P1) All Flumps are Pink
P2) Some Roops are Flumps

C1) All Roops are Pink
C2) Some Roops are Pink

C1 is invalid and C2 is valid. What specifically is your problem?

For a position to be valid it doesn’t have to work from true premises. Meaning, as long as the conclusion is true the statement is valid. If the premises and the conclusion is true then the statement is sound and valid. A statement - as far as I know! - cannot be invalid and sound though.

Validity obtains when

(1) It's impossible that premises are true

and/or

(2) It's impossible that a conclusion is false,

or

(3) If the premises true, then it's impossible that the conclusion is false.

Anything other than that is invalid.

The "or" above between (1) and (2) is intuitively weird, because it means that if it's impossible for premises to be true--for example, say that the premises are contradictory --then it doesn't matter what the conclusion is; the argument is valid. (This is why "anything follows from a contradiction." )

Likewise, if the conclusion is a tautology, then it doesn't matter what the premises are. The argument is valid.

With (3), it doesn't imply that the premises are true. The idea is "If it would be the case that these premises were true (then due to them, it would be impossible for this conclusion to be false) ."

A rejection of the "or" between (1) and (2) is what lead to relevance logics ( (3) being a relevance interpretation--in a nutshell, under relevance, the premises and conclusion are required to be related to each other implicationally). But traditional logic, traditional validity, has the "or" between (1) and (2) (or is at least interpreted that way).

A statement - as far as I know! - cannot be invalid and sound though. — I like sushi

Correct. Soundness is defined as a valid argument with true premises.

An argument is valid if and only if it's impossible for the premises to be true and the conclusion false. Validity is determined by the form rather than content of an argument.

Specific argument X1:
1. All dogs are mammals
2. All alsatians are dogs
Therefore
3. All alsatians are mammals

Form of argument X1:
1. All A are B
2. All C are A
Therefore
3. All C are B

The form of the argument, if correct, ensures validity of any argument that uses that form. The form of an argument is like a template which ensures that whatever you use to replace the statement variables A, B and C you can be sure to have a valid argument. The specific argument X1 is a substitution instance of the form described above.

An invalid argument is one in which it is possible for the premises to be true and the conclusion false. It happens when we have an invalid form of an argument and this may be difficult to detect:

Invalid argument X2:

1. All dogs are mammals
2. All collies are mammals
Therefore
3. All collies are dogs

Form of invalid argument X2:
1. All A are B
2. All C are B
Therefore
3. All C are A

We can check for invalidity of an argument by first extracting its form (which is done above) and creating a substitution instance where the premises are true and the conclusion false. This is called a counter example.

Let's create a counter example for X2:

1. All cats are mammals
2. All dogs are mammals
Therefore
3. All dogs are cats

The counter example reveals that with the form of argument X2 it is possible to have true premises and a false conclusion. In other words the argument is invalid.

We are dealing with arguments that are deductive, or logical entailments between statements. The first thing you have to do is decide if an argument is deductive or not. Only deductive arguments can be valid in the usually discussed sense. With that in mind:

Validity is a property of deductive arguments. An argument is valid when and only when the truth of its premises ensures the truth of its conclusions. "P implies Q, P, therefore Q" is a valid argument. "P implies Q, Q, therefore P" is an invalid argument. Valid arguments connect truths or stipulations to their logical consequences.

Particularly common invalid argument patterns are given names as 'formal fallacies', particularly common valid arguments are given names as syllogisms. The invalid argument above is called affirming the consequent, the valid argument is called modus tollens.

A sound argument is a valid argument with true premises. A valid syllogism is a sound argument when and only when its premises are true.

This talk about contradictions in the premises of an argument ensuring validity is complete nonsense. Note that the definition of validity requires us only to consider cases where the argument's premises are true. When they are false, other concepts are in play.

Particularly, in the usual introductory logical systems of propositional and predicate logic, assuming a falsehood allows you to derive arbitrary conclusions. EG "apples can't be red, therefore Santa exists". The reason this works is that in these systems an implication turns out to be forced to be true when its premise is established as false - called the 'principle of explosion'.

