Using Truth Tables and Proofs for a Proposition
Hey everyone, I'm posting here and in r/askphilosophy .
I am new to the study of logic, and reading online forums, I found an archived post where someone snuck a bad proposition into their argument that no one caught. I could not reply there as it was archived, but I realized I didn't know how I would explain the problem anyway.
The proposition in question is:
(P∨Q) → (P∧Q)
Resulting in the truth table (to the best of my knowledge):
| P | Q | (P∨Q) | → | (P∧Q) |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
This is obviously a "false" conditional as the antecedent can be true without the consequent being true. But I can't seem to find what to call a bad proposition, or how to articulate that the proposition is bad using a truth table. Can anyone help me close the gap between knowing this is a bad proposition and articulating/proving it?
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For context to the original post, the person's argument used the sentence in question as a premise. Since this sentence is "false", the argument was not sound. I'm using "false" since I am not sure what to call a bad sentence like this.
The argument went:
- (P∨Q) → (P∧Q)
- (P∧Q)
C. (P∨Q)
They were using this to show that Affirming the Consequent could be a valid argument in some cases. Commenters pointed out that premise 2 already entails the conclusion without premise 1. Other commenters pointed out other errors in reasoning as the post was very long. However, I didn't see any comments simply stating that the first premise was incorrect, but in my estimation that was the most significant error here, and I wanted to learn how to properly articulate that.
Post here: https://www.reddit.com/r/askphilosophy/comments/13wbqzc/making_sense_of_affirming_the_consequent_fallacy/
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The best I can do is something like, "A conditional is true iff there is no case where the antecedent is true and the consequent is false. There is a case where the antecedent is true and the consequent is false. Therefore, the conditional presented is not true."