false dilemma: a systematic exposition

False Dilemma: A Systematic Exposition

Taeda Tomic

Published online: 21 January 2013

� Springer Science+Business Media Dordrecht 2013

Abstract False dilemma is a specific form of reasoning: despite the fact that it is

based on a deductively valid argument form, it is rightly depicted as fallacy.

A systematic exposition of false dilemma is missing in theoretical approaches to

fallacies. This article formulates six criteria for a well-grounded exposition of a

fallacy, suggesting also a systematic exposition of false dilemma. These criteria can

be used to both explain, and categorise, the various false dilemma fallacies. The

article introduces distinction between four types of false dilemma (and the

respective subtypes): (1) False quandary, (2) Defeasible sound quandary, (3) False

obstruction and (4) Defeasible sound obstruction. The types of criticism appropriate

for each variant of false dilemma are suggested. Being able to discover false

dilemma in situations when it is important to make good and free choices is a

significant dimension of critical thinking. It may liberate us from accepting the

consequences that necessarily follow from prearranged alternatives, when these are

not the only alternatives possible. It may also liberate us from refraining from

actions on the basis of obstructive disjunctive statements. In that respect, the faculty

of critical thinking directed at uncovering reasoning based on false dilemma may

also initiate the discovery of new alternatives, or discover the ways of unifying

seemingly opposing alternatives. Thus, a well-developed ability to understand the

tricky argumentative moves of false dilemma may be a good step towards initiating

some features of creative thinking.

Keywords False dilemma � Constructive dilemma � Destructive dilemma �Fallacies � Fallacy � Argumentation structure � Defeasible argument �Critical thinking � Argument criticism � Argumentation theory

T. Tomic (&)

Department of Philosophy, University of Uppsala, Box 627, 751 26 Uppsala, Sweden

e-mail: [email protected]

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Argumentation (2013) 27:347–368

DOI 10.1007/s10503-013-9292-0

1 Why Bother?

Since Hamblin’s initial work (1970), the failure of deductive validity as the essential

flaw in fallacies (i.e. types of arguments that are not logically correct but may

misleadingly appear to be so) has been rightly challenged. Want of deductive

validity cannot systematically explain why fallacies are flawed, as many fallacies do

not even pretend to deductive validity (Hamblin 1970, 90–223). Consequently, the

reason why they are flawed cannot be their lack of deductive validity. Good

examples of such fallacies are the so called informal fallacies: types of arguments

that are logically incorrect due to aspects other than their deductive invalidity. Some

of informal fallacies are known as the ad fallacies (e.g. ad verecundiam and ad

hominem arguments); other examples are informal fallacies based on some missing

aspects of inductive strength (e.g. some features of weak analogy, sequndum quid or

some forms of causal fallacies), or some missing aspects of plausibility strength

(e.g. weak analogy, slippery slope, straw man, appeal to ignorance, avoiding the

burden of proof, shifting the burden of proof, begging the question, equivocation,

many questions).

Even other aspects of logical incorrectness, different from deductive invalidity,

improper inductive or plausibility strength, and falsity or unacceptability of

premises, are discussed in theories of logic and argumentation. For instance, pragma

dialecticians define some of such aspects, e.g. preventing each other from advancing

standpoints or from casting doubt on standpoints; presenting a standpoint as self-

evident; falsely presenting something as a common starting point (van Eemeren and

Grootendorst 1992, 208–211). Walton’s approach to informal fallacies also points

out some of the above mentioned aspects of logical incorrectness: for instance, a

fallacy may consist in explicit or implicit attempts to avoid answering or to block

critical questions, even if the reasoning in itself might be plausibly strong and

comprise highly acceptable premises (Walton 1997, 231–257). Even in the classical

theories of formal logics, aspects of logical incorrectness other than deductive

invalidity or falsity of the premises are discussed, e.g. rules of definition or

relevance of premises. In Sects. 3.3 and 3.5 of the present article, some aspects of

logical incorrectness specific for dilemma reasoning and not limited only to

deductive invalidity, missing aspects of inductive or plausibility strength, and falsity

of premises are suggested.

As a matter of fact, besides formal fallacies such as affirming the consequent and

denying the antecedent, none of the fallacies from the standard lists of today can be

explained by applying the criterion of missing deductive validity. For some standard

lists of today’s fallacies, see for instance Copi and Cohen 1986 (2005); Hamblin

1970; Grootendorst and van Eemeren 1992; Walton 1987; Hansen and Pinto 1995;

Tindale 2007. Thus, it is proper to define only formal fallacies as types of arguments

that are not deductively valid but may misleadingly appear to be so.

False dilemma is, though, a rather specific type of reasoning. Despite the fact that

it is usually based on deductively valid argument forms, it comprises logically

incorrect reasoning based on other aspects than deductive invalidity—even if due to

its very deductive validity it may misleadingly appear to involve only logically

correct reasoning—and is thus rightly depicted as fallacy. Apart from circular

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argumentation/begging the question, the ways of reasoning which despite their

deductive validity are rightly defined as fallacies have not been much explored in

theories of fallacies. An exception is an article of Boone (1999). The phenomenon

of deductively valid fallacies deserves attention since it provides additional

evidence for the thesis that deductive validity may not always be a proper, nor a

sufficient criterion, for explaining the principles of logically well-grounded

thinking.

Another reason why a systematic exposition of false dilemma is important is that

some interesting theories have been initiated by implicitly searching solutions for

specific examples of false dilemma reasoning. One of them is the solution for the

Liar paradox developed in one version of paraconsistent logics (Priest 2006,

particularly part I.1 and II).

A third reason is that many established theoretical approaches to fallacies do not

at all deal with the fallacy of false dilemma (see for instance Sidgwick 1886;

Hamblin 1970; Walton 1987; Tindale 2007). Other works, such as Copi and Cohen

1986 (2005, 297–300) and Govier (2001, 293), do not explicitly distinguish between

dilemma and false dilemma and do not classify false dilemma as a fallacy, even if

they provide a systematic analysis of a number of cases in which the conclusion of

dilemma-reasoning should be rebutted, as well as a clearly specified corresponding

methodology of rebuttal. According to Copi and Cohen’s definition of formal

fallacies [1986 (2005, 126)], false dilemma cannot be defined as formal fallacy since

the forms of false dilemma discussed in their work are deductively valid arguments.

