Fallacious Analogical Reasoning and the Metaphoric Fallacy to a Deductive Inference (MFDI) GIUSEPPE SERGIOLI – CLAUDIO TERNULLO Abstract: In this article, we address fallacious analogical reasoning and the Metaphoric Fallacy to a Deductive Inference (MFDI), recently discussed by B. Lightbody and M. Berman (2010). We claim that the authors' proposal to introduce a new fallacy is only partly justified. We also argue that, in some relevant cases, fallacious analogical reasoning involving metaphors is only affected by the use of quaternio terminorum. Keywords: analogy, argument, fallacy, MFDI, metaphor, quaternio terminorum. 1. Introduction In this article, we discuss Lightbody and Berman's proposal to introduce a new fallacy of analogical reasoning (Lightbody Berman (2010)): the Metaphoric Fallacy to a Deductive Inference (MFDI), which, in the authors' opinion, should be seen as a special case of false analogy fallacy. We view the introduction of the MFDI as only partly justified. Moreover, we argue that, in some relevant cases, the kind of fallacy involved can be more aptly (and more simply) described as an equivocation fallacy, that is, quaternio terminorum. Consequently, as far as these latter cases are concerned, there are no sufficient grounds to introduce a new fallacy. Given our purposes, we also present a formalisation of fallacious analogical reasoning that uses a set-theoretic framework. Our exposition is self-contained: in Section 2 we summarise some basic notions. In Section 3, we describe the structure of analogical reasoning and fallacious analogical arguments, whose examination is crucial for our purposes. In Section 4 we discuss quaternio terminorum. In Section 5 we address Lightbody and Berman's proposal and set forth our full argument. 2. Some preliminaries In this preliminary section, we briefly summarise some terminology, in order to provide the reader with all the notions required for the understanding of this work. All readers knowledgeable about logic may entirely skip this section. 2 The first, fundamental notion is that of argument. Arguments are made of declarative sentences, that is, sentences which assert something. The classical definition of declarative sentence dates back to Aristotle's De Interpretatione, 4: Every sentence (logos) signifies, but not every sentence is declarative (apophantikos): only those sentences in which one can be right or wrong are declarative. For example, a prayer may be a sentence, but it is neither true nor false.1 Hence, a sentence is a declarative statement which can be assigned a truth-value. Arguments are collections of a certain number n of sentences (n-1 premises and one conclusion). In particular, as Epstein and Kernberger explain [...] an argument is an attempt to convince someone that a particular statement, called the conclusion, is true. The rest of the argument is a collection of claims called premises, which are given as the reasons for believing that the conclusion is true.2 Logicians are especially interested in checking whether an argument is valid, that is, that the conclusion really follows from the premises. If the premises are also true, the argument is said to be sound. In other terms: • an argument is valid if its conclusion is true, whenever its premises are true • an argument is sound if it is valid and all its premises are true.3 The argument in footnote 2 is both valid and sound, as the reader can easily realise. One can also verify the formal validity of a certain argument by checking that it fits a valid argumentation scheme.4 The notions of validity and soundness are crucial for evaluating arguments, but they also have some shortcomings. On the one hand, mere reliance on the semantic definition of validity expounded above may lead to the acceptance of unintuitive arguments (consider, for example, the cases of a fortiori, or ex falso sequitur quodlibet-style arguments). On the other hand, it is not uncommon to come across fallacious arguments which are only prima facie valid.5 1 See Needham Harbsmeier (1998), p. 182. 2 Epstein Kernberger (2006), p. 5. For the uninitiated, consider the following trite example of a twopremise argument (a syllogism, to be precise): (P1) Socrates is a man (P2) All men are mortal, hence (C) Socrates is mortal. For an exhaustive overview of these and other basic notions, see, for instance, Paoli Crespellani Sergioli (2012). 3 See, e.g., Walton (2005), p. 49. 4 However, this latter is a sufficient, but not necessary condition for an argument to be valid. Formal validity is a stronger form of validity (see Groarke -Tindale (2004), pp.144-150), as an argument can still be valid, even if it is not formally valid. For example, consider the following argument: (P1) John is a bachelor, hence (C) John is unmarried. This is not a formally valid argument, since it is not an instance of any known valid argumentation scheme. However, it is obviously a valid argument, since the premise and the conclusion contain equivalent assertions. 