# A MISCONCEPTION OF LOGIC IN ITS DIFFUSION ACROSS NON-MATHEMATICAL ENVIRONMENTS: THE EMBARRASSING MISTAKE OF RELYING ON NATURAL LANGUAGE.

1. IntroductionArgumentation is taught in many ways. Even though argumentation could be taught informally, there are a number of careers in the humanities, especially philosophy, for which formal logic is the preferred way to set a standard for good arguments. Insofar as the teaching of argumentation is implemented by means of formal logic, one of the instruments would be a textbook on first-order logic. There are a number of textbooks that fill this role, and they are in the business of presenting their subject matter in as accessible a way for a non-mathematical audience as possible. Despite their purpose of being accessible, however, they cannot waive the requirement of doing a proper presentation of their contents. Such a propriety is always up for assessment, since the quality of a textbook depends upon it. In this paper we set out to criticize a number of textbooks that are widely used nowadays (and, incidentally, also some influential texts in the history of logic). All of them draw upon a common practice of teaching (or disseminating) first-order logic. This practice consists in the presentation of the semantics of first-order logic (SFOL for short) by means of the use of natural language (NL for short) as if it constituted a semantics. We shall argue that this is not a proper presentation of SFOL, and that it has negative consequences for the dissemination of logic in nonmathematical environments.

It is important to emphasize at the outset that our discussion is not restricted to a note on what kinds of texts teachers of logic should use to prepare their courses outside mathematical environments. It also makes part of larger discussions about what logic is, how we should apply formal tools in philosophy, and what the meanings of NL sentences are. We will underline some connections with these larger discussions along the way, some of them in the footnotes.

An example of the practice in question is the following. Suppose 'a' is an individual constant and 'B' a monadic predicate. In order to explain the truthconditions of the formula 'Ba' one can appeal to some translation keys, such as 'a' is translated as/means/stands for Andrew, and 'Bx' is translated as/means/stands for x is bald. (1) Hence, 'Ba' is true if, and only if, Andrew is bald, or so the explanation goes. We shall refer to this practice as translational semantics (TS for short).

From a historical point of view, the most pristine example of a TS, although not restricted to first-order logic, comes from Carnap's semantical systems, developed first in Carnap (1942) and later expanded in Carnap (1947). For Carnap, a semantical system is "a system of rules, formulated in a metalanguage and referring to an object language, of such a kind that the rules determine a truth-condition for every sentence of the object language" (Carnap, 1942: 22). The object language is a formal language, and the metalanguage is a natural language (expanded with logical symbols). Sentences in the metalanguage are used to provide truth-conditions for sentences (formulas) in the object language. We can use a small example to illustrate this idea (Carnap, 1942: 23f). Suppose that there are individual constants (a, b, c), predicates (B, H), and auxiliary symbols ((,)). Suppose further that a designates Chicago; b designates New York; c designates Carmel; B designates the property of being large; and H designates the property of having a harbor. Now, let [alpha] be an individual constant and [beta] a predicate. Then the rules of truth of the system determine that [beta]([alpha]) is true (in the system) if, and only if, the designatum of [alpha] has the property designated by [beta]. In particular, the system provides the following truthconditions for 'H(b)':

'H(b)' is true (in the system) if and only if New York has a harbor.

In general, this system provides truth-conditions of the form

's' is true iff p,

where 's' is a formula and 'p 'a natural language sentence, for every formula in the object language.

Another very influential kind of TS consists in the practice of a step-by-step transformation of a sentence in natural language into a formula in first-order logic. A paradigmatic case in this regard is Quine's book Methods of Logic (Quine, 1950). Quine leads us to entertain the idea that this transformation makes explicit the sentence's underlying logical form, preserving its truth-conditions.2 For instance, if the schematic letter 'Fx 'represents the open sentence 'x is a book' and 'Gx' represents 'x is boring', then 3x(Fx [and] Gx) is true iff some books are boring (Quine, 1950: [section]16).