Surprisingly, neither do they define it as an informal fallacy, despite their definition

of informal fallacies as ‘‘the types of mistakes of reasoning that arise from the

mishandling of the content of the propositions constituting the argument’’ [Copi and

Cohen 1986 (2005, 126)], and the specific classification of the possible ways of

‘‘mishandling the content’’ as those that either (1) use irrelevant premises, (2) use

inadequate premises, (3) make unjustified assumptions, or (4) shift the meaning of

words used in the premises. Govier (2001, 293), on the other side, provides a nice

example of a false dilemma argument based on the false disjunctive statement (or

false dichotomy as Govier calls it), but does not define the very false dilemma

fallacy. Govier also rightly points out that false dichotomy is not an argument but a

false belief (Govier 2001, 441), which is the main reason why it cannot be equal

with false dilemma fallacy. Kelley (1998, 372–373) defines clear argumentative

schema of complex dilemma, as well as the fallacy of false alternative (1998,

141)—which corresponds to false disjunctive proposition, or false dichotomy in

Govier’s terms, but does not combine these two issues, nor defines the fallacy of

false dilemma. For elaboration on the distinction between on the one side false

dichotomy or false alternative (which is a false proposition) and on the other side

false dilemma (which is a wrong way of arguing), see next paragraph. Yet other

works on fallacies include false dilemma in their fallacy collections but explain only

one or two variants of the fallacy; thus providing only partial explanations of it, e.g.

Boone 1999.

Finally, some authors make an easily made classification mistake in analyzing (1)

unsound disjunctive syllogism (having an incomplete or misleading disjunctive

premise) as if it is fallacy of false dilemma, or (2) false-dichotomy statements as a

False Dilemma: A Systematic Exposition 349

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fallacy of false dilemma; e.g. Van Eemeren and Grootendorst 1992, 190–194.

However, these types of reasoning actually do not rely on the argumentation

schemas/argumentation structures of dilemma-reasoning. For a clear formulation of

argumentation schemas of false dilemma, see Argumentation schemas 1 and 2 in

Sect. 3.1 and Argumentation schemas 4 and 5 in Sect. 3.4. Thus, reasoning

mentioned in (1) above relies instead on argumentation schema of disjunctive

syllogism which considers only the logical relation between the disjuncts involved

in a disjunctive statement, without considering the consequences following from

each of the disjuncts contained in the disjunctive premise, or the logical

consequences of the disjunction between the negated consequents of two or more

implicative statements, which are the differentia specifica of the two basic

argumentation schemas of false dilemma. Reasoning mentioned in (2) above is a

false disjunctive statement, not an argument, whereas false dilemma is an argument;

a distinction nicely made in Govier’s discussion on false dichotomy (Govier 2001,

234–235, 441). That points out the importance of criterion (1) defining a clear

exposition of a fallacy and provided in Sect. 2 below. Unfortunately, this robs van

Eemeren and Grootendorst’s correct and intelligent analysis of the ways in which

different types of disjunctive premises may be incomplete or misleading, of a fair

amount of its theoretical grounding. Indeed, the incompleteness or falsity of the

disjunctive premise is a necessary but not a sufficient condition for classifying a

type of reasoning as false dilemma. False dilemma also (implicitly or explicitly)

involves the argumentation schemas of dilemma reasoning, as elaborated below.

2 What Is a Systematic Exposition of a Fallacy?

A systematic exposition of a fallacy fulfills the following six criteria:

1. It defines the argumentation schema(s)/argumentation structure(s) implicitly or

explicitly comprised in the reasoning characterizing the fallacy.

2. It defines an adequate type of argument evaluation relevant for the fallacy.

Thus, it explains (a) what type of argument standard (deductive, inductive or

plausible) the analyst can most charitably use in analyzing the reasoning

involved in the fallacy, and consequently (b) what type of the argument

evaluation is suitable for the reasoning actually contained in the fallacy.

3. It also explains in which way the criteria for the adequate reasoning schema and

the type of reasoning evaluation are not fulfilled and thus why the given

reasoning is fallacious.

4. It distinguishes between eventual variants—i.e. various types—of the fallacy.

5. A systematic exposition of a fallacy supplies some possible reasons why the

given way of thinking may seem a good argument even though it actually is not

(Walton 1987, 93–96).

6. It suggests types of criticisms proper for handling the fallacy. (For some

interesting works about strategies for argument criticism, see Krabbe, 1992,

1999; Walton et al. 2008, 220–274; Krabbe and van Laar 2010).

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3 Four Types of False Dilemma

It is possible to distinguish between four types of false dilemma: (1) False

quandary, (2) Defeasible sound quandary, (3) False obstruction and (4) Defeasible

sound obstruction. A common feature of all four types is that: (a) they all appeal to

one of the deductively valid argumentation schemas of dilemma reasoning (see

Argumentation schema 1 and 2 in Sect. 3.1, and Argumentation schema 4 and 5 in

Sect. 3.4); and (b) they all in one way or another misleadingly appeal to a

disjunctive premise specific for false dilemma. A difference between the four

variants is that: (a’) they appeal to different types of the deductively valid schemas

of dilemma reasoning, and (b’) some of them are deductively valid arguments being

fallacies on the basis of their masked unsoundness; the others are deductively sound

arguments but still fallacies since they involve intentional or unintentional

obscuring of the information relevant for the given reasoning.

Before we continue with analysing each of the four types of false dilemma, it is

important to clarify the relation between the argumentation schemas and the

examples used in the article. The schemas are stated in the language of propositional

logic, whereas some statements used in the examples (particularly in Examples 4, 5

and 6) involve expressions about possibility and necessity, best captured by modal

logic, or expressions about moral obligations, best captured by deontic logic.

Nevertheless, the schemas only formalize the aspects of the examples insofar as the

standard language of propositional logic enables one to do, even though the

examples are richer, and also have the modal/moral element. Moreover, the

schemata make it clear how the disjunction functions within arguments and make

thus the analysis of false dilemma in the examples easier than if the formalization

based on modal or deontic logic was involved.