5 See Tindale (2006), p. 2 or Walton (2010). 3 Furthermore, validity and soundness are too strong for the purposes of the ordinary language. In particular, since we will be concerned with analogical reasoning, it can be shown that validity and soundness are too restrictive for the purpose of assessing the legitimacy of analogical arguments. For all these reasons, we propose introducing weakened versions of these notions. For a start, consider the following argument: (P1) John is 80 years old hence (C) John will be dead within 40 years Strictly speaking, the argument is not formally valid (nor is it valid tout court, for that matter) since it is not an instance of a valid argumentation scheme: in principle, it might happen (although it is quite unlikely) that John will die at the age of 121 years. Yet, one feels some pressure to concede that, if the premise is true, the conclusion should also be held to be true. In other terms, one would feel that the mentioned argument, although not valid, is fully legitimate. But, if its legitimacy amounts to its validity, then the argument is not legitimate. To fix this uncomfortable state of affairs, we introduce the notion of strength,6 which is a weakening of that of validity: • an argument is strong if it is very likely that its conclusion is true, whenever its premises are true. By the definition just given, the argument above is strong. The concept of strength pairs with that of goodness, in the same way as soundness pairs with validity. We say that: • an argument is good if it is strong and all its premises are plausible.7 An argument can be strong and not good. Consider one last example: (P1) Rio de Janeiro is covered with snow therefore (C) it must be quite cold in Rio Although, conceding the truth of the premise, the conclusion is likely, such a premise is definitely not plausible. 6 See Groarke Tindale (2004), p. 134, or Epstein Kernberger (2006), Ch. 3. 7 Epstein Kernberger, op. cit., p. 37. We refer the interested readers also to Turner (1984), Bonissone (1987) and Borwein Bailey (2008). 4 In the next section we will introduce and discuss the features of analogical reasoning. 3. Comparisons and Analogical Reasoning Comparisons are very frequent in everyday language, and play an important role in our reasoning. Analogical reasoning is based on comparisons, in particular on statements such as: "A is like B", "A is analogous to B", or "A is to B as C is to D" etc.8 Epstein and Kernberger propose the following definition of analogical reasoning: A comparison becomes reasoning by analogy when it is part of an argument: on one side of the comparison we draw a conclusion, so on the other side we should conclude the same.9 Let us briefly consider an example. Let us assume that (P1) Yesterday was very cold and today is very cold too and (P2) Since yesterday was very cold, I came home with a headache hold. Given the premises (P1) and (P2), we derive the conclusion: (C) Chances are that also today I will come home with a headache This is an example of analogical reasoning. It should be noticed that the argument is strong. Furthermore, according to our personal experience, the statements (P1) and (P2) are plausible, hence, the aforementioned argument can also be considered good. However, the argument is far from being valid. If (P1) and (P2) are true, it does not follow that (C) is necessarily true. This shows that strength and goodness are particularly suitable for assessing the legitimacy of analogical arguments.10 Analogical arguments may have different forms, each corresponding to a particular way of creating analogies. However, all analogical deductions have the same structure: the comparison of two cases, A and B, that yields an argument with two premises, the 8 However, in recent years there has been a debate about whether similes and/or metaphors are best defined in terms of comparisons or categorisations (see, e.g., Bowdle Gentner (2005), Glucksberg (2001), (2008)). The debate has especially focussed on similes and metaphors of the form "T is/are like S", and "T is/are S". 9 Epstein Kernberger (2006) p. 37. 10 For the sake of precision, following the terminology of Govier (Govier (1987), § 4; Govier (1999), § 9), one may say that the argument is an instance of inductive analogical reasoning, i.e. an argument containing a prediction based on our knowledge of an analogous situation which has previously occurred. 