A more recent example of a TS can be found in the first five editions of The Logic Book (Bergmann et al., 2008), which present the semantics of first-order logic by means of symbolization keys. Consider the following quote:

We can view the symbolization keys for sentences [...] as embodying interpretations for those sentences. That is, the truth-conditions of sentences of [first-order logic] are dependent upon the choice of universe of discourse and upon how each of the predicates and individual constants in the sentences are interpreted. [...] Let us start with [the] example 'Fa'. Whether this sentence is true depends on how we interpret the predicate 'F' and the individual constant 'a'. If we interpret them as follows: Fx: x is human a: Socrates then Fa' is true, for Socrates was human (Bergmann et al., The Logic Book, 379).

It is important to emphasize that the metatheoretical concepts of first-order logic e.g., tautologies, equivalences, valid arguments, etc.--are defined on the basis of such a semantics. This becomes clear when we note (in the fifth edition) that interpretations by means of symbolization keys are introduced in section 8.1, the metatheoretical concepts in sections 8.2-8.6, and the extensional interpretations (i.e., the model-theoretic definition) only until section 8.7. Therefore, the use of symbolization keys was not just for the sake of presenting only a few introductory examples--which in itself is prone to error--but constitutes a semantics on its own. (3)

Now, in the 6th edition (Bergmann et al., 2013), the authors changed their presentation. Instead of saying that symbolization keys 'embody' interpretations, they say that "[t]here is a sense in which the symbolization keys [...] provide interpretations" (ibid., 330). They now state that "interpretations only need to specify the things of which the predicates are true, rather than provide English-language readings for the predicates" (ibid., 331). Moreover, they now only include examples with the set of positive integers. It appears, then, that the authors have realized the pitfalls of using natural language to interpret first-order formulas. We feel relieved about this change. Nonetheless, a justification for it is nowhere to be found in the book. But such a justification is required. Firstly, because they are moving away from a presentation that should be exposed for what it is: a problematic presentation of the semantics of SFOL that should no longer be disseminated. And secondly, because despite the change in presentation, the fact that The Logic Book still features exercises with interpretations in natural language tells that the root of the problem may still be alive and grow again.

To bring this brief recount to a close, we just want to mention that there are other textbooks that are examples of TS, such as Magnus (2010) and Paez (2007).

A comment about our terminology is in order, since there is another way in which the term translational semantics has been used. What we have in mind is the widely read exposition of Evan's and McDowell's (Evans and McDowell, 1976) about theories of meaning and truth, which also features the term translational semantics. It is important to separate our use of this term from theirs. Translational semantics, in Evan's and McDowell's exposition, refers to a kind of theory of meaning that provides clauses such as (1), instead of clauses such as (2), where the latter are provided by theories of meaning of the kind proposed by Donald Davidson (cf. Davidson, 1967):

(1) 's' is translated as 'p'

(2) 's' is true iff p.

Our use of the term does not correspond to practices using (1). Rather, the practice that we refer to by means of the term TS is one in which a clause like (2), in which 's' is replaced by a formula in first-order logic and 'p' by a sentence in natural language, is used to provide the truth-conditions of 's'. (4)

Now, even though there are several widely used textbooks promoting TS, it is worth mentioning that not every textbook promotes it. There are also widely used textbooks that present the semantics of first-order logic by means of Model Theory, some at the undergraduate others at the graduate level, such as Gamut (1991); Sider (2010); Smith (2003); de Zwart (1998). These are examples of textbooks that people outside mathematical environments are better off using in order to understand the semantics of first-order logic.

As a final introductory note, we would like to point out that earlier public presentations of our critique of this practice have been received in different ways by mathematicians and philosophers. The former tend to belittle the range of dissemination of this practice, as if it were confined to "some (poorly-conceived) introductory texts". The latter, on the other hand, tend to feel that it is a normal and convenient practice. But it should be clear to both mathematicians and philosophers that this is a widespread practice in non-mathematical fields, and that this fact does not dismiss the need for a proper treatment of the subject.