3.1 False Quandary

The first type of false dilemma, false quandary, consists of two subtypes; namely,

false simple constructive dilemma appealing to deductively valid schema of simple

constructive dilemma, and false complex constructive dilemma relying on deduc-

tively valid argumentation schema of complex constructive dilemma. Despite its

deductive validity, false quandary is a fallacy since as a matter of fact the target type

of the argument is a deductively sound argument, whereas the actually provided

argument is deductively valid but has a false disjunctive premise and is therefore not

sound. Moreover, the falsity of the disjunctive premise in false quandary is usually

well hidden beyond the argument’s deductive validity. Simple constructive dilemma

is symbolically represented in the following way:

(p1) A ∨ B(p2) A → C(p3) B → C

C

Argumentation schema 1 Simple constructive dilemma


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The symbolic argumentation schema says that if claim A or B is true, as stated in

premise (p1), then, if it is also true that C follows both from A and from B [as

claimed in premises (p2) and (p3)], it deductively follows from the premises that C

is the case.

Complex constructive dilemma is based on a similar argumentation schema,

except that each of the disjuncts implies different consequent and the conclusion is

thus another disjunctive claim:

(p4) A ∨ B(p5) A → Q(p6) B → R(p7) Q ∨ R

Argumentation schema 2 Complex constructive dilemma

Let us now consider two examples so as to realise in which way simple and

complex constructive dilemma are transformed into the specific variants of false

dilemma.

(s1) Either you tell the truth, or you lie.(s2) If you tell the truth, you force your friend into a social tragedy; and therefore, are an immoral person.(s3) If you lie, you are an immoral person (since it is immoral to lie).You are an immoral person (whatever choice you make in the given situation).

Example 1 The false quandary type of false dilemma based on false simple constructive dilemma

(q1) You may either confirm that your friend likes the positive aspects of liberal politics, or you may deny that your friend likes the positive aspects of liberal politics. (q2) If you confirm that your friend likes the positive aspects of liberal politics, your friend is going to be sent to prison. (q3) If you deny that your friend likes the positive aspects of liberal politics, someone else (who heard you two talking about the liberal politics) is going to contradict you, so that both of you are going to be sent to prison. Either your friend is going to be sent to prison or both of you are going to be sent to prison.

Example 2 The false quandary type of false dilemma based on false simple constructive dilemma

As we may see, both examples deal with deductively valid arguments, having

the argumentative structure of the dilemma-reasoning. Nevertheless, the arguments

are unsound because they have a false disjunctive premise and therefore fail to

meet the implicit aim of the reasoning in both examples, namely the aim of being

deductively sound arguments (i.e. deductively valid arguments with actually true

premises).

Even if the premises (s2) and (s3) in the first example, and the premises (q2)

and (q3) in the second example were true, the falsity of the disjunctive premises

(s1) and (q1) is the specific characteristic of the false quandary type of false

dilemma. Typically, the possibilities stated in the disjunctive premise are falsely

presented as the only available alternatives; the disjunctive claim is thus

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incomplete and therefore false. In the first example, it is possible both to avoid

saying the truth and to avoid lying, for instance by staying silent. In the second

example, it is not true that the two given alternatives are the only two possibilities

available, since there are ways of avoiding to testify and thus of avoiding to

accept the violent force of either telling the truth or lying in a politically corrupted

and biased court. It is therefore not necessary to accept the consequences of the

given two alternatives, since the disjunctive claim may involve at least three

alternatives and the third alternative might suggest a different and more preferable

consequent.

3.2 Intermezzo: What is a Defeasible Argument?

Before we go into details of the second type of false dilemma, defeasible sound

quandary, we need to define a defeasible argument. Following the concept of

epistemological defeasibility (e.g. Lehrer 1965; Annis 1973; Swan 1974), the

concept of defeasible argument has started to figure in argumentation theory (e.g.

Bowell and Kemp 2005, 229; Walton et al. 2008, 223).

According to Bowell and Kemp’s clear definition of defeasible arguments, there

is a distinction between refuting an argument and defeating it. As in the traditional

meaning, an argument is refuted either if it is shown that something is wrong with

its inference steps (that is, if the argument is shown to be deductively invalid,

inductively weak or eventually presumably weak); or that the argument relies on

false premises; or both. In difference to a refuted argument, Bowell and Kemp

suggest, when an argument is defeated, neither the logic of the argument, nor the

truth or acceptability of the premises is challenged. Instead, new relevant

information is added on the basis of which the conclusion of the initial argument

is not any more acceptable, although the initial argument is in itself still logically

well-grounded and still has true or acceptable premises. Bowell and Kemp (2005,

229) use a simple example to illustrate their point:

(d1) Almost all residents of Inverness own at least one item of woolen clothing. (d2) Jane is a resident of Inverness. (Probably) Jane owns at least one item of woolen clothing.

Despite the fact that the argument is inductively strong and that there are reasons

for a person to accept both premises (d1) and (d2), additional information about

Jane’s allergy to wool suggests that the conclusion of the argument is not

acceptable, even if it still holds with a high level of probability from the true

premises. By means of the additional relevant information, the initial inductively

strong argument with acceptable true premises is therefore defeated, even if not

refuted (since the initial argument is still inductively forceful and still has

acceptable premises).

According to Bowell and Kemp, only inductively forceful arguments may be

defeated, but not deductively valid arguments (they do not talk about plausible

arguments). The authors define a defeated inductively forceful argument in the

following way (italics added):


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To say that an inductively forceful argument is defeated for a person is to say:

The person reasonably believes the premises, but, nevertheless, reasonably

rejects the conclusion. (Bowell and Kemp 2005, 229)

The reason why, according to Bowell and Kemp, a deductively valid argument

cannot be defeated is that once an argument’s deductive validity has been proven

the only remaining strategy is to refute it by challenging the acceptability of its

premises. However, if the premises of the argument are true or reasonably

acceptable (as the above definition of defeated argument implies), there is no

additional information that may be added to the argument so that the initial

argument is still deductively valid and has true or acceptable premises, but that the

new information might challenge the acceptability of the conclusion.

Walton et al. (2008, 223) seem to relate the concept of defeated argument to

inductively strong and to plausible arguments:

Many common arguments are defeasible (fallible), meaning that even if the

premises give reasons to support the conclusion as true, new evidence may

later come in that shows that the conclusion is false. Such arguments hold only

tentatively, and may have to be retracted later if new evidence shows that the

original argument is no longer tenable.

Walton et al. (2008, 223) seem to take a strategy of defeating as one of the argument-

refuting strategies. According to them, there is a distinction between defeated and

refuted arguments: not every refuted argument is a defeated argument, but all defeated

arguments are refuted. Their point is that the strategy of defeating an argument consists

in adding new relevant information to e.g. a presumably strong argument that, due to its

presumptive logical strength, may always be a subject to discussion. The initial

presumably strong argument holds only tentatively and new evidence may always be

added so as to defeat the initial argument and to show that it does not hold any more.