5 similarity, and the base premise, respectively, and a conclusion containing an inference from the premises.11 More precisely: • the similarity premise asserts that A and B share certain features a, b, c...; • the base premise asserts that A presents an additional characteristic x; • the conclusion states that, by virtue of the similarity established by the premise 1, B also possesses the quality x. In the previously mentioned argument, A and B are, respectively, "today" and "yesterday"; the feature a which A (today) and B (yesterday) have in common is "to be cold", and the additional characteristic x is "to come home with a headache". The similarity premise is (P1); the base premise is (P2); the statement "Today I will come home with a headache", in which the additional characteristic x is transferred by the argument from A to B, is the conclusion drawn by analogy. In order to understand what a weak or fallacious argument consists in, we now proceed to introduce a formalisation of analogical reasoning which will help us elucidate these two notions. As said, analogical reasoning is based on the use of comparisons. Although comparisons resemble standard declarative sentences, they are not declarative sentences. When we assert that "A is like B", we are not committing ourselves to stating that all the features of A are also features of B. A comparison only implies that some features of A are also features of B. In other terms, whereas "A is B" implies that whatever is predicated of A is also predicated of B, "A is like B" implies that there exist some features of A which can also be predicated of B. A comparison, therefore, is not a class-inclusion statement of the form: ∀x x ∈ A → x ∈ B , but rather a class-intersection statement of the form: ∃x x ∈ A ∧ x ∈ B . By asserting that "A is like B", we, thus, take it for granted that the intersection of A and B is non-empty. Now, what and how many properties are shared by A and B, whenever we assert that they are analogous? Unfortunately, there is no other way to respond to this the question but to check, each time a property of A is taken into account, that also B has it. Let |A|=i and |B|=j be the cardinality of the sets A and B, whose members are, respectively, the properties of 'A' and 'B'. We claim that, when we assert "A is like B", intuitively we are fixing a threshold T such that the sentence "A is like B" is true iff T ≤ A∩B < i, j. 12 Let us, now, resume the model of analogical reasoning which we have described above, which consists of a base premise, a similarity premise and a conclusion. Let us suppose that the similarity premise asserts that "A is like B", that is, A ∩ B ≠ ∅. This means that there are at least as many features as posited by T, which A and B share. However, as 11 Walton (2005), p. 96. 12 T = min(n), where n≥1, is the smallest number of properties shared by A and B, such that "A is like B" is true. The existence of a threshold is a necessary condition in order for the truth-functionality of analogical statements to be defined. It turns out that, in most cases, T = 1 and |A ∩ B| = T. 6 said, we do not know what these features are, and, at some point, we may stumble across features which are not shared by both sets. In other terms, the similarity premise does not guarantee that, if we find any additional characteristic of A, say x, x also always belongs to B. In this latter case, the analogical argument is weak (that is, non-strong). We are, therefore, ready to give the following definition of weak analogical argument: Weak Analogical Argument. An analogical argument is weak iff the similarity premise does not guarantee, for a particular x in A, that the conclusion: "x is in A implies that x is also on B" is true (or, in other terms, that x ∈ A ∩ B). In the argument examined above, the additional characteristic = "I came home with a headache" is likely to belong to both A = "cold day1 (yesterday)" and B = "cold day2 (today)". This is the reason why the argument is strong. As we said, in many cases the set of properties which account for the truth of "A is like B" has cardinality equal to 1. Any argument whose similarity premise implies that A ∩ B = 1 will, of necessity, be weak. The reason is, no additional characteristic will be shared by the two sets. Let us now proceed to examine fallacious analogical reasoning. It is generally assumed that a fallacious argument, as almost every account from Aristotle onwards tells you, is one that seems to be valid but is not so.13 Hamblin's definition, although very appropriate, is unsuited for our purposes, as we do not care about the validity, but rather about the strength of analogical arguments. The following one seems more adequate: A "fallacy" is a particular kind of egregious error, one that seriously undermines the power of reason in an argument by diverting it or screening it in some way. But