Our attack on TS consists in advancing three arguments. First, there are a number of constructions in NL that do not behave in the way required by SFOL ([section]2). Second, even if we could restrict our attention only to those constructions that are 'well-behaved', the use of NL to interpret logical formulas is not able to provide an explanation of the semantics of individual constants and predicates, and therefore, it has negative consequences for the dissemination of first-order logic in nonmathematical environments ([section]3). Third, this practice incites the idea that the semantics of NL--at least for those constructions that are 'well-behaved'--is structurally identical to SFOL, and does so unreflectively and without justification ([section]4). In the sequel we will develop each of these arguments in turn.

2. Touchy Constructions

There are well-behaved constructions, such as the following:

(3) Every human is mortal.

Socrates is human.

Therefore, Socrates is mortal.

This construction has been a paradigm when it comes to the elaboration of valid arguments. Intuitively, we know that it is necessary that if its premises are true, so is its conclusion. This construction has been formalized by means of the following valid argument in first-order logic (FOL for short):

(4) [for all]x(Hx [right arrow] Mx)

Hs

[??]Ms

We can connect (3) and (4) using a TS such that 's' represents Socrates, 'Hx' represents that x has the property of being a man, 'Mx' represents that x has the property of being mortal, '[for all] represents every, and '[right arrow]' represents if...then.

Let's try to make explicit the connection between (3) and (4) so that it can be applied to any kind of argument. Let X be an argument in NL and Y an argument in FOL such that there is a TS that links Y with X--i.e., individual constants in Y represent proper names in X, predicates in Y represent properties in X, etc. We are going to assume that, when we apply the standard procedure for formalizing X into an argument in FOL, we obtain Y. (5) We say that X is 'well-behaved in FOL' iff the following condition holds: Y is valid in FOL iff there is a truth-preserving entailment from the premises of X to its conclusion. (6)

Thus, for example, (3) is a well-behaved construction in FOL since the TS specified earlier constitutes a link between (4) and (3), (4) is the standard formalization of (3) in FOL, there is a truth-preserving entailment in (3), and (4) is valid in FOL.

Now, the literature in philosophy of language is rife with constructions that are not well-behaved in FOL. Let's consider the following construction:

(5) Bruce Willis has 0 hairs in his head and is bald.

Jim Morrison has 1 million hairs in his head and is not bald.

If someone with 0 hairs in his head is bald, someone with 1 hair is still bald.

If someone with 1 hairs in his head is bald, someone with 2 hairs is still bald.

If someone with n hairs in his head is bald, someone with n + 1 hairs is still bald.

If someone with 999.999 hairs in his head is bald, someone with 1 million

hairs is still bald.

Therefore, Jim Morrison is bald.

Let's consider the TS such that 'b' represents Bruce Willis, 'j 'represents Jim Morrison, 'Bx' represents that x has the property of being bald, '[N.sub.n]x 'represents that x has the property of having n hairs in his head. By means of this TS, we can link (5) and (6):

(6) [N.sub.0]b [and] Bb

[N.sub.1million] j [and] [logicaly not]Bj

[for all]x, y ([N.sub.0]x [right arrow] Bx) [right arrow] ([N.sub.1]y [right arrow] By)

[for all]x, y ([N.sub.1] x [right arrow] Bx) [right arrow] ([N.sub.2]y [right arrow] By)

[for all]x, y ([N.sub.n]x [right arrow] Bx) [right arrow] ([N.sub.n+1]y [right arrow] By)

[for all]x, y ([N.sub.999.999]x [right arrow] Bx) [right arrow] ([N.sub.1million]y [right arrow] By)

[??]Bj

It is not difficult to see that (5) is not well-behaved, since there is no entailment in (5) (the predicate 'bald' is vague, so the chain of inferences must break at some point, because Jim Morrison is not bald), whilst (6) is valid in FOL (cf Hyde, 2014).