Bowell and Kemp suggest that the initial inductively forceful argument does not

lose its inductive strength when new information leading to rejection of the

argument’s conclusion is obtained; it remains inductively forceful with reasonably

acceptable premises, but the conclusion is rejected due to the additional

information. On the other side, Walton, Reed and Macagno suggest that the whole

initial argument holds only tentatively and is, as a whole, retracted by the additional

information. Let us consider an additional example of a defeated argument:

(k1) 78% of Russians are orthodox Christians.(k2) Oleg is Russian. (C1) Oleg is orthodox Christian.

Argument a

(k3) No Russian communist is religious. (k4) Oleg is Russian communist.(C2) Oleg is not religious.(C3) Oleg is not orthodox Christian.

Argument b Relates to Argument a

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According to Bowell and Camp (2005, 229) Argument a is in itself still an

inductively forceful argument. The conclusion (C1) still follows from the premises

(k1) and (k2), with probability of 0.78. The new information only enables the

conclusion that, regardless of the 0.78 argumentative probability of Oleg being

orthodox Christian on the basis that he is Russian, Oleg does not belong to the 78 %

of Russians who are orthodox Christians. But it is still reasonable to expect that his

orthodox Christianity follows from (k1) and (k2) with probability 0.78.

In contrast, Walton et al. (2008, 223) suggest (see their definition above) that the

initial argument as a whole, and not only the conclusion of the argument, is retracted

when the additional information is provided. It is an interesting point since Walton,

Reed and Macagno would like to keep the distinction between refuted and defeated

arguments. But the distinction relies on the very fact that defeating strategies of

argument criticism succeed in challenging the conclusions of certain arguments,

even if the arguments themselves are not refuted, that is, even if the arguments’

logical strength or the truth or acceptability of the premises is not challenged. This is

the case in the above argument that is defeated because its conclusion is, due to the

new information, no longer acceptable; even if the initial argument in itself keeps its

high level of inductive strength and is thus not refuted. In brief, Bowell and Kemp’s

definition of defeated arguments is better at upholding the distinction between

refuted and defeated arguments.

Nevertheless, it is interesting that Bowell and Kemp plead for the impossibility of

defeating a deductively valid and sound argument. Walton et al. analysis (2008,

366) suggests that it is possible to distinguish between, on the one hand, a strict form

of a deductively valid argumentation schema which holds necessarily and, on the

other hand, a corresponding less strict and thus defeasible argumentation schema

that simply resembles the strict schema but in a non-valid way. Consequently, the

authors distinguish between strict modus ponens (SMP) and defeasible modus

ponens (DMP). SMP is the deductively valid argumentation schema, whereas DMP

merely resembles SMP but in a non-valid way. DMP holds generally, in most

situations (and may thus be qualified as inductive or presumptive type of argument),

but there might be situations in which it is not applicable. By means of additional

information it is then possible to clarify why the rule of DMP is not applicable in

some particular case. It is therefore obvious that Walton, Reed and Macagno’s

approach does not allow defeat of deductively valid arguments.

In this article, the principles of Bowell and Kemp’s definition of a defeasible

argument are accepted as far as it concerns defeasibility of inductively or plausibly

strong arguments. However, a new approach to defeasibility of deductive arguments

is suggested. In difference to Bowell and Kemp’s approach, I would like to suggest

that it is possible to defeat deductively valid arguments with true premises (that is,

deductively sound arguments), without challenging their validity or the truth of their

premises. In difference to Walton, Reed and Macagno, I would like to suggest a

defeating strategy that does not consist in challenging the universal meaningful

applicability of a plausible argumentation schema that only resembles to a

corresponding deductively valid schema but in a non-valid way. According to the

idea of the present article, a deductively sound argument is defeated if and only if:


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1. New true and for argument relevant information is added to the initial set of

premises;

2. The initial argument is still deductively sound, even if the initial set of premises

is augmented;

3. The new information leads to at least one additional relevant conclusion that

logically follows from the augmented set of premises;

4. The initial conclusion is not withdrawn on the basis of the new additional

relevant conclusion(s) that logically follow(s) from the augmented set of

premises; the new conclusion(s) rather supplement the initial conclusion;

5. The new conclusion is either:

(5a) not compatible with the initial one, or

(5b) compatible with the initial one but comprises a new relevant information

that (intentionally or unintentionally) was obscured in focusing only to

the initial set of premises and the initial conclusion; and

6. Preferences between the conclusions following from the augmented set of

premises are such that the new conclusion is preferred to the initial one.

Let us now go further, to see how the suggested definition is implemented in the

second type of false dilemma—defeasible sound quandary.

3.3 Defeasible Sound Quandary

The second type of false dilemma, defeasible sound quandary, is a deductively

sound type of quandary argument and has two subtypes: defeasible sound simple

constructive dilemma and defeasible sound complex constructive dilemma. The

subtypes of sound quandary are fallacies, since they involve intentional or

unintentional overemphasized focus on the negative consequences of the disjuncts.

In that way, they obscure other information that might be relevant for evaluation of

the given reasoning and its consequent actions, e.g. information about other,

positive consequences of the disjuncts. Consequently, despite their deductive

soundness, the subtypes of sound quandary may be defeated due to the possibility of

preferring some other, positive consequences of each of the disjuncts, in the

implicative premises of the arguments.

Example 3 illustrates how the complex dilemma in Example 2, even if all the

three premises were actually true, is turned into defeasible sound complex dilemma

(and thus into false dilemma) since the consequents of the premises (q2) and (q3) do

not have to be the only relevant consequents following from the true antecedents of

(q2) and (q3). In other words, additional information may imply that it is not

necessary to act in accordance with and thus to accept the conclusion of the

argument presented in Example 2, even if the argument in itself is still a deductively

sound quandary. The new information opens up a possibility to prefer other positive

consequences that also follow from the same disjuncts; and thus to defeat the

conclusion from the original argument. The focus on negative consequences of

disjuncts of a true disjunctive premise is not necessary, if there are other positive

consequences following from the disjuncts; this being the reason why a deductively

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sound quandary may be defeated. Similar examples relating to simple constructive

dilemma may easily be constructed. [For the principles of defeating strategy

described above, see also Copi and Cohen 1986 (2005, 299–300)].