a more precise definition is difficult to give and depends on a range of considerations.14 In what follows, we have tried to summarise "the range of considerations" Tindale mentions in the quoted passage. Basically, we claim that, since we do not know much about A ∩ B, we may be deceitfully led by the analogies used in the argument to ascribe to it a certain x that, for instance, could be in A, but not in B, and vice versa. In that case, we may say that the argument is fallacious: it is prima facie strong, but, in fact, it is weak. A fallacious analogical argument is, therefore, one which is made in such a way as to lead us astray in evaluating the strength of the argument. We can summarise this through the following definition: 13 Hamblin (1970), p. 12. 14 Tindale (2006), § 3. Tindale also distinguishes between structure fallacies (which fall under Hamblin's definition) and fallacies related to language problems (Tindale, op. cit., § 4), such as fallacies of equivocation and vagueness (also known as informal fallacies). 7 Fallacious Analogical Argument. An analogical argument is said to be fallacious if and only if it seems strong, but is, in fact, weak. Any such argument is fallacious, inasmuch as it leads us to ascribe to A ∩ B a property which is not in A, but is in B and vice versa. A fallacious analogical argument can also be described as one which establishes a faulty analogy as its conclusion. Fearnside and Holther explain the notion along the same lines as ours: Faulty analogy consists either in assuming that shared properties will continue indefinitely to be found in new members, or in assuming that it is highly probable there will be some other shared property in a class so wide that there is only a low initial probability of finding any other shared properties relevant to the purpose at hand.15 In our opinion, the formalisation we have presented helps clarify some crucial points concerning analogical reasoning and we will use it in what follows to elucidate further aspects of the question. Before examining in depth Lightbody and Berman's arguments, let us briefly review the fallacy named quaternio terminorum and its features. 4. A fallacy of equivocation: quaternio terminorum16 Ambiguity may affect single words, statements, or even entire arguments.17 Statements or arguments can be ambiguous in two main different ways. A sentence (argument) will be: • lexically ambiguous: if it contains terms whose usage is ambiguous; • structurally ambiguous: if it is the structure of the sentence (argument) itself that yields ambiguous interpretations. In other words, lexically ambiguous statements and arguments are ambiguous in virtue of the terms that they contain, whereas structurally ambiguous arguments "create" their own ambiguity. As an example, the ambiguity of the sentence: "The Rabbi married my sister" depends on the double meaning of the single word 'married' (which means both 'celebrating a marriage', or 'getting married'). On the other hand, the ambiguity of the statement: "The man saw the boy with the binoculars" 15 Fearnside Holther (1959), p. 4. 16 Readers with some training in logic may skip also this section. 17 See, for instance, Kroeger (2005), Subsection 3.1 and Quine (1960), §§ 27-31. 8 is due to the fact that the expression 'with the binoculars' may be interpreted as referring either to the man's or the boy's using the binoculars. In the former case, ambiguity is a property of a term in the statement; in the latter, ambiguity is a global (holistic) property. Arguments affected by equivocation fallacies may appear prima facie strong. Quite often, deductions of this sort also seem formally valid. Closer inspection reveals their concealed weakness, in that they are based upon either form of ambiguity (lexical, or structural). Quaternio terminorum is based on lexical ambiguity. Here follows a simple example of an argument containing the fallacy: (P1) A star is a massive luminous ball composed of plasma in hydrostatic equilibrium (P2) George Clooney is a star hence (C) George Clooney is a massive luminous ball composed of plasma in hydrostatic equilibrium As one quickly realises, the reasoning involved in this argument hides a trap. The term 'star' has different meanings in the premises. While in the first premise it is assumed to refer to the celestial body, in the second one it means movie celebrity. Historically, the name of this fallacy, quaternio terminorum, was coined in the context of the Aristotelian syllogistic theory.18 More precisely, as is widely known, according to Aristotle, the basic syllogism consists of three sentences: two premises (major and minor, respectively) and a conclusion, which, in turn, contains three terms: the subject, the predicate of the conclusion, and