25 upvotes, 4 direct replies (showing 4)
View submission: What makes an argument valid?
You pretty much got it. An argument is valid if and only if it's not possible for the premises to be true and the conclusion false. Read that several times and really try to understand it.
Take this example:
1. 2+2=5
2. Therefore the moon is made of cheese.
Now, consult the above definition. Is it possible for the premises to be true and the conclusion to be false? No, because it's not possible for the premise to be true. 2+2=5 is necessarily false, it cannot be true, it's not possible for it to be true.
1. The moon is made of cheese.
2. Therefore 2+2=4.
It's not possible for the conclusion to be false. So, "it's not possible for the premise to be true and the conclusion to be false." And so it's valid.
Comment by crank12345 at 27/01/2025 at 22:20 UTC
7 upvotes, 0 direct replies
This explanation is great!
Just keep focusing on "Is it possible for the premises to be true and the conclusion to be false?".
Do NOT focus on:
Only focus on u/drinka40tonight's key question!
Comment by JackOnTheBox_ at 27/01/2025 at 22:19 UTC
6 upvotes, 3 direct replies
Ok this makes sense. Thank you for reassuring me! I was wondering though what if there was no necessary truth nor false in an argument with unrelated premises and conclusions. Would an argument like that be automatically invalid since the premises and conclusion are unrelated?
Comment by ConceptOfHangxiety at 27/01/2025 at 22:33 UTC
2 upvotes, 1 direct replies
I like this framing for a 1-premise argument because it casts validity according to a conjunction rather than a conditional (the latter being how I was taught).
But how would one prove such an argument through natural deduction?
Comment by NemeanChicken at 28/01/2025 at 00:10 UTC
2 upvotes, 0 direct replies
My logic is a bit rusty, but I wonder if a bit more context on what's happening underneath the hood would help. The crucial move is 2+2=5 is *necessarily* false. So we can do a bit of craziness using a contradiction.
1. 2+2=5
2. ¬(2+2=5)
3.(2+2=5) ∧ ¬(2+2=5)
4. (2+2=5) ∨ The moon is made of cheese
5. The moon is made of cheese
Annotations.
1. This is one of our premises
2. The little ¬ symbol means "not". In other words, this line is asserting that 2+2 does not equal 5. This is provable using ordinary facts about arithmetic.
3. The little ∧ symbol means "and". Basically this line combines together line 1 and 2 into the statement that 2+2 does equal 5, and 2+2 does not equal 5. Logicians call this "conjunction introduction". This line is kind of fluff, but it clearly shows that we've arrived at a contradiction.
4. The little ∨ symbol means "or". Basically I've used my premise and added on a statement, either (2+2=5) or The moon is made of cheese or both. Logicians call this disjunction introduction.
5. To get at the conclusion, I essentially said, either the 2+2=5 or the moon is made of cheese, and it we know form line two that ¬(2+2=5). So the only way for the or statement to be true is if the moon is made from cheese. This is called a disjunctive syllogism.
In fact, you can prove anything from a contradiction in something called the Principle of Explosion. A *necessarily* false statement is functionally a contradiction because you are assuming one part (2+2=5 in this case), but you can prove it's negation (otherwise it wouldn't be necessarily false), and this lets you form a contradiction. Then, from a contradiction, you can prove anything.
u/ConceptOfHangxiety I wonder if this addresses you question