Logically Incorrect Arguments

Logic can be, of course, conceived more broadly—for example so that it includes all normative approaches to the study of reasoning. Unlike formal logic these more inclusive accounts of logic normally do not aspire at providing rigorous methods for deciding which arguments are correct (valid) and which are not.

The terminology within logic as well as within argumentation studies is somewhat unsettled. Sometimes the term logically incorrect argument is used as synonymous with fallacious argument. Walton (1986), however, points out that this is improper as there are arguments which are weak but reasonable (to some extent) and hence such arguments are “not really fallacies at all in the sense of being arguments that are incorrect”. On the other hand sometimes the term “fallacious” is used to denominate arguments that are not just incorrect but also misleading in the sense of making a delusive impression of correctness and thus being tokens of common errors in reasoning (c.f. Finocchiario 1981). An argument which is clearly incorrect thus would not be classified as fallacious.

Another, related problem emerges when we do not want to accept that every argument is either correct or incorrect. (This may happen when we accept logic other than the classical one—an intuitionist, for example, may hesitate to classify the argument “It is not true that it does not rain, hence it rains” as logically incorrect. Moreover, when we concentrate on natural language, we can hardly avoid acknowledging a grey zone of arguments that are not determinately classifiable in either of these groups). In general, lumping together all arguments that do not instantiate the favoured form may be problematic.

Or, on the pragmatic level, from meaningful utterances to a meaningful utterance. We will, however, ignore the pragmatic level of argumentation in this article.

It was Lewis Carroll (1895) who taught us that trying to articulate all such ‘covert premises’ explicitly leads us into the blind alley of an infinite regress.

We, however, believe that the delineation of the concept of logical incorrectness we propose could be easily adjusted even to deontic arguments.

Also, arguments with no premises count as containing only premises that are true (under any circumstances)—for it holds for any sentence that if it is a premise of the argument then it is true.

Here it is worth pointing out that the terms valid argument and correct argument are often used interchangeably in the literature. This terminological promiscuity is usually harmless (we could also speak about valid instead of correct arguments in this paper). We, however, think that some terminological conventions may be useful, so in this paper we will speak about correctness and incorrectness in case of arguments (series of full-fledged statements) and about validity and invalidity in case of argument forms.

Thus, there cannot be a fully-fledged sentence without a logical form just as there cannot be a sentence without a grammatical form. Which particular form (or forms) a particular sentence ‘has’ is often a controversial issue.

While Frege (1879) gives the impression that reaching a logical form is a matter of merely “forgoing expressing anything that is without significance for the inferential sequence”, already his later writings present finding logical forms as a much more complex and much less transparent process. Russell then turned the uncovering of logical forms (or logical analysis) into a true ‘art’.

See footnote 9.

Among the philosophers who pointed to the problem picture prominently Bolzano (1837, Band II, §148, p. 84) and Tarski (1936, pp. 188–190).

Though we do not expect that there is any definite, ‘tangible’ reality that is to be described, we are sure that there is a lot more to reveal and understand in this area.

Indeed, we are convinced that if it were not determinate enough the project of building logic could not have been properly launched at all.

What we mean by this are not systems of logic that purport to be ‘natural’ in the sense that they reflect some intuitions of their creator with respect to natural language, but rather true empirical research of representative samples of inferential practices of speakers of natural languages that would gather statistical data allowing the comparison of different systems–e.g., classical and intuitionist logic—as concerns their faithfulness to common speaker’s standards.

Let us note in passing that this is far from an unproblematic presupposition, but it is not a topic that we can go into in the present paper.

This kind of interpretation may even develop into a form of relativism with respect to correctness.

Note that accepting this category of correct arguments is not essential for the explication of logical incorrectness that we pursue in this paper; we include them because it makes, we think, the depiction of the landscape of arguments we present more realistic. Note also that saying that we treat All dogs live on Mars as necessarily false (within the given state of affairs) implies that adding this sentence as an additional premise to (A5) does not change the argument into an incorrect one. (As it is not possible that All dogs live on Mars is true, the resulting argument has premises that cannot be true.) An alternative possibility, which we do not consider in this paper, would be to consider arguments like (A4) or (A5) as correct only ceteris paribus and to assume that the addition of a premise may turn a correct argument into an incorrect one.

The arguments classified as “incorrect” simpliciter by this picture are not incorrect in any strong sense. For example arguments like Fido sleeps hence Fido is at home would belong to this area.

We can, of course, imagine that for example the words sister or woman might change their meanings in such a way that the above analytically correct argument becomes incorrect; but then we would say that it is not the same argument anymore.

It is obvious that any conclusion is inconsistent with inconsistent premises.

In footnote 18 we noted that the argument with the premises Fido is a dog and All dogs live on Mars and the conclusion Fido does not live on Mars turns out status quo correct. Now we see that it will not be incorrect—insofar as we treat its premise All dogs live on Mars as impossible and hence inconsistent.

It seems that the concept delineated by DefSAnInCor is coextensive with the concept of an argument having a “counter-valid” argument form, i.e. the form that has only incorrect instantiations (see Woods and Irvine 2004, p. 68).

We might think of various alternatives—for example, in view of the fact that (AF10) is equivalent to the premiseless argument form with the conclusion A₁ → (…(A_n → B)…), we might think of its opposite counterpart as the premiseless argument with the conclusion ¬(A₁ → (…(A_n → B)…)). But it is readily seen that this would not work at all.

Super-invalid argument forms have premises that are formal tautologies and a conclusion that is a formal contradiction, see Cheyne (2012, p. 51).