Clear Language, Clear Mind

Validity is defined in a couple of ways. I like to define it like this: An argument is valid iff the superconjunction¹ of all the premises and the negation of the conclusion is impossible. It sounds a bit unusual at first but it is worded that way to prevent confusion about modalities. A more common definition is: An argument is valid iff it is impossible for all the premises to be true and the conclusion false. I contend that mine is clearer when understood. Consider this argument:

n	Proposition	Explanation
1	P	Premise
2	P→Q	Premise
3	Q	From 1, 2, MP

Anyone trained in logic will immediately recognize that this argument is valid (and the form valid too) for it has the form of MP and all arguments of that form are valid. To see this using my definition of valid above we can simply make the superconjunction of the argument and put it into a truth table:

P	Q	¬Q	P→Q	P∧(P→Q)∧¬Q
T	T	F	T	F
T	F	T	F	F
F	T	F	T	F
F	F	T	T	F

Note that I skipped a conjunction step.

The superconjunction comes out as impossible.

And here comes the tricky part. Suppose that the conclusion of an argument happens to be a necessary truth. Consider this argument:

n	Proposition	Explanation
1	P	Premise
2	¬(Q∧¬Q)	From 1

Is that argument valid? It is according to my definition and the usual definition of valid. A truth table will show that:

P	Q	(Q∧¬Q)	P∧(Q∧¬Q)
T	T	F	F
T	F	F	F
F	T	F	F
F	F	F	F

Note that (Q∧¬Q) is equivalent to ¬¬(Q∧¬Q) which is the negation of the conclusion in the argument above. I skipped a double negation step.

Realize that whatever is conjoined with an impossibility such as (Q∧¬Q) will come out as false. So no matter the argument structure, an argument with a necessary truth in the conclusion is valid. This doesn’t seem to bad again, after all, we already know that anything implies a necessary truth and that an impossibility implies anything. Truth tables show that:

Necessary truth as consequent:

P	□Q	P→□Q
T	T	T
F	T	T

Note that Q is true on all rows.

An impossibility as antecedent:

□¬P	Q	□¬P→Q
F	T	T
F	F	T

Note that P is false on all rows.

But now consider that in the realm mathematics all propositions are either necessary truths or impossibilities. Thus, any mathematical argument that happens to have a necessary truth in its conclusion is valid, no matter the form of it. That seems like an odd conclusion.

I’ve read many times and places that an argument is sound iff

1.      The argument is valid.

2.      All the premises of the argument are true. [1]

Let’s call this the standard definition of a sound argument. Now consider this argument:

1.      If the Earth is a planet, then there is a satellite above Denmark this moment.

2.      The Earth is a planet.

3.      Thus, there is a satellite above Denmark this moment.

The argument is valid. (Form: Modus ponens.) Is this argument sound? The second premise is true. The first is true from time to time but I guess it is false most of the time. It follows then, given standard definition, that this argument is sound from time to time.

But still this argument should never convince anyone. Why not? It should not because the first premise is never justified for anyone. Even though the argument is sound from time to time given the standard definition, it should not be convincing for anyone at anytime. This flies in the face of the normal practice in discussions of trying to establish that one’s arguments are sound.

Consider this revised definition of a sound argument. An argument is sound iff

1.      The argument is valid.

2.      All the premises of the argument are justified.

This seems to solve the above problem but it’s still not good enough. Here is why. In the second condition, justified for whom? Oneself? If so, then soundness becomes subjective. I take this as a reductio. Soundless is an objective matter. Thus, the revised definition is false.

Perhaps a new definition of soundness is not what is needed but rather a distinction between the soundness and convincingness of arguments. There are at least two kinds of convincingness of arguments. One kind of convincingness is that the argument convinces someone to change their beliefs. The argument may be completely bogus (e.g. invalid) but the person does not realize that.

But philosophers are usually not interested in rhetoric. Philosophers usually, thus, talk about arguments being rationally convincing. This is the second kind of convincingness. I take it to mean that an argument ought to convince a rational person. Let’s work with this.

What are the criteria for an argument being rationally convincing? Soundness is neither a sufficient nor a necessary condition of rationally convincingness. I wonder what is. Maybe a subjective definition works here. Consider this definition of a rationally convincing argument. An argument is rationally convincing for person p iff

1.      The argument is valid.

2.      All the premises of the argument are justified for person p.

A person may not recognize when something is justified for him to believe, and so he can fail to be convinced (i.e. change beliefs) by an argument that is rationally convincing for him.

A person may know of an argument that is rationally convincing for him. Suppose that I don’t know that argument and it contains some proposition that I’ve never heard of. Then it follows that the argument is not rationally convincing for me. Further, suppose that I know the argument and one of the premises are not justified for me. Then, the argument is rationally convincing for him but not for me. I’m happy with the implications above. They seem to capture what is meant with rational convincingness.

So, next time when we are in a discussion, we are not really trying to prove that our arguments are sound. We are trying to prove that they are rationally convincing for the other person. The confusion might arise because the two, soundness and rationally convincingness, overlap much of the time.

The reason why I started thinking about this is a fine tuning argument. I don’t particularly recall the argument but let’s suppose that it has three premises. I think that two of them are true but the last one I don’t know about, and neither does the proponent or anyone else. The last premise is unjustified for everybody. The argument might be sound but no one knows. The interesting thing seems to be whether it is rationally convincing not whether it is sound.

Readers of this essay may want to read this essay as it is relevant.[2]

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[1] Here’s two: www.fallacyfiles.org/glossary.html, en.wikipedia.org/wiki/Soundness#Sound_arguments,

[2] deleet.dk/2009/02/23/jtb-and-the-first-person-perspective/

As Wikipedia notes,ⁱ the term is used in everyday speech to deny a conditional (if-then statement), and it is also used in logic to mean an invalid argument. This together with the fact that in normal language people do not write the complete arguments when they argue, makes it very easy to confuse things.

Suppose that someone says “Since that global warming is undesirable, we ought to not burn more coal.” Another person then responds to that with “Non-sequitur“. What did the second person criticize? The argument structure or validity? That structure is hidden and we have to guess at what it is. (Se Note 1.) The conditional? Maybe.

We should therefore at least not use the term ‘non-sequitur’ without also clarifying what the problem is. But we ought not to use the term at all since there are better alternatives. If one means that the argument is invalid, then say “The argument is invalid”. If one means that a conditional is false, then say “The conditional is false.”. This recommendation is for discussions of a certain quality only. Chances are that if you argue with a random member of the mob, then that person will not know what it means to say that an argument is invalid or what a conditional is.

There are also similar phrases such as ‘That doesn’t follow’. Non-sequitur is latin for ‘it does not follow’.

Note 1

The principle of charity applies in that case. We don’t know which argument structure the person had in mind when making the argument. If it is member of the mob, then he or she probably doesn’t know either which structure it has. But if we should follow the aforementioned principle, then we ought to make an interpretation of the argument in question that is valid. This is sometimes a task in itself.

ien.wikipedia.org/wiki/Non_sequitur_(logic)