Another example: Consider the following argument in FOL:

(7) B(g, b)

b = w

[??] B(g, w)

This argument can be linked, by means of a suitable TS, with (8):

(8) Inspector Gordon seeks Batman.

Batman is Bruce Wayne.

Therefore, inspector Gordon seeks Bruce Wayne.

But clearly, (8) is not well-behaved, since there is no entailment in (8) (because Gordon does not know that Bruce Wayne is Batman and for him to seek the former is not the same as to seek the latter), whilst (7) is valid in FOL.

These are not isolated examples, on the contrary, there is an extensive literature about the limitations of FOL to provide what is traditionally known as the logical form of sentences in NL; we can find discussions about 'and', 'or', 'if...then', quantifiers and determinants, adjectives, proper names, adverbs and events, etc. (compare the compilation of arguments in Bach, 2002, and the references therein).

At this point it is important to address a relevant criticism of our argument, according to which we fail to distinguish between two different tasks: (a) the translation of sentences in NL into formulas in FOL; and (b) the provision of truthconditions to formulas in FOL by means of sentences in NL. (Observe that TS, as we have defined it here, coincides with task (b).) The previous examples containing vague and intensional predicates create problems for (a), not (b). These examples show that FOL is not able to capture the implicit semantic phenomena in these constructions in NL; but they do not show that it is impossible or inappropriate to provide truth-conditions to constructions such as (6) and (7) by means of sentences in NL, or so goes the criticism of our argument. Now, we do not deny that it is possible to come up with constructions that are well-behaved in FOL in order to provide truth-conditions for (6) and (7). However, we sustain that the naive use of a TS is a recipe for making logical mistakes. Without any prior restriction on the constructions in NL that we can take into account in order to interpret formulas in FOL, counterexamples to tautologies and valid arguments are not difficult to come by. This is more conspicuous when we recall two things: (i) in the practice of TS these metatheoretical concepts--i.e., tautologies and valid arguments--are defined on the basis of interpretations presented in natural language (see [section]1); and (ii) these two concepts ask us to take into account the totality of possible interpretations of formulas in FOL. Consequently, in TS, a formula is a tautology if it is true in every possible interpretation in natural language; an argument is valid if every interpretation in natural language that makes the premises true, also makes the conclusion true.

The following considerations about the difference between errors in task (a) and errors in task (b) can be useful to further convince the reader that our examples can be brought to bear on task (b). On the one hand, it makes sense to study the typical mistakes that students make in translating a sentence in NL into a formula in FOL--i.e., errors in task (a)--by producing a taxonomy of the kind of possible mistakes and relating them with certain constructions in NL (cf. Barker-Plummer et al., 2008). For instance, there is convincing evidence that a construction such as 'only if' in a sentence in NL is likely to elicit the error of reversing the antecedent and the consequent when translating it into FOL. This sort of study can give rise to a teaching plan that takes into account these difficulties in translating sentences from NL into FOL (cf. Palau and Coulo, 2011). On the other hand, regarding errors in task (b)--i.e., errors in the use of TS--, we have a whole different story. The error in which a student incurs if he or she provides truth-conditions to a formula such as B(g, b) by means of the sentence 'Inspector Gordon seeks Batman' is of a different nature than that of a translation error. We should tell the student that he or she cannot use intensional verbs, such as 'seek', because they cannot be used to represent truthconditions in FOL. This is not an error on the student's part, but rather it is an inadequacy of the tool--i.e., the TS.

