn. In other words, if we have premises P1,...,Pn, then P follows as a conclusion.
ii. Similarly, Q logically follows from Q1,...,Qm.

Is it true that the conjunction (P and Q) follows from P1,...,Pn, Q1,...,Qm?
In other words, if we have premises P1,...,Pn, Q1,...,Qm, does (P and Q) follow as a conclusion?
I voted yes. But, I’d rather it didn’t.
The answer is correct. But why would you'd rather it didn't follow?

4. That’s hard to explain. The words that come to mind are “it doesn’t track well.”

Given the construct (or the foundation of) the logic, I understand the importance of keeping the truth of any premise in the P chain independent of the truth of any premise in the Q chain, but because of how validity and truth are separated, the evaluation of the argument is unecessarily postponed.

The problem with
P
Q
P and Q

Is the same for
P
Q
P or Q

When it comes to an evaluation ‘on the go’.

Suppose n-min, n-max, m-min, and m-max is 5. Let P1, P2, P3, P4, P5 for first conclusion P and P1, P2, P3, P4, P5 for second conclusion Q be illustrated as:

P1
P2
P3
P4
P5
C1

P6
P7
P8
P9
P10
C2

Therefore, C1 and C2

Under the construct of this logic, there’s no interim screams of “Houston, we have a problem” when conflict is detected. It’s not a problem with weakening per se but rather the instances where inconsistencies, contraries, or contradictions arrive. The logic doesn’t lend itself to a p by p on the fly problem detection.

The issue isn’t necessarily a failure to not keep truth separate from form, although it occasionally appears that way. In some instances, I could be completely oblivious to the truth condition of a premise and recognize conflict in the making, but instead of screams of caution, I hear, “it’s valid!” Sure, in the end, later, after the paper is turned in, the grade is calculated, but there’s no sign I’m failing even after not answering 8 questions of a 10 question test.

P1) If A, then B (which is true)
P2) A (which is true)
P3) any nonconflicting P (which is false)
Therefore, B

The falsity of that premise does not affect the truth preservation of that argument —yet it’s unsound! All because not all premises are true. And the need for that is a function of the construct

Weakening isn’t contentious when it’s nonconflicting. We readily accept the argument is valid despite the additional premise, but let it conflict and still call it valid, then well, there ya have it.

5. The problem that you seem to have is that new information might allow you to discard some of the previous premises because you realize they're false. But that's not a problem with whether the conclusion follows. Rather, in that case, you have a new argument, and the new premises aren't the same as the previous ones, so this causes no trouble in real cases.

On the other hand, there are cases in which you know some premises P1,...Pn are true, and you want to prove that another one - say, Q1 - is false. However, there is no obvious contradiction. What you do is you add premise Q1 to the previous ones, and then from all of the premises together, you derive a contradiction. That way, you prove that Q1 is false. You could not do that if having contradictory premises invalidated an argument.

6. I apologize for not having much information, but when I used to work around artificial intelligence researchers (I was never one myself) there was a fair bit of talk on "non-monotonic logic" and "belief revision." The SEP has an article on it. The rough intuition, as I understood it, was that agents that go about learning more and more about the world may realize mistakes in their past conclusions as they acquire more information. There are models for this which change the way that logical entailment works, and weakening is implicated in a way that it is not implicated by linear and relevant logics.

Mathematical logic doesn't have any need of this, since there isn't room for mistakes or revision in the accumulation of mathematical theorems, and so we don't need those dynamics as we formalize proofs.

7. So, the real-world advantages of the construct (of this system of logic) outweigh whatever perceived disadvantages the construct might have. It might not be perfect but perhaps the best system of logic we have—kind of like how the JTB Theory of Knowledge has room for improvement but heralded as the best theory we have. What’s the proper name that describes the particular type of logic in the recent discussions?

8. Originally Posted by A Toy Windmill
I apologize for not having much information, but when I used to work around artificial intelligence researchers (I was never one myself) there was a fair bit of talk on "non-monotonic logic" and "belief revision." The SEP has an article on it. The rough intuition, as I understood it, was that agents that go about learning more and more about the world may realize mistakes in their past conclusions as they acquire more information. There are models for this which change the way that logical entailment works, and weakening is implicated in a way that it is not implicated by linear and relevant logics.

Mathematical logic doesn't have any need of this, since there isn't room for mistakes or revision in the accumulation of mathematical theorems, and so we don't need those dynamics as we formalize proofs.
Hey, that's good info. Thanks.

I should have added for further clarity that I'm talking about deductive entailment exclusively.

9. Originally Posted by fast
So, the real-world advantages of the construct (of this system of logic) outweigh whatever perceived disadvantages the construct might have. It might not be perfect but perhaps the best system of logic we have—kind of like how the JTB Theory of Knowledge has room for improvement but heralded as the best theory we have. What’s the proper name that describes the particular type of logic in the recent discussions?
We are talking about deductive logic. There are systems of non-deductive logic that might match most kinds of realistic reasoning better. But I would not call deductive logic a "system of logic". There are several different deductive systems, and several different empirical questions, such as which one fits traditional intuitive usage in mathematics better (for example), or which one matches common usage of the words (if there is one), when used in an exclusively deductive manner, etc.

In any event, the debates in all of these threads are about deductive logic, so that leaves other types of reasoning aside.

10. How is that a question? Conjunction or AND is a definition. If a is true and b is true than a & b is true. The sequences of premises are irrelevant.

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