(q1) You may either confirm that your friend likes the positive aspects of liberal politics, or you may deny that your friend likes the positive aspects of liberal politics. (q5) If you confirm that your friend likes the positive aspects of liberal politics, your friend’s open expression of the attitudes is going to encourage other people with similar attitudes to express their opinions openly, which may lead to challenging the political habits of censuring common opinion. (q6) If you deny that your friend likes the positive aspects of liberal politics, you might save your friend from the negative risk of being sent to prison.Either your friend’s attitudes are going to encourage other people with

similar attitudes to express their opinions openly, which may lead to challenging the political habits of censuring common opinion; or you might save your friend from the negative risk of being sent to prison.

Example 3 Defeating of the (here assumed to be) sound quandary presented in Example 2: the focus onother, positive consequences of disjuncts is also possible

To make our point more transparent, let us: (a) symbolize the propositions

involved in Examples 2 and 3; (b) formulate Argumentation schema 3 that

represents the structure of reasoning in the defeating strategy considered above; and

(c) explain the defeating strategy by relating it to the definition of defeated

deductively sound argument given in the end of Sect. 3.2. We first introduce the

following symbolisation:

A: You may confirm that your friend likes the positive aspects of liberal

politics.

B: You may deny that your friend likes the positive aspects of liberal politics.

Q: Your friend is going to be sent to prison.

R: Both you and your friend are going to be sent to prison.

Z: Your friend’s open expression of the attitudes is going to encourage other

people with similar attitudes to express their opinions openly, which may lead

to challenging the political habits of censuring common opinion.

W: You might save your friend from the negative risk of being sent to prison.

We now formulate Argumentation schema 3 that structures the reasoning of the

present defeating strategy:

(q1’) A ∨ B(q2’) A → Q(q3’) B → R[(q5’) A → Z][(q6’) B → W](Con a) Q ∨ R[(Con b) Z ∨ W]

Argumentation schema 3 The strategy of defeating the reasoning given in Argumentation schema 2


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Let us now explain the defeating strategy represented by Argumentation schema

3, in relating it to the definition of the defeated deductively sound argument given in

the end of Sect. 3.2. As we may see, the initial argument from (q10), (q20), (q30) to

(Con a) is a deductively sound complex constructive dilemma argument, under

assumption that the premises are true. We see that the initial argument corresponds

to Argumentation schema 2 and Example 2. So, the condition (2) from the definition

is fulfilled. Condition (1) from the definition is also fulfilled, since the initial

argument is augmented by new relevant information represented here by premises

(q50) and (q60). To point out the fact that the additional information is (intentionally

or unintentionally) obscured in the initial argument, we bracket the corresponding

premises (q50), (q60) and (Con b). A crucial aspect of the defeating strategy is then to

make the bracketed information explicitly available. We see that argument from

(q10), (q50) and (q60) to (Con b) corresponds to Example 3. Now, condition (3) from

the definition is also fulfilled since the new set of premises leads to at least one

additional conclusion, namely (Con b). Additionally, we do not withdraw (Con a); it

still logically follows from the augmented set of true premises, which fulfills

condition (4) of the definition. Moreover, (Con b), even if compatible with (Con a),

comprises a new relevant information that (intentionally or unintentionally) was

obscured in focusing only to the initial set of premises and the initial conclusion;

this fulfills the condition (5b) of the definition. Finally, in defeating the reasoning

presented in Argumentation schema 2 and Example 2, we do not challenge the

deductive validity of the argument nor the truth of the premises, but defeat (Con a)

as the only reasonable guidance for acting since we—in accordance with condition

(6) of the definition—prefer (Con b) as a more reasonable guidance for acting in the

given situation, even if both conclusions logically follow from the augmented set of

true premises. This defeats the reasoning given in Argumentation schema 2 and in

Example 2.

It is interesting to point out that both the false quandary and the defeasible sound

quandary seem to be logically well-grounded arguments due to their deductively

valid logical form, despite their failure in this respect! When implemented in natural

language and in existential contexts in which, for instance, political or social inertia

or diverse political violence forms are dominating, deductive validity of the two

types of false dilemma easily leads the focus away from their pertinent soundness or

defeasibility. This may easily lead to forgetting about the potentially false premises

dictating the alternatives available (in case of false quandary), or to neglecting other

relevant information, e.g. the possible positive consequences of the given disjuncts

and thus to overemphasizing either the chance or dis-utility of certain hazards (in

case of defeasible sound quandary). It is worth noticing that both false quandary and

defeasible sound quandary may also work the other way, overemphasizing thus

either the chance or utility of the positive consequences following from the disjuncts.

3.4 False Obstruction

The third type of false dilemma may be called False obstruction. It relies on

deductively valid argumentation schema of destructive dilemma. There are two

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subtypes of false obstruction: false simple destructive dilemma appealing to

deductively valid schema of simple destructive dilemma; and false complex

destructive dilemma relying on deductively valid argumentation schema of complex

destructive dilemma. (For an established definition of destructive false dilemma and

its subtypes, see for instance Pospesel 2000, 108; complex destructive dilemma is

also defined in Kelley 1998, 372–373). Despite the fact that it is based on

deductively valid argumentation schema, false obstruction is a fallacy since

(similarly as in false quandary) the target type of the argument is a deductively

sound argument whereas the actually provided argument is deductively valid, but

not sound because it has a false disjunctive premise, or at least the evidence for the

truth of the disjunctive premise is missing. Additionally, the falsity or the lack of

evidence for the truth of the disjunctive premise in false obstruction is usually well

hidden beyond the deductive validity of the underlying argumentation schema of

destructive dilemma.

Simple destructive dilemma is symbolically represented by following argumen-

tation schema:

(a1) E → G(a2) E → H(a3) ¬G ∨ ¬H

¬E

Argumentation schema 4 Simple destructive dilemma

The deductively valid argumentation schema says that if it is true that E implies

G, and if it is true that E also implies H, as stated in premises (a1) and (a2), then, if it

also is true that at least one of the consequents of the premises (a1) and (a2) is false

[that is, if either G is false or H is false, as claimed in premise (a3)], it deductively

follows from the premises that E is false.