Despite the fact that no textbook is careful enough to warn the reader of such perils, an advocate of TS might argue that with a prior delimitation of the constructions in NL to those that are well-behaved, it is legitimate to use TS without risk of logical error in the presentation of SFOL. This suggestion has two problems. First, which constructions should make part of the list of well-behaved constructions is not exempt from controversy. To be sure, there are constructions that are paradigmatic and which indisputably could go into the list. For instance, construction (9) is well-behaved:

(9) Lightning McQueen is a red car.

Therefore, Lightning McQueen is red.

In order to capture this inference from premise to conclusion we need to use the conjunction of two predicates: 'red', represented by 'Rx', and 'car', represented by 'Cx'. Let's assume that 'm' represents 'Lightning McQueen'. Thus, (9) can be represented by (10), which is valid in FOL:

(10) Cm [and] Rm

[??] Cm

On the other hand, there are constructions that are not so paradigmatic and which are up for discussion. Take (11) for instance; is it well-behaved or not?

(11) Doraemon is a plastic cat.

Therefore, Doraemon is a cat.

If we are of the opinion that science determines the extension of predicates in NL, in a way such that in the extension of 'cat' there are only creatures made of flesh, (7) we would say that (11) is not well-behaved--since (11) would be invalid in NL, whilst (12), its representation in FOL, is valid:

(12) Gd [and] Pd

[??] Gd

But if we think that it is the use of predicates in everyday life what determines their extension, we could be fine with the idea that a plastic cat is a cat. Therefore, (11) would be well-behaved. How can we solve this dispute? We believe that there are no theoretical tools that are solid enough to solve the dispute in one way or the other. So there is no uncontroversial way of producing the list of well-behaved constructions.

The second problem is that, even if we could build such a list, it would be far too cumbersome a tool that it would create more problems than it would solve. TS might be an effort to facilitate the learning of SFOL, but its correct usage requires a proper use of the list of well-behaved constructions. In order to properly use it, a lot of effort should be put in memorizing it, which is even harder if we are not explained why this list is what it is. However, if we try to provide reasons, and since we have to differentiate between well-behaved and ill-behaved constructions in FOL, we need to provide a prior explanation of FOL. If such an explanation were semantic (i.e., by explaining the validity of arguments by means of models), we would defeat purpose; if such an explanation were syntactic, we would require to explain one of the cumbersome deductive systems of FOL. In both cases, the use of TS, enhanced with the list of well-behaved constructions, does not make it any easier to teach SFOL as compared to the learning of the standard model-theoretical account, and rather increases the amount of information to learn and memorize.

We have shown that TS requires a delimitation of the interpretations that can be used to provide truth-conditions for formulas in FOL. Without such a delimitation, TS is prone to logical error. However, the list providing such a delimitation is not exempt from controversy, and even if it were, the mastering of it would require additional work and thus would defeat purpose. Nevertheless, for the sake of the remaining two arguments, we are going to concede that such a list is feasible and conspicuous to use, and hence we are not going to dispute that it is not logically problematic to provide a TS in an attempt to explain SFOL. Even in this stronger position, we will show that TS does not explain what it purports to explain, and that it makes far-reaching, unjustified assumptions.

To bring this section to a close, we hasten to emphasize that we have not aimed our arguments against the very possibility of providing formal models of the semantics of natural language. The examples provided in this section are well-known arguments that show the limits of task (a)--i.e., the translation of sentences in NLinto formulas in FOL. Such examples show that this task requires logical tools that go well beyond FOL. What we have proposed so far is to adapt such arguments in order to bring out the initial limitations of task (b)--i.e., the provision of truthconditions to formulas in FOL by means of the use of NL. If this section shows anything is that FOL and NL do not fit quite well with each other, and therefore, if we are to relate them, it is necessary to restrict ourselves to the fragments of NL that -at least at first glance--do not give problems. As we have seen, such a restriction is not exempt from controversy and is not trivial, and nevertheless our critique so far has not been addressed to the very possibility of carrying out task (b). What we want to show now is the impropriety of TS by pointing out how it lacks the required explicitness when it comes to the elements composing the semantics of FOL, and by arguing that it incites, without reflection, the belief that NL has, at least, a fragment whose structure coincides with the structure of FOL. This is what we set out to show in the next two sections.