a third term (the middle term), which connects the subject of the first premise to the predicate of the second premise. If the third term assumes different meanings in the premises (like 'star' in the aforementioned argument), then the syllogism contains a fourth, hidden term: this fact gives rise to the fallacy. To summarise: A valid standard-form categorical syllogism must contain exactly three terms, each of which is used in the same sense throughout the argument. [...] If a term is used in different senses in the argument, it is being used equivocally, and the fallacy committed is that of equivocation [quaternio terminorum].19 However, it should be noted that, although historically this fallacy is related to Aristotle's theory of syllogism, it is by no means necessary that the argument containing the fallacy actually be a syllogism. 18 See Smiley (1973), pp. 136-154. 19 Copi Cohen (1990), p. 206. 9 5. The Metaphoric Fallacy to a Deductive Inference We finally proceed to discuss the Metaphoric Fallacy to a Deductive Inference (MFDI) proposed by Lightbody and Berman. In their article, the authors define the MFDI as follows: The MFDI is [...] committed when the following two conditions are fulfilled: (i) a faulty comparison is made between two things (false analogy); and (ii) this faulty comparison is then used as premise in a sub-argument that is supposed to prove some conclusion which is believed to follow deductively.20 The MFDI would be a special case of the false analogy fallacy. The authors' treatment of this notion is fairly similar to ours, except that we call it fallacious analogical reasoning. Let us consider the argument presented by Lightbody and Berman in order to illustrate it: (P1) Human communities are analogous to beehives (P2) All beehives need a queen hence (C) All human communities need a queen The argument can be re-translated in the following way, according to our formal template: (P1) Human communities (A) are like beehives (B) [similarity premise] (P2) Beehives need a queen (the additional characteristic x is 'to need a queen') [base premise] (C) Human communities need a queen (also B has x) The authors say that (C) is a faulty analogy, as the fact that x ∈ A is not sufficient to guarantee that x ∈ B. All this is in accordance with the results of our examination of fallacious analogical reasoning.21 Now, the MFDI would be a variant of the false analogy fallacy and would occur in a wider and more articulated reasoning context. In order for the fallacy to take place, one does not only require that a faulty analogy is created, but also that 1) the analogy be used as a premise in a further sub-argument, and that 2) the faulty analogy derives from relating metaphors. According to the authors, the invalidity of the whole argument would, 20 Lightbody – Berman (2010), p. 191. 21However, it would seem that Lightbody and Berman are not keen on distinguishing, as we have done, between weak and fallacious arguments. In view of our definitions, the argument proposed above may simply be viewed as weak. However, given the patent blurredness of the notion of fallacy, one can still say, as the authors do, that it is, in fact, fallacious. Note that the authors define fallacious analogical arguments as 'those wherein the similarity between the two components being compared is questionable or irrelevant.' (op. cit., p. 187). 10 then, be specifically dependent upon the use of metaphors, a fact which would warrant the use of the label 'metaphoric fallacy'. In order to illustrate the MFDI, the authors use the following example:22 (P1) The heart is like a mechanical pump (P2) The heart is like a red, red rose From (P1) and (P2), we conclude that (C1) A mechanical pump is like a red, red rose Then, we use (C1) in a sub-argument, whose other premise is: (P3) A mechanical pump can be fixed Hence, from (P2) and (P3), we infer that (C2) A red, red rose can be fixed.23 The argument above is also formally translated by the authors as follows (P1) Hx is analogous to Px (P2) Hx is analogous to Rx (C1) Px is analogous to Rx (Inferred from 1 and 2) (P3) (x)(Px→ Fx) (C2) (x)(Rx→ Fx) (MFDI) where: the domain is unrestricted, and Hx= "x is a heart"; Px= "x is a mechanical pump"; Rx= "x is a symbol of love"; and Fx= "x is a fixable entity". Let us pause a moment to examine the argument and the authors' claim that it represents an example of a new fallacy. First, let us see what the authors themeselves say about the first bit (P1-C1) of the aforementioned argument: The structure of MFDI proceeds from analogously relating two metaphors and then claiming that a property (quality or function) from one compared predicate of the analogy is contained by the other predicate. That is, the predication is treated as being transitive across an analogy between metaphors.24 22 Lightbody Berman, op. cit., p. 189-90. 