Complex destructive dilemma relies on a similar argumentation schema, except

that the implicative premises do not have the same antecedent, but still do have

different consequents; the conclusion is thus a disjunctive claim with two negative

disjuncts that are the corresponding negations of the implicative premises’

antecedents:

(b1) F → J(b2) K → L(b3) ¬J ∨ ¬L

¬F ∨ ¬K

Argumentation schema 5 Complex destructive dilemma

The deductively valid argumentation schema for complex destructive dilemma

says that if antecedent F implies J, and antecedent K implies L [as claimed in

premises (b1) and (b2)], then, if it is true that at least one of the consequents of the

premises (b1) and (b2) is false [as claimed in premise (b3)], it deductively follows


123

from the premises that at least one of the antecedents is false. Let us now consider

two examples so as to see how simple destructive dilemma and complex destructive

dilemma are transformed into the specific types of false obstruction and thus into the

specific variants of false dilemma.

(a1’) If you teach young people how to be social, to have fun and to reach calmness, harmony and happiness without using alcohol, it will diminish many various forms of alcohol abuse.

(a2’) If you teach young people how to be social, to have fun and to reach calmness, harmony and happiness without using alcohol, it will reduce the enormous profit of the alcohol industry. (a3’) However, either many various forms of alcohol abuse will never be diminished, or the enormous profit of the alcohol industry will never be reduced. You should not teach young people how to be social, to have fun and to reach calmness, harmony and happiness without using alcohol.

Example 4 The false obstruction variant of false dilemma based on the false simple destructivedilemma

(b1’) If many more or less wealthy people from the rich part of the world regularly give away a small amount of money to the governments of poor countries, the world’s poverty will diminish.(b2’) If many countries from the rich part of the world work on the development of industry, health care and education in the poor countries, the poor countries will gradually succeed in developing their own economy.(b3’) Nevertheless, it is either impossible to diminish world’s poverty, or it is impossible for the poor countries to gradually succeed in developing their own economy. Either the wealthy people from the rich part of the world should not regularly give away a small amount of money to the governments of poor countries, or many countries from the rich part of the world should not work on the development of industry, health care and education in the poor countries.

Example 5 The false obstruction variant of false dilemma based on the false complex destructivedilemma

Obviously, both examples are based on deductively valid arguments since they

are based on the argumentation schemas of destructive dilemma. However, the

arguments are unsound since they contain a false disjunctive premise or at least a

disjuntive premise without evidence for its truth, and therefore, similarly to the false

quandary arguments, fail to meet the implicit aim of the reasoning, namely to

provide deductively sound arguments.

Under assumption that the premises (a10) and (a20) in Example 4 and the

premises (b10) and (b20) in Example 5 are true, the specific characteristic of false

obstruction is the falsity of the disjunctive premises (a30) and (b30). In the first

example, it is not at all obvious that many various forms of alcohol abuse will

never be diminished, nor that the enormous profit of the alcohol industry will

never be reduced, which is the reason why it is not necessary to follow the

obstructive reasoning strategy leading to acceptance of the false (even if logically

well-deduced) and the positive-action-blocking conclusion. Example 5 illustrates a

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similar point: if the falsity or the lack of evidence for the truth of the disjunctive

premise is well hidden beyond the argument’s deductive validity, it might be

difficult to realize that it is not reasonable to accept the false and the positive-

action-blocking conclusion suggesting that at least one of the antecedents of the

implicative premises has to be abandoned. It is of course also possible to use false

obstruction in the opposite way, so as to obstruct refraining from actions with

undesirable consequences.

3.5 Defeasible Sound Obstruction

The fourth type of false dilemma, defeasible sound obstruction, is—similarly to

defeasible sound quandary—based on deductively sound argument. It has two

subtypes: defeasible sound simple destructive dilemma and defeasible sound

complex destructive dilemma. We start with considering characteristics of the

defeasible sound complex destructive dilemma, since its corresponding defeat-

ing strategy is straighter than the one for the defeasible simple destructive

dilemma.

The main reason why defeasible sound complex destructive dilemma is a type of

false dilemma, and thus fallacy, is that it (intentionally or unintentionally)

overemphasises only the initial set of premises and thus leaves out the relevant

additional true information hidden beyond the initial argument’s soundness. It thus

forces the way of thinking according to which the initial set of premises is allegedly

the only relevant information on the basis of which therefore seems to be necessary

to consider only the initial conclusion.

Despite the fact that sound complex destructive dilemma is a deductively valid

argument with true premises, it is possible to defeat it by means of strategy

suggested in the definition of the defeated deductively sound argument, in Sect. 3.2.

We do not attempt to refute the argument since the strategy is not to challenge either

the truth of the premises or the logical validity of the argument. Instead, due to

relevant information added to the premises of the initial argument, we realize that

another conclusion also logically follows from the augmented set of premises.

Moreover, we realize that we may choose between the two conclusions in such a

way that we prefer to act in accordance with the new conclusion and not in

accordance with the initial one. This defeats the initial argument, in accordance with

the definition provided in Sect. 3.2.

Let us consider an example which illustrates the defeasible sound complex

destructive dilemma and the corresponding strategy of defeating it. Assume that all

premises in Example 5 are true. Under this assumption, the example turns into a

sound complex destructive dilemma. Let us symbolize the argument’s conclusion by

(Con 1) and use the following symbolization for the main propositions involved in

the argument:

F: Many more or less wealthy people from the rich part of the world

regularly give away a small amount of money to the governments of poor

countries.


123

K: Many countries from the rich part of the world work on the development of

industry, health care and education in the poor countries.

J: The world’s poverty will diminish.

L: The poor countries will gradually succeed in developing their own

economy.

By means of this symbolization, Example 5 corresponds to the Argumentation

schema 5. Again, under assumption that the argument’s premises are true, the

argument turns into a deductively sound complex destructive dilemma and we

cannot refute it. Nevertheless, we may defeat it, in accordance with the definition of

the defeated deductively sound argument given in Sect. 3.2. Consider the logically

valid Argument c given bellow. It augments the set of premises in Argumentation

schema 4 by additional relevant and true information given in premise (b4), and

fulfills thus condition (1) from the definition. It deduces (Con 2) from the

augmented set of premises, keeping also conclusion from Argumentation schema 5

(now, Con 1) that still logically follows from the augmented set of premises. It

therefore fulfills condition (2), (3) and (4) from the definition. Similarly as in

Argumentation schema 3 discussed in Sect. 3.3., we bracket the information

presented in (b4) and the (Con 2), since this information is (intentionally or

unintentionally) obscured in the initial argument. A crucial point of the defeating

strategy, as we shall see below, is to make the obscured information explicitly

available.