If you pay attention to how people actually reason out in the wild, you'll find formal calculi and syllogisms provide rules of thumb and interpretive heuristics rather than good descriptions of argument. One reason for this is that most instances of reasoning are compositions of heuristics (see Kahnneman and Tversky for some good examples) which, strictly speaking, aren't even deductive nevermind valid. Another reason is that we are often in situations where formal argument from premises will not allow us to derive a good explanation; eg 'there is no smoke without fire' is not automatically true in a logic, it has to be encoded as the entailment "smoke => fire'. An even stranger example is the statement 'red things are coloured' - this isn't true by definition in logical calculi, and has to be stipulated or encoded in the logic to use it.

In this regard, formal calculi are only ever restricted models of human reasoning, and while no models are true some are useful.

This talk about contradictions in the premises of an argument ensuring validity is complete nonsense. — fdrake

You're giving misinformation here. You're favoring a relevance logic interpretation, which is fine (I favor that, too), but that's a far more recent interpretation. The traditional interpretation is that validity can (also, in addition to a relevance interpretation) obtain when either it's impossible that the premises are true OR when it's impossible that the conclusion is false.

It's irresponsible to answer with misinformation.

You're giving misinformation here. You're favoring a relevance logic interpretation, which is fine (I favor that, too), but that's a far more recent interpretation. The traditional interpretation is that validity can (also, in addition to a relevance interpretation) obtain when either it's impossible that the premises are true OR when it's impossible that the conclusion is false. — Terrapin Station

Unrestricted explosion is not a feature of relevance logics. I've no idea what you're talking about. The example "apples can't be red => Santa exists' is the kind of thing relevance logics are designed to block - so I'm explicitly not talking about relevance logic, and talking about standard propositional and predicate calculus.

For the notion of validity, what I'm saying is exactly what the IEP is saying on the topic:

A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Otherwise, a deductive argument is said to be invalid.

A deductive argument is sound if and only if it is both valid, and all of its premises are actually true. Otherwise, a deductive argument is unsound.

If you're being incredibly pedantic, yes, the following argument is valid:

Things that are eggs are always made of cheese
My spoon is an egg
My spoon is made of cheese.

Validity doesn't care about the truth or falsity of the premises, it only cares whether the conclusions necessarily follow from the premises. This is precisely why the principle of explosion should not be seen as a consequence of the definition of validity, as we must stipulate a specific truth value of the premises.

Edit: yes, you demonstrate explosion through a valid argument, but the thing which is doing all the work is the assumption of falsehood or contradiction.

Unrestricted explosion is not a feature of relevance logics. — fdrake

It's as if you didn't read or couldn't comprehend what I wrote. You are giving misinformation if you're saying that under traditional (NOT relevance-logic) validity, contradictory premises do not produce a valid argument.

It's as if you didn't read or couldn't comprehend what I wrote. You are giving misinformation if you're saying that under traditional (NOT relevance-logic) validity, contradictory premises do not produce a valid argument. — Terrapin Station

Just to be clear.

If you have P & ~P. Then you can instantiate to P. Then you can disjunction introduce to P or Q, then you can disjunctive syllogism to Q. So whatever the contradiction is, you can derive an arbitrary conclusion. The 'truth' assumed here is P and ~P, but unfortunately that is never true. The argument form of disjunctive syllogism is of course valid, assuming a contradiction lets you derive anything through the syllogism through a valid argument.

The reason this works is because you input a contradiction to an already valid argument structure, the contradiction does not make the argument valid. This can easily be seen from the conditional nature of validity - it is indifferent to the truth value of the premises! Whereas explosion works precisely by specifying the premises as a particular species of falsehood, a contradiction.

I haven't had much luck using google. — Josh Alfred

Probably because you're using the wrong keyword, "non-validity". Google "invalid reasoning", and that should help.

Invalidity is a property of reasoning. What is characteristic of reasoning is that we produce reasons as evidence for a certain conclusion we wish to establish. Reasoning is closely connected with inferring. The reasons we provide allow us to infer a certain conclusion.

Deductive logic deals with reasoning that attempts to establish conclusive inferences. To say that an inference is "conclusive" means that if the reasons given are true, then it would be impossible for the inference based upon these reasons to be false. Such reasoning is called "valid" reasoning. "Invalid" reasoning is where the conclusion isn't necessarily inferred from the reasons, or premises. In other words, the conclusion doesn't necessarily follow from the premises.

Just to be clear. — fdrake

That was anything but clear.

It's very simple. Contradictory premises are sufficient for a valid argument (in non-relevance logics) due to the definition of validity.

It's very simple. Contradictory premises are sufficient for a valid argument (in non-relevance logics) due to the definition of validity. — Terrapin Station

To anyone reading this, please to not listen to Terrapin, and instead look at this excellent account from a citable resource.

Edit: Terrapin was right! Look at his original post or here for some explanations.

A summary: a valid argument means 'true premises implies true conclusion', which means 'not true premises or true conclusion', contradictions are never true, so the implication always holds when the premises are contradictory, so the argument is valid. Sub in a tautology into the conclusion part of the disjunction defining validity and it is valid too.

To anyone reading this, please to not listen to Terrapin, and instead look at this excellent account from a citable resource. — fdrake

You're giving misinformation. I don't know why.

It's not even clear at this point if you agree that traditionally (that is, not in relevance logics, which aren't the traditional interpretation), any argument with contradictory premises is valid. This is the case for a very simple reason having to do with the definition of validity.

Do you agree with that or not?

If you don't agree with it, I can give you a bunch of citations from academic phil sources for it. I can explain it to you, too, if you need me to explain it to you.

If you don't agree with it, I can give you a bunch of citations from academic phil sources for it. I can explain it to you, too, if you need me to explain it to you. — Terrapin Station

Ok! Do this for me please.

Sure, so here are some things I found very quickly online for you.

https://www.uta.edu/philosophy/faculty/burgess-jackson/Technical%20Validity.pdf
That's a paper by philosopher Keith Burgess Jackson at UT Arlington. See the third paragraph

cstl-cla.semo.edu/hhill/PL120/handouts/exam%201%20partial%20answer%20key.docx
(That's an answer key from a quiz in a logic course from Southeast Missouri State University. See question #5)

http://media.podcasts.ox.ac.uk/conted/critical-reasoning-2012/2012-11-05-week-4.pdf
That's from a presentation on logic/validity that someone gave at Oxford. See slide #21

Do you need more?

A valid argument is an argument that preserves truth. To say that an argument is valid, therefore, is to say that it is impossible for its premises to be true while its conclusion is false. To say that an argument is invalid is to say that it is possible for its premises to be true while its conclusion is false. But if an argument has inconsistent premises, then, by definition (of “inconsistent”), it is impossible for
its premises to be true. Therefore, if an argument has inconsistent premises, it is impossible for its
remises to be true while its conclusion is false. This is the definition of “valid argument.” It follows that any argument with inconsistent premises is valid.

That makes more sense. A valid argument means 'true premises implies true conclusion', which means 'not true premises or true conclusion', contradictions are never true, so the implication always holds when the premises are contradictory, so the argument is valid. Sub in a tautology into the conclusion part of the disjunction defining validity and it is valid too.

This is funny.

What's the best way to reason? To say nonsense or nothing at all!

That makes more sense. A valid argument means 'true premises implies true conclusion', which means 'not true premises or true conclusion', contradictions are never true, so the implication always holds when the premises are contradictory, so the argument is valid. Sub in a tautology into the conclusion part of the disjunction defining validity and it is valid too. — fdrake

That's what I said at the start though. Validity obtains when it's impossible for premises to be true and(/or--I add or for reasons I detailed in my first post) (It's impossible for) the conclusion to be false.

Therefore, when it's impossible for the premises to be true, as is the case when the premises are a contradiction (or are inconsistent), we have a valid argument.

That's what I said at the start though. Validity obtains when it's impossible for premises to be true and(/or--I add or for reasons I detailed in my first post) (It's impossible for) the conclusion to be false. — Terrapin Station

Yeah I saw that and edited a post above to link to yours and the references, and included my explanation of why I was wrong.

Cool. I hadn't noticed any changes. The important thing is that we get the right info to anyone trying to understand this stuff.

Cool. I hadn't noticed any changes. The important thing is that we get the right info to anyone trying to understand this stuff. — Terrapin Station

I'm convinced that the three ways you outlined in your original post are appropriate. But I think they work in very different ways. It's extremely weird that one can conclude from a definition relating premises to a conclusion that sometimes the relation doesn't matter at all, even when it's true.

Yeah, but that's a start. I think I will widen my search. You guys have been helpful. Thanks.