23 The italics throughout are all ours. 24 Lightbody Berman (2010), p. 185. 11 We agree with the authors that incorrect analogical reasoning is at work in (P1-C1). However, it is far from clear that the reason is that some sort of 'analogical transitivity' across the terms in P1, P2 and C1 is fallaciously assumed. To begin with, if P1, P2 and C1 were not analogical statements, the whole argument would simply be an invalid syllogism, and it is precisely because there would be no transitivity across the terms in the three statements that the syllogism could not possibily work. 'The heart is a red rose' and 'The heart is a mechanical pump' obviously do not imply that 'A red rose is a mechanical pump'. However, the authors think that P1, P2 and C1 would deceive us into thinking that there might be some sort of 'analogical transitivity' across the terms involved. In particular, some property, let us say x, belonging to 'red rose' would also be predicated of 'mechanical pump' as a consequence of relating the metaphors contained in P1 and P2. But it seems to us that, in the example proposed, there is no necessity to view the faulty analogy as the product of an invalidly assumed 'analogical transitivity'. One could simply relate the weakness of the analogy to the invalid syllogistic structure of (P1-C1). Coming to the second part, (C1-C2), this bit is clearly an instance of fallacious analogical reasoning, and our formalisation helps us establish this fact very easily. Let us assume that: A= the set of properties of mechanical pumps B= the set of properties of red roses Similarity premise: 'Mechanical pumps are like red roses' Base premise: 'Mechanical pumps can be fixed' Conclusion: 'Red roses can be fixed'. The argument is fallacious according to our very definition, since, although it does not seem that there is any T ≥1, shared by 'mechanical pumps' and 'roses', which satisfies A ∩ B ≥ T, the analogy drawn in (C1) would deceive us into inferring the opposite. Therefore, as far as (C1 C2) is concerned, we agree with the authors that this bit is affected by faulty analogical reasoning. However, there is no special analogical fallaciousness at work here. It is irrelevant for the fallaciousness of the argument whether C1 is a faulty analogy derived from relating metaphors. Any analogy may lead to a fallacious argument, since the requirements for the strength of an analogical argument can be very easily violated. These considerations lead us to view the introduction of the MDFI as dubious. However, let us concede that the authors are right and that the fallacy in the invalid analogical syllogism (P1-C1) is specifically related to assuming an 'analogical transitivity' across P1, P2 and C1 induced by metaphors. The crux of the authors' argument is that (C1) is a peculiar faulty analogy, inasmuch as it results from relating two metaphors ('the heart is like a red rose' and 'the heart is like a mechanical pump'). We wish to argue, instead, that the kind of fallacy committed in the example proposed would be quaternio terminorum. In other terms, what would happen is that one of the terms involved in the statements has two different meanings. 12 Let us resume for a moment steps (P1-C1). In particular, let us examine the two premises: (P1) The heart is like a mechanical pump. (P2) The heart is like a red rose. The 'heart' to which the two premises refer seems to be the same object. However, the two hearts (heart1 and heart2) are clearly different denotata. Heart1 is the 'muscular organ that pumps the blood through the circulatory system', whereas heart2 is 'the center of a person's thoughts and emotions'.25 Heart2 is already the result of a process of metaphorisation, which has been so strong and successful as to create, as it were, a new entity: a metaphorical heart, which does not share any feature with the physical heart. In our interpretation, the speakers who use 'heart' in the aforementioned argument, are clearly aware of the differences between heart1 and heart2, and, are, in principle, able to tell them apart. Thus, if they use 'heart' in its two different meanings but, at the same time, assume 'analogical transitivity' across the terms in P1, P2 and C1, they might inadvertently fall upon an equivocation fallacy, that is, quaternio terminorum. However, even accepting our interpretation, there would still be cases where metaphors are mutually related in such a way as to produce faulty analogies wherein there is not even the least hint of equivocation. For instance, consider the following argument: (P1) The swordfish is like a fencer (P2) The swordfish is like a serial killer hence (C) A fencer is like a serial killer26 In the argument above, the term 'swordfish' is totally unambiguous and, thus, cannot engender equivocation. If we use (C) in a sub-argument with one more premise, we might now commit the MFDI in the following way. Assume: (P3) Fencers can win or lose games. hence (C2) Serial killers can win or lose games What was, maybe, lacking in the authors' discussion of metaphors is a distinction between two relevant kinds of metaphors. The distinction has been crucial for us. The authors say: 25 Oxford English Dictionary, s.v. 