(b1) F → J(b2) K → L(b3) ¬J ∨ ¬L[(b4) F ∨ K](Con1) ¬F ∨ ¬K[(Con2) J ∨ L]

Argument c A strategy of defeating the argument given in Argumentation schema 5 and Example 5

The defeating strategy presented in Argument c does not refute the initial

argument presented in Argumentation schema 5 and Example 5, since it does not

challenge its deductive validity nor the truth or acceptability of its premises.

However, the strategy defeats the argument: the additional relevant information

leads to a possibility of focusing on the positive aspects of the disjunctive premise

(b4) and the corresponding disjunctive conclusion (Con 2), which might be more

preferable than only focusing on the initial set of premises and conclusion (Con 1).

This also fulfills the condition (5) and (6) from the definition. Shifting the focus to

the augmented set of relevant true premises and preferring to act in accordance with

(Con 2) defeats (Con 1) and the initial (in this section assumed to be) deductively

sound argument presented in Argumentation schema 5 and Example 5. Example 6

illustrates the strategy of defeating (here assumed to be sound) argument presented

in Example 5:

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(b1’) If many more or less wealthy people from the rich part of the world regularly give away a small amount of money to the governments of poor countries, the world’s poverty will diminish.

(b2’) If many countries from the rich part of the world work on the development of industry, health care and education in the poor countries, the poor countries will gradually succeed in developing their own economy.

(b3’) Nevertheless, it is either impossible to diminish world’s poverty, or it is impossible for the poor countries to gradually succeed in developing their own economy.

(b4’) However, either the wealthy people from the rich part of the world should regularly give away a small amount of money to the governments of poor countries, or many countries from the rich part of the world should work on the development of industry, health care and education in the poor countries.

(Con1) Either the wealthy people from the rich part of the world should not regularly give away a small amount of money to the governments of poor countries, or many countries from the rich part of the world should not work on the development of industry, health care and education in the poor countries.

(Con2) The world’s poverty will diminish, or the poor countries will gradually succeed in developing their own economy.

Example 6 Defeating (in this section assumed to be) deductively sound complex destructive dilemma ofExample 5: the focus on additional relevant premises and preference of additional conclusions is alsopossible

It is now time to consider a strategy for defeating sound simple destructive

dilemma. Assume that all premises in Argumentation schema 4 and the

corresponding Example 4 are true, which turns the exemplified reasoning into the

sound simple destructive dilemma. Now, the reason why the sound simple

destructive dilemma might still be a type of false dilemma (and thus the fallacy of

defeasible sound simple destructive dilemma) is that it overemphasises only the two

consequents of the antecedent in the implicative premises (a1) and (a2) in

Argumentation schema 4 and (a10) and (a20) in Example 4. The argument’s

soundness possibly blocks a possibility of thinking about other relevant information.

The fact is that the consequents of the implicative premises in Argumentation

schema 4 (and in Example 4, respectively) do not have to be the only relevant

consequents following from antecedent E. Additional relevant information may

show that there are other positive consequences that also follow from the antecedent

E. This may lead to acting from motivation to achieve these other consequences and

therefore to preferring to do E. Consequently, the new relevant information may

diminish motivation for not doing E only because at least one of the consequences G

or H is impossible to fulfill. This defeats the initial simple destructive dilemma

argument (according to the corresponding definition provided in Sect. 3.2), even if

the initial argument is still deductively sound and its conclusion therefore still true.

Let us reflect over an example that illustrates the strategy of defeating sound

simple destructive dilemma, as described in the paragraph above. Assume that all

premises in Example 4 are true and that we thus work with a sound complex

destructive dilemma. Let us symbolize the argument’s conclusion by (Con 3) and let


123

us use the following symbolization for the main propositions involved in the

argument:

E: You teach young people how to be social, to have fun and to reach

calmness, harmony and happiness without using alcohol.

G: Various forms of alcohol abuse will be diminished.

H: The enormous profit of the alcohol industry will be reduced.

By means of this symbolization, the Argumentation schema 4 corresponds to

Example 4. Let us add symbols M and T to express some additional propositions:

M: The young people will grow up into physically and psychologically

healthy persons.

T: An amount of aggression and violence in the world will be diminished.

We are now able to present the defeating strategy:

(a1) E → G (a2) E → H (a3) ¬G ∨ ¬H [(a4) E → M (a5) E → T (a6) Assume E (a7) M (from a4 and a6) (a8) T (from a5 and a6) (a9) M ∧ T (from a7 and a8) (a10) E → M ∧ T (from a6-a9)]

(Con 3) ¬E (from a1, a2 and a3)

[(Con4) E → (M ∧ T) (from a10)]

Argument d A strategy for defeating the argument presented in Argumentation schema 4 andExample 4

Similarly as in Argumentation schema 3 and Argument c, we use the brackets in

Argument d so as to point out the fact that the information in brackets is obscured in

the initial argument and that a crucial point of the defeating strategy is to make the

obscured information explicitly available. A natural-language example that

illustrates the strategy of defeating a (here presumably sound) argument in Example

4 is easily constructed if the symbols used in Argument d are replaced by their

natural language meaning given above.

Figure 1 presents all four types of false dilemma, together with the related

subtypes.

4 Types of Argument Criticism for False Dilemma

Argument criticism has been an implicit subject of argument analysis; however,

some recent works concentrate explicitly on theoretical principles and classifica-

tions of argument criticism (e.g. Krabbe 1992, 1999; Walton et al. 2008, 220–271;

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Krabbe and van Laar 2010). These works analyse general types of argument

criticism and are not primarily focused on false dilemma or other particular types of

fallacies. Therefore, it is interesting to see what types of argument criticism—and

thus different types of focus in critical thinking—might adequately be applied to

false dilemma.

4.1 Refutation or Opposition by Tenability Criticism

False quandary (related both to simple and complex constructive dilemma) is most

suitably approached by tenability criticism of the argument’s disjunctive premise.

Tenability criticism (Krabbe 1992, 1999) may rely on the speech act of questioning

the factual truth or other criteria of acceptability of the disjunctive premise in a

quandary argument. In that case, tenability criticism is used only to express

opposition to the initial argument. A stronger variant of tenability criticism refutes

the argument since it is based on the (warranted) speech act of asserting that the

disjunctive premise is false and hence that the argument is false quandary—a

variant of false dilemma. For a nice distinction between opposition to and refutation

of an argument, see Walton et al. 2008, 220–271.

