# Suppose you know the premises of an argument are inconsistent. Do you have to do a truth table to know whether it is valid or invalid?

Suppose you know the premises of an argument are inconsistent. Do you have to do a truth table to know whether it is valid or invalid?

• Posting the separate parts of your homework question one by one, I presume? Apr 21, 2015 at 4:41
• If the premises of an argument are inconsistent you can conclude anything, and thus the argument is automatically valid. You don't need to use a truth table to know it is valid, since a truth table checks for an interpretation when the premises are all true and the conclusion false, i.e checks for invalidity. Since inconsistent premises can never be all true, we get that invalidity can never be attained, and thus the argument is valid.
– john
Dec 22, 2019 at 20:38

# The Answer You're Probably Looking For

Under a common "critical thinking" or "intro to logic" in philosophy approach, the following definitions apply:

validity: an argument is valid if it is the case that the conclusion cannot be false when all of the premises are true.

consistency: it is possible for all of the premises to be true.

The answer is that you do not need a truth table on these definitions, because inconsistency in the premises means that it is impossible for all of the premises to be true. In turn, this means the argument is valid.

Behind this is that the definition of validity is this: were the premises all to be true then the conclusion could not be false. Since an inconsistent argument can never have all of its premises true, it can never attain a state with all premises true and a false conclusion.

# The Answer if You are Doing Formal Semantics

(please upvote the answer by Badrinath if this is what you were seeking) Note that if you are referring to Tarskian model-theoretic semantics and some other advanced contemporary approaches to logic that this no longer obtains -- because validity and invalidity only apply to models, and models only occur when:

A set T of sentences is called a (first-order) theory. A theory is satisfiable if it has a model \mathcal M\models T, i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set T. Consistency of a theory is usually defined in a syntactical way, but in first-order logic by the completeness theorem there is no need to distinguish between satisfiability and consistency. Therefore model theorists often use "consistent" as a synonym for "satisfiable". (wiki)

On such an account, no theory could be simultaneously inconsistent and valid, because only consistent theories are valid or invalid.

If the premises are inconsistent, then you can conclude anything from it. This is called Principle of explosion.

I follow Tarski in defining valid or invalid to be statements made vis-a-vis the existence of models for the given premises. For inconsistent premises, there are no models. In this framework, no argument can be both inconsistent and valid.

As a side note, In FOL, to handle such kind of logics, you need to go for non- classical logic methods like Paraconsistent logic, defeasible logic, Auteespistemic logic and default logic.

• No, I think that seems fair enough. The "strict" definition of validity is "whenever the premises are true, the conclusions are true"; but in fact that "strict" definition glosses over the modality involved in the "whenever". As Badrinath Jayakumar points out, this modality is generally determined by what models exist that satisfy the premises. Apr 21, 2015 at 10:05
• @BadrinathJayakumar to notify someone you need to use @ in front of their name ... / That's interesting, but see also jimpryor.net/teaching/vocab/validity.html#consistency Apr 21, 2015 at 12:56
• I've made some slight edits to make clear this is referring to a Tarskian approach. I'm not sure that's the level the OP is being asked to perform on -- also the "I believe" language originally threw me off. Apr 21, 2015 at 13:00

Suppose you know the premises of an argument are inconsistent. Do you have to do a truth table to know whether it is valid or invalid?

No. Truth tables give sometimes very counter-intuitive results and there is no convincing proof or argument that these results would be nonetheless correct.

Truth tables are consistent with the following definition of logical validity, which is used in Mathematical Logic, and is indeed the definition most often quoted on the Internet:

In logic, an argument is 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. -- https://en.wikipedia.org/wiki/Validity_(logic)

This definition is formally different from the one given by Aristotle, and indeed from various formulations proposed by philosophy websites. I don't know of any attempt to justify that all these definitions would be equivalent and mean the same thing.

Thus, I don't know of any good reason to accept that truth tables correctly represent the logic of the logical arguments that human beings may want to consider.

That being said, if you can't decide by yourself, you still have a range of possibilities. You can indeed rely on truth table logic. But you can also ask l logicians if you know one!

Short of that, you can also discuss the validity of the argument with other people you know and seem reasonable to you to see if you together can reach a sensible consensus.

I would recommend that last method if at all available to you.

• This is irrelevant and possibly misleading to a student studying modern formal logic, but nonetheless interesting from a historical point of view (assuming it is accurate). So how did Aristotle define a valid argument? Does it matter how Aristotle defined a valid argument, or for that matter anyone before Frege, Boole, and Tarski (maybe Leibniz to a degree as well). Asking people you know if an argument seems reasonable is probably the worst option.
– john
Dec 22, 2019 at 20:49
• @John (a) I provided an answer to the question asked. And there is nothing misleading in it. If you disagree, please articulate. (b) Aristotle characterised what we think of now as logic. The logic of the Stoics and the logic of the Scholastic agrees with Aristotle's (except for one Scholastic). (c) Why would what Frege, Boole or Tarski said matter more than what Aristotle, the Stoics, the Scholastic said? Or indeed any idiot picked at random on the street? (d) If asking people is the worst option, then who are you going to ask? Machines? Aliens? God? Superman? Dec 31, 2019 at 10:55
• "Why would what Frege, Boole or Tarski said matter more than what Aristotle, the Stoics, the Scholastic said? Or indeed any idiot picked at random on the street?" Because modern logic, the logic developed by Frege, and Tarski, is the logic we use when discussing 'truth tables', not Aristotle or the Stoics. As I said earlier, it is interesting from a historical point to observe the evolution of logic, but can be misleading to a student working on a truth table. And yes, asking random people is the worst option. It is better to ask an expert on logic.
– john
Jan 6, 2020 at 10:33
• @John No, the question is not about truth tables as such, but about whether we need them for arguments whose premises are inconsistent. Do we need truth tables for that? No. Truth tables are not any correct model of the kind of deductive logic humans use in their arguments. Jan 13, 2020 at 12:08
• @John As to experts on logic, sorry, no, no mathematicians is an expert on logic. Merely calling something "logic" doesn't make it so. Mathematicians have proved themselves unfit for investigating logic. They have been working on it for more than 166 years and still can't make head or tail of it. Mathematical logic is essentially mathematics, it is not logic. Not at all. Far from it. Jan 13, 2020 at 12:09