Is it true that if an argument is invalid, any argument of that logical form must be invalid?



I am stuck over whether these statements are true:

First: "If an argument is invalid, any argument of that logical form must be invalid."

Second: "There may be invalid argument with inconsistent premises."

I thought that both of them are false but I'm not sure. Can you correct me if I'm wrong?


Posted 2020-08-22T17:50:48.907

Reputation: 125

The technique of counter example is what you first question proves. That is if your argument is valid then I can change the content to any topic I desire with the same form and my new argument must ALSO BE VALID regardless of the subject matter. Think about how often something can be true in one subject but false elsewhere. It can happen and often. One counter example proves the truth is not 100 percent. At best you have a half truth. The 2nd question is confusing. Yes you can have an invalid argument with inconsistent premises but not sure that is what you really mean. Clarify. – Logikal – 2020-08-23T02:43:15.597



Hint for the first question:

An argument scheme being valid means that all instances of sentences of this form are valid; if the form is invalid, then not all instances are valid. According to this definition, could it be the case that there exist valid instances of an invalid form?

Hint for the second question:

An argument is valid iff in all structures, either at least of the premises is false or the conclusion is true, and invalid iff there exists at least one structure (a counter model) under which all premises are true but the conclusion is false. If the premises are inconsistent, i.e. true in no possible structure, can there be such a counter model that makes the premises true and the conclusion false?


Posted 2020-08-22T17:50:48.907

Reputation: 2 099

then the first question is FALSE i.e we can find counterexample but the answer of second must have been true i.e we can find an example for invalid argument with inconsistent premises. Am i right? – Bulbasaur – 2020-08-23T07:42:13.977

Your first answer is correct. Though note that we only might find a counter example in the general case; it could still be that the invalid argument is indeed not only invalid in some, but possibly even all instances. For the second one, by a counter example I mean a counter valuation that proves the invalidity of the argument, which makes the premises true but the conclusion false. If there is no interpretation which makes the premises true because they are inconsistent, can there be an interpretation which makes the premises true but the conclusion false? – lemontree – 2020-08-23T07:45:24.900

then there is a possiblity when all premises are inconsistent, there may be invalid argument,so the second is true. Thanks for your helps – Bulbasaur – 2020-08-23T07:53:35.007

No. If the premises are inconsistent, then this means there is no way to make all of them true, so in particular there is no way to make the premises true but the conclusion false, so the argument can not be invalid. An argument with inconsistent premises is said to be vacuously valid. See also here:

– lemontree – 2020-08-23T07:59:23.420

@lemmontree, if an argument is NOT valid then we have two options: either the argument is invalid or meaningless. By meaningless I mean the language being used doesn't meet proper syntax to count for any type reasoning. Here is an argument with inconsistent premises: if outer space aliens exist, then 4 is an even integer. There are no existing outer space aliens. Therefore 4 is not an even integer. That example is fallacious correct? So we can have an invalid argument with inconsistent premises. – Logikal – 2020-08-24T18:31:57.607

@Logikal If one of the statements involved is not even a well-formed expression at all (Something like "Oranges the beautifully sleep and", one would best not call it an invalid argument, but not a proper argument at all, assuming that the notion of an argument presupposes well-formed sentences. – lemontree – 2020-08-24T21:32:30.410

In your example, there is nothing syntactically wrong or nonsensical -- it just so happens that the proposition "Outer space aliens exist" is not true in the actual world, but it is perfectly grammatical and meaningful -- and your premises of the form "If A then B", "not A", (where A = "Outer space aliens exist", B = "4 is an even integer") are consistent. The argument is invalid as an instance of the fallacy of denying the antecedent (from "If A then B" and "not A" you may not conclude "not B"). – lemontree – 2020-08-24T21:33:18.193

The issues are several in Mathematical logic. First the premise the logician is not concerned with whether the premises are true or false is falsified here. You are concerned if the premises are true. You already know one is false. For the logician not to care is inconsistent already. Secondly, the premise I gave is deemed an assumption in math. I assert the assumption which is positive & the second premise I used denies the 1st assumption I just made; therefore the premises are inconsistent by definition. We agree the argument is invalid but it's form makes it invalid. It is not trivial. – Logikal – 2020-08-25T00:46:03.040