'heart'. 26 The objection above, that this is simply an invalid syllogism, may also apply to this argument, but can be ignored for our purposes. 13 [...] a metaphor can indicate a transferring of information from one particular (predicate) to another particular (subject), that is, the ascription of some property, quality or function to the target occurs.27 [...] The MFDI also assumes that a metaphor is the description of one thing as something else. It need not be taken as a factual claim insofar as such is subject to truth conditions. Rather, a metaphor can provide an expression of insight which elicits or prompts thought in new directions.28 We agree with the opinion expressed above, but we also distinguish between metaphors that are so strongly lexicalised as to give rise to different denotata and, thus, to ambiguities which lead to quaternio terminorum, and live metaphors that really provide new insights concerning the relationships between two different items.29 Whether or not words like 'heart' or 'swordfish' give rise to live metaphors depends upon their degree of lexicalisation. In the authors' example, the degree of lexicalisation of 'heart' is so high that the equivocation is unavoidable. In the argument about the 'swordfish', on the contrary, live metaphors are created and, therefore, fallacious analogical reasoning may really be at work. To conclude, we claim that, even conceding that certain analogical arguments are specifically fallacious in virtue of: 1) relating two metaphors via the assumption of a sort of 'analogical transitivity' which creates a faulty comparison; 2) using the faulty comparison in a fallacious sub-argument, that is, even conceding that there is a point in the introduction of the MFDI, we would encourage a re-consideration of its range of application: it seems to us that the MFDI would only be at work whenever metaphors really induce a faulty comparison. Whether or not this happens depends upon their satisfaction of the following principle: Principle of Lexicality A metaphor is a live metaphor iff it is not an already established lexical item. In simpler terms, it is live iff it is not listed among the different meanings of a dictionary item. All metaphors satisfying the Principle of Lexicality may, therefore, be good candidates to give rise to faulty analogies. Metaphors which do no satisfy the Principle of Lexicality would, instead, engender equivocation and, in particular, fallacies such as quaternio terminorum. 6. Concluding Remarks In this paper, we have tried to examine whether fallacious analogical reasoning based on metaphors can lead to what Lightbody and Berman have identified and described as the Metaphoric Fallacy to a Deductive Inference (MFDI). We have presented three main objections. The first two concern the relationship between standard and analogical reasoning, and the last one aims to bring to light that some arguments based on 27 Lightbody Berman, op. cit., p. 188. 28 Ibid., p. 188-9. 29 See Ervas – Ledda (submitted for publication). 14 metaphors which seemingly lead to faulty analogies are, in fact, affected by quaternio terminorum. We have also presented a formalisation of fallacious analogical reasoning which, in our opinion, helps elucidate the topic significantly. 7. Acknowledgements We wish to express our gratitude to the following people, who read earlier drafts of this paper, discussed its contents and suggested significant improvements: Vincenzo Fano, Francesco Paoli, Elisabetta Gola, Marco Giunti, Francesca Ervas, Antonio Ledda. An early draft of this paper was presented and discussed at the conference "Metafora e Argomentazione", which took place in June 2012 at the University of Cagliari. Thoughtful suggestions for a significant re-structuring and improvement also came from the many discussions we had with the organisers and the invited speakers in that occasion. 8. References Blackburn, S. (1996) The Oxford Dictionary of Philosophy, Oxford: Oxford University Press. Bonissone, P. Plausible Reasoning: Coping with Uncertainty in Expert Systems in Shapiro, S. C. (ed.) Encyclopedia of Artificial Intelligence, pp. 854-63, New York: John Wiley and Sons. Borwein, J. Bailey, D. 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