3. A Building with Papier-mache Foundations

In this section and the next one we will argue that, even granting that we have a list of well-behaved constructions, it is not proper to use TS to teach SFOL. We start out from the observation that (13) and (14) are not the same thing:

(13) Hs is true iff Socrates is human.

(14) Hs is true in M = < A, I > iff I(s) [member of] I(H) (where I(H) [[subset].bar] A).

In this section we will argue that (13) cannot be used to introduce (14); in the next section we will criticize the far-reaching assumption that (13) and (14) are structurally identical.

Let's look back at our example (3) and its representation in FOL, (4). The TS that connects them consists of, among others, the following rules:

(a) 's' represents Socrates.

(b) 'Hx' represents that x has the property of being human.

The first thing to which we should draw our attention regarding (a) and (b) is that the expressions on the left-hand side of 'represents' do not have a meaning on their own, while the ones on the right-hand side do, since this is assumed in the practice under discussion--that is, they belong to the natural language that we are using to explain SFOL. The second thing to which we should draw our attention is that the use of language is not the same as the providing of an explanation of its own semantics. In other words, we are appealing to our prior understanding of some expressions in natural language in order to say something about the formal expressions, but we are not explaining what that prior understanding consists in. And from here we cannot reach the elements required in (14).

Another way of bringing the point home is the following: (a) says that 's' represents Socrates, but it does not explain the meaning of Socrates, let alone explain that 's' is to be interpreted as I(s)--the same goes for (b). Any semantics that purports to explain the meaning of Socrates should appeal to at least three elements: the name ('Socrates'), the person (the Greek philosopher), and a relation between them. All by itself, the name 'Socrates' does not give us the other two elements. When the name is not used (i.e., when it is mentioned), it gives us the first element. But when the name is used, it gives us only the second element. (8) Moreover, the relation between the two is utterly opaque and implicit.

What a SFOL should provide in the case of an individual constant such as 's' is a tripartite structure as described before. That is to say, it should give us marks, objects, and a link between the two. 'Socrates' by itself, even when used, does not provide any kind of tripartite structure, much less the tripartite structure that explains the interpretation of 's' required in (14). (9)

Thus, even though the expressions used in the right-hand side of the rules can provide (by way of stipulation) the meaning of the expressions in the left-hand side (which are devoid of meaning), the use of these expressions in the right- hand side is not able to provide an explanation of the meaning of individual constants and predicates. Thus, TS is not really a semantics, in as much as it does not provide the required tripartite structure. (10)

4. Counterfeit Money of Logic

So far we have criticized the practice of presenting SFOL by means of TS on the grounds that it is besieged by difficulties. In particular, in the previous section we have argued that TS (in the guise of (13)) cannot be used to teach SFOL (in the guise of (14)). However, an advocate of TS could argue that SFOL represents the extensional aspects of the semantics of natural language (SNL for short). This defense is based on the idea that a rule of a TS is a way to link the meaningless signs on its left-hand side with some aspect of the world, and that this link is formed by means of the use of NL (enriched with variables). To bring the point home, let's look again at (a) and (b) from section [section]3, repeated here for ease of reference:

(a) 's' represents Socrates.

(b) 'Hx' represents that x has the property of being human.

They both are rules, so claims the advocate of TS, that link meaningless signs with some aspect of the world by means of the use of natural language: 's' is linked with the person Socrates by the use of his name; 'Hx' is linked with all human beings by the use of the expression 'x has the property of being human' in natural language (enriched with variables). Consequently, TS--i.e., the use of natural language to explain the truth-conditions of formulas in FOL--is for him quite acceptable. For instance, he maintains that when we use TS to explain the truth-conditions of 'Hs', a clause such as (b) gives the extension of 'Hx', namely the set of all x that are human beings. Thus, this practice is based upon the idea that FOL provides the extensional aspects of NL, insofar as both of them have semantics that designate the same referents for corresponding expressions ('s' and 'Socrates'; 'Hx' and 'that x is human'; etc.)--at least when it comes to well-behaved constructions. We will call this idea the logical-core-of-language hypothesis.

It is very important to bear in mind that our purpose in this section is not to criticize formal semantics--i.e., the study of SNL by means of logical tools. We do not want to provide empirical data that cannot be explained by such-and-such a theory (or by any theory of the formal kind). Rather, our target is the practice of teaching SFOL by means of NL. Even though what we argue in the sequel can give rise to a critique of formal semantics, to read our argument in this light would be to miss the point that we want to make here. What we want to argue is that the practice of TS incites an unwarranted link between SFOL and SNL and makes its practitioners prone to confusing two kinds of objects (i.e., SFOL and SNL). More precisely, we argue that the practice of presenting the relation between an object and a set to which it belongs (the '[member of]' relation) by means of an example such as 'Socrates is human'--compare our earlier claim that TS uses (13) to introduce (14) --does not make sense if we do not presume that such a mathematical relation is implicit in this natural language sentence. That is, a practitioner of TS assumes that 'Socrates is human' and 'I(s) [member of] I(H)' have the same structure. Thus, even if the practitioner of TS--e.g., the author of a textbook or the professor giving this kind of presentation in a lecture--would not avow his or her agreement with this assumption--i.e., what we called the logical-core-of-language hypothesis--, the practice itself presupposes it. (11) Hence, many students, who are eager to absorb what is being presented to them as a distilled piece of knowledge, are led to believe such a hypothesis.

When the advocate of TS asserts that a formula in FOL, such as 'Hs', is true iff Socrates is human, he is taking for granted that Socrates is human iff the entity Socrates belongs to the set of all human beings. Let's take a closer look at this assertion. The truth-conditions of 'Hs' can be obtained from interpretation rules for 's' and 'Hx' (rules (a) and (b) above). Now, rule (b) assumes that the referent of 'Hx' is the same as that which is given to the expression 'x has the property of being human' by means of its use in natural language (enriched with variables). Moreover, since the purpose is to explain SFOL, the advocate assumes that the referent of the expression 'x has the property of being human', given to it by its use in natural language, is a set in the mathematical sense. This is, by all means, an imposition on this natural language expression of far-reaching consequences. We will criticize the imposition of the logical-core-of-language hypothesis on two accounts: in [section]4.1 we will argue that it is unjustified on the face of the actual body of knowledge about language; and in [section]4.2 we will show that it confuses two kinds of study.

4.1 The hypothesis is unjustified

The assumption that the referent of the expression 'x has the property of being human', given to it by its use in natural language, is a set implies that this reference is: (i) a collection of things; (ii) something of a binary character--that is, for all objects they either belong to the collection or not--; (iii) whether an object belongs to the collection or not does not depend on any kind of parameter--for instance, 'x has the property of being human' applies to any human throughout history, past, present, and future; and (iv) there is a prior and fixed delimitation as to which objects belong to the collection--for instance, there is no doubt whether an imaginary human belongs to the collection or not (cf. our discussion about the predicate 'cat' in section 2). Such impositions clash with the fact that the body of knowledge about natural language is not yet consolidated, nor are there any warrants that it will consolidate around the logical-core-of-language hypothesis. Take (i) for instance. For some scholars, such as Frege and many others in the tradition of formal semantics, it is evident. But it is not so for some interpreters of Wittgenstein, such as Travis, and for all those who Recanati refers to as proponents of 'Meaning Eliminativism' (Recanati, 2003)--for it is their contention that to which things a predicate can be applied depends upon the interests of the participants. Thus, there is no consensus around there being the extension of a predicate as a set in the mathematical sense. Consequently, there is no consensus about (ii) either, inasmuch as the existence of binary predicates depends upon the existence of the extension of a predicate, of which we can say that objects belong to it or not in a binary way. Moreover, to assume (ii) means to take position against a great amount of literature in linguistics and cognitive science that goes under the umbrella term of 'Prototype theory', according to which lexical categories do not have clear borders (cf. Taylor, 2003; Rosch, 1975; Molesworth et al., 2005). As for (iii) and (iv), there are positions in the literature asserting that no predicate that is not an indexical requires any parameters; on the other hand, there is a diversity of positions adhering to different extents to the idea that the context and the speaker's intentions are required to determine the content of any predicate (Recanati, 2003).

The advocate of TS, when seeing himself forced to consider that (i)-(iv) are substantial impositions, could try to rebut the foregoing arguments by claiming that: (v) every predicate has an extension; (vi) the restriction to well-behaved constructions, inasmuch as their referential properties coincide with those of sets in the mathematical sense, avoids the need for making far-reaching suppositions about natural language; (vii) that the prior definition of a universe of discourse (UD) allows for an implicit role of the parameters in natural language--for instance, the extension of 'Hx' can be made to depend upon any UD containing humans that are alive at the present moment; and (viii) that the prior definition of a UD allows for the consideration only of uncontroversial interpretations. Now, our reply is the expected one: (v)-(viii) are themselves far-reaching impositions upon natural language, regardless how plausible they may seem. To begin with, claim (v) only recasts the same position once again; one among many positions in the aforementioned debate. Claim (vi) presupposes (iii) and (iv) (but it is difficult to overcome qualms such as those discussed in section 2 about the predicate 'cat'). As for (vii), the advocate must assume that UD is a set in the mathematical sense, and hence UD also lacks parameters. Last but not least, (viii) presupposes that there are uncontroversial cases in NL and that we can build a semantics restricting our attention to them (but the qualms exposed in section 2 seem to deter the feasibility of this idea).

In the foregoing we have shown that the logical-core-of-language hypothesis rests on far-reaching suppositions about natural language. We have also shown that there is no consensus upon these suppositions. Moreover, in the contemporary scene in philosophy of language there is no consensus upon either the method of study or the object of study. For instance, not every scholar gives truth-conditions the same role in the study of natural language: Chomsky considers that meaning is a sociocultural aspect of language which is not even part of linguistics; Wittgenstein (in Philosophical Investigations) seems to claim that meaning is use and that 'use' must be understood as a practice which is not reducible to individual intentions; and Heidegger (in Being and Time) claims that language is the form in which meanings, which are already present in day-to-day practices, are expressed (compare Dreyfus, 1991: 12). Even among the approaches to language for which truthconditions are nuclear, there are at least eight different positions, differing on whether truth-conditions can be attributed to sentences or to their utterances (Recanati, 2003). There is no escape to the conclusion that the advocate of TS is projecting the properties of the mathematical tools underwriting SFOL onto thereferents of natural language expressions, and that he does not provide any justification to do so (and the burden of proof certainly falls on him).

At this point it is important to clarify our position so that we do not give the erroneous impression that what we are doing here is to criticize the translation of sentences in NL into formulas in FOL. Indeed, if this was what we were doing here, it would seem that our point is reduced to the claim that such translations are terribly simplified, since the semantics of NL exhibits complicated phenomena that lie beyond the scope of FOL. And since we are discussing introductory texts, simplification in the translations is not something to be avoided, but rather it seems to be the proper way to go about it, or so goes this misunderstanding of our position. Now, it is important to bear in mind that the object of our critique is the practice of presenting the truth-conditions of FOL by means of sentences in NL. The gist of our critique here is that this practice induces in the students the belief that NL possesses properties (i)-(iv) and that, in view that there is no clear and reasonable consensus about these properties, to promote such a