Thus, if a given quandary argument attempts to be a deductively sound argument,

it has to be deductively valid and to have true premises. As shown in Sect. 3.1, both

variants of false quandary are deductively valid arguments, but at least the

disjunctive premise suggesting that there are only two given alternatives to choose

between is typically false. Refutation by tenability criticism is in that case the most

relevant type of argumentation criticism since it calls attention to the relevance of

the true premises and points out that at least the crucial, disjunctive premise is not

true.

False obstruction (related both to simple and complex destructive dilemma) is

also best approached by opposition or refutation by tenability criticism. Opposition

by tenability criticism challenges the false obstruction argument in asking for

evidence to support the truth of the disjunctive claim in which falsity of at least one

of the consequents of the implicative premises in the related argumentation schema

is asserted. Refutation by tenability criticism is based on a warranted assertion

Fig. 1 Variants of false dilemma


123

pointing out that the disjunctive claim which negates at least one of the consequents

is not true.

4.2 Defeating by Active Criticism

Defeasible sound quandary (related both to constructive and simple dilemma) is best

approached by active criticism. Active criticism is discussed by Krabbe (1999) and

it may appropriately be combined with analyses of defeated arguments (e.g. Bowell

and Kemp 2005; Walton et al. 2008, 220–230). A specific feature of active criticism

is that it requires some inventive thinking. Usually, it does not focus only on

checking the stipulated logical structure of the argument or on checking the

evidence that is typically expected to prove the truth of the given premises. Instead,

it commonly involves realizing that some information relevant for the argument is

missing, but also shows the relevant ways of influencing the initial argument when

the new relevant information is added.

The principles of the active criticism involved in the strategy of defeating

deductively sound arguments are defined in Sect. 3.2. Their application to defeasible

sound quandary and to defeasible sound obstruction is analysed through Sects. 3.3

and 3.5. A common, and the crucial, feature of the strategies is to make the obscured

relevant information explicitly available. The strategies differ though in the type of

obscured information most suitable to look for so as to make it explicitly available,

as described below.

Consequently, defeasible sound quandary (related both to simple constructive

dilemma and to complex constructive dilemma) is best approached by active

criticism that shifts the focus on other relevant consequences of the disjuncts

constitutive for the disjunctive premise in constructive dilemma arguments. This

shifts the focus on other, more preferable, conclusions that, equally as the initial

conclusion, logically follow from the augmented set of premises.

Slightly different type of information is made explicitly available in the strategies

used when active criticism is applied to defeasible sound obstruction. As described

in Sect. 3.5, defeasible sound simple destructive dilemma is best handled by active

criticism that shifts the focus on other positive and desired consequences following

from the antecedent of the implicative claims and not negated by the disjunctive

statement. Then, motivation to bring about these other positive consequences leads

to preferring to act or think in accordance with what is suggested in the antecedent,

even if at least one of the initially formulated consequents is truly negated in the

initial argument.

Active criticism suitable for approaching defeasible sound complex destructive

dilemma points out that, since the initial disjunctive claim only suggests that at least

one of the consequents of the two implicative claims is false, and that therefore at

least one of the antecedents has to be false, we are still free to act in accordance with

the conclusion following from in the initial argument concealed possibility of

having at least one of the antecedents true.

To conclude, in strategies of active criticism suitable for defeasible sound

quandary and defeasible sound obstruction, the main thing is: (a) to point out that

some information relevant for the initial arguments is missing; and (b) to show in

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which way adding the missing relevant information relates to the initial arguments.

Usually, a possibility of deducing a new relevant conclusion becomes much more

obvious. Moreover, preferences for acting in accordance with the new conclusion

are apparently stronger than the preferences for acting in accordance with the

conclusion of the initial argument, whereas both conclusions follow from the

augmented set of premises.

Table 1 summarizes all types of false dilemma defined in the article, the related

types of argument evaluation, argumentation schemas and the relevant types of

criticism.

5 Concluding Remarks

In trying to provide a systematic exposition of the fallacy of false dilemma, we have

suggested a group of criteria that can be used to both explain, and categorise, the

various false dilemma fallacies. We have distinguished between four general types

(and the matching subtypes) of false dilemma, the related types of argument

evaluation, as well as the argumentation schemas appealed to in each of them. Some

reasons why the variants of false dilemma may seem to be logically good arguments

are indicated and we have suggested the types of criticism most appropriate for each

variant of the fallacy.

Being able to discover false dilemma in situations when it is important to make

good and free choices is a significant and important dimension of critical thinking. It

may liberate us from accepting the consequences that necessarily follow from

prearranged alternatives, when these are not the only alternatives possible. It may

also liberate us from refraining from actions on the basis of the false obstructive

Table 1 Different types of false dilemma with adequate types of argument evaluation, argumentation

schemas and types of criticism

Different types of false dilemma Related argument evaluation

Related argumentation schema

Adequate types of criticism

False quandary False simple constructive dilemma

simple constructive dilemma

False complex constructive dilemma

complex constructive dilemma

Defeasible sound quandary

Defeasible sound simple constructive dilemma

simple constructive dilemma

Defeasible sound complex constructive dilemma

complex constructive dilemma

False obstruction False simple destructive dilemma

simple destructive dilemma

False complex destructive dilemma

complex destructive dilemma

Defeasible sound obstruction

Defeasible sound simple destructive dilemma

simple destructive dilemma

Defeasible sound complex destructive dilemma

complex destructive dilemma

deductively valid but unsound

refutation or opposition by tenability criticism

deductively sound, still defeasible

defeating by active criticism

deductively valid but unsound

refutation or opposition by tenability criticism

deductively sound, still defeasible

defeating by active criticism


123

statements about the impossibility of the desired consequences of the actions. In that

respect, the faculty of critical thinking directed at uncovering reasoning based on

false dilemma may also initiate the discovery of new alternatives, or discover the

ways of unifying seemingly opposing alternatives. Thus, a well-developed ability to

understand the tricky argumentative moves of false dilemma may be a good step

towards initiating some features of creative thinking.

Acknowledgments I would like to express my deep gratitude to two anonymous reviewers of the

article, engaged in the valuable peer-review process provided by the journal Argumentation. Their

detailed, analytical, responsible and intelligent comments, grounded in a high level of expertise in the

field of argumentation, have been of a great advantage for improving the quality of the article. Many

thanks to George Masterton, for proofreading of the first draft of the article, and to Martin Korpi for

proofreading of the accepted version. I am grateful to my family for outdoor life, math and love.

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