I’m in a Logic class and this seems like a really simple concept but all the explanations given are so wordy that I can’t wrap my head around it.
In: Other
A valid argument is where the premises necessitate the conclusion. This means that if the premises are true, it would impossible for the conclusion to be false.
An example of a valid argument is this:
* P1: Spot is a dog
* P2: All dogs can fly
* C: Spot can fly.
There is no way that the conclusion can be false if the premises are true. If all dogs can fly, then spot cannot be a dog which does not fly. That is simply not possible according to the argument.
As I am sure you notice, the argument overall is wrong. Dogs cannot fly. However, that does not matter when determining the validity of the argument. Validity only cares if the formation of the argument makes sense, not the accuracy of the argument. The above argument is valid, but it is not sound.
The following is an invalid argument:
* P1: Spot is a dog
* P2: Spot cannot fly
* C: All dogs cannot fly
Nothing individually I say here is wrong. However, the formation of the argument is not valid. It is true that Spot is a dog that cannot fly. However, just because one dog cannot fly, you cannot conclude that all dogs cannot fly. Although the conclusion is correct, the argument to get the conclusion is not valid.
You can make the argument valid by saying this instead:
* P1: Spot cannot fly
* P2: All dogs have the same flying ability as Spot
* C: All dogs cannot fly.
This argument is now valid because the conclusion is made necessary from the premises. The conclusion cannot possibly be wrong if we assume the premises are true.
Yeah, this sort of thing tends to get wordy. The simplest way I can come up with is that a valid logical argument demonstrates the truth of the conclusion through it’s premises. In other words, it’s sound or just good logic.
Where it gets wordy is when you get into the fact that it doesn’t have to be true to be valid. Validity is only based on whether the premises lead to the conclusion or not.
So I could start with a flawed premise like “All men have x-ray vision.” Then go from there: “I’m a man. Therefore I have x-ray vision.”
Is that true? Of course not because it starts with a false premise. But it is valid logic because it leads to a logically sound conclusion.
I hope that makes sense.
**TL;DR: Validity is about *the form* of the argument. Does each step proceed from the previous by a valid rule of inference? If you have an unbroken chain of valid inferences, you have a valid argument.**
A logical argument is *valid* if it only uses valid rules of inferences. E.g., *modus ponens* (affirming the antecedent), *modus tollens* (denying the consequent), law of the excluded middle / double negation / proof by contradiction (unless you’re an intuitionist). Another way to put it is an argument is valid if its conclusion follows from its premises.
An example of a valid argument:
1. All dogs are mammals (i.e., if an animal is a dog, it is a mammal)
2. Fido is not a mammal.
3. Therefore, Fido is not a dog.
This argument is *valid* in that it uses all valid rules of inference. The conclusion (3) is arrived at via *modus tollens* (contraposition).
The opposite of this would be a fallacy, which makes for an invalid argument, because an invalid form of argumentation was used.
An example of an invalid argument is the following argument:
1. All dogs are mammals.
2. Fido is not a dog.
3. Therefore, Fido is not a mammal.
This is not a valid argument, because it makes use of an fallacy (denying the antecedent, also sometimes called the fallacy of the inverse). I.e., each step does not proceed as a valid inference from the previous.
—
There’s another concept called soundness. Soundness is concerned with if an argument is both valid *and* its premises are true. Here is a valid but unsound argument:
1. All birds are mammals.
2. Barry is not a mammal.
3. Therefore, Barry is not a bird.
This argument is *valid* in that it uses all valid rules of inference. The conclusion (3) is arrived at via *modus tollens*. But premise (1) is not true, so it is not sound.
So soundness means the conclusion follows from the premises, *and* the premises are also true.
There’s a handy quote that captures the difference between validity and soundness: “If my mother had wheels, she would’ve been a bike.” In keeping with the spirit of the quote, if we assume bicycles are the only object with wheels, then yes, if your mother had wheels, she indeed would’ve been a bike. When the premises are true, the conclusion is true. But she doesn’t actually have wheels. The premise isn’t true.
Logic is, at its very core, irrefutable truth and subsequently deriving further irrefutable truths from it. But because truth can be very nuanced, It is very easy to make a statement that is only highly likely to be correct, but not definitively so. Logic is difficult, because as humans we often forget about edge cases where our statement might not be an absolute truth.
For example, I know Bob always cries when his favourite team loses. I also know his favourite team was playing today, and that he was at the stadium for the match. Bob came home crying. Therefore, his team lost.
This sounds very reasonable, but it is not ironclad logic. Maybe something else happened that made Bob sad. Maybe he saw something very sad while he was driving home. Maybe a family member died and he just got the phone call. We cannot state for the fact that Bob’s team lost just because we saw him crying, even if it is irrefutably true that Bob cries when his team loses.
However, other example: I know Bob tries when his favourite team loses. I know Bob is watching his team play. I don’t see Bob, but I do watch the match on TV and I see that his team has lost. Therefore, Bob is crying.
This follows formal logic. We first established that when his team loses, he will definitely cry. We then established that his team lost. The first statement means that the second statement means that Bob will now cry.
(I’m ignoring “clever” answers like “Bob died before the game finished so he’s not actually crying now” because this is just a simple example)
Logic is one of those things that is very difficult to get right, because humans are so innately good at approximating things that they stop to consider the tiny edge cases. We take a 99% certainty as a 100% certainty. Logic refuses to do this, instead only allowing for 100% logical validity and nothing less.
Logic mandates validity in the same sense that mathematics mandates that 3 means _precisely_ 3, not _roughly_ 3 (i.e. an approximation).
Formal logic has specific structures for an argument. As an example, one of them is the following:
If P, then q
P
Therefore, q.
No matter what you substitute for p and q, the argument is valid because it follows that structure.
However, an argument can be valid, but not cogent in formal Logic. So if my dog is outside, then it will rain. It is a valid argument, but it is not cogent because it does not always rain when my dog is outside.
Logical and true are not the same thing when discussing formal logic.
In formal logic. The conclusion must be based on valid premises.
The premises do not have to be true.
All dogs are blue.
Juliette is a dog.
Therefore Juliette is blue.
That is a valid argument of formal logic. However the premise all dogs are blue isn’t true.
If you’re taking a logic class, then the meaning of words becomes very specific very fast, and don’t always match colloquial English – be aware.
In this case, an argument is valid if (and only if) it is impossible for the premises to be true and the conclusion to be false. For example:
Premises:
* A
* A and B -> C
* not C
Conclusion:
* not B
If the premises are true, the conclusion must be true, and you’ve got various things you can do to show this.
Note that like many mathy things, there are things that sound funny that nevertheless meet the definition:
Premise:
* All crocodiles are dead
Conclusion:
* All blue things are blue.
Premise:
* the empty set contains an element
Conclusion:
* the moon is made of green cheese
Both of these are valid, because it is not possible for the premises to be true and the conclusions to be false (in the first case because the conclusion is always true, and in the second because the premise is always false).
So be aware.
Another way that arguments can be unsound is if that, if you have premises which lead to a contradiction, then any conclusion from those premises is unsound.
Suppose my premises are:
* A: All Creteans have purple hair.
* B: I am a Cretean.
* C: All Creteans are liars.
It would be unsound to conclude that i have purple hair due to the contradiction produced by premises B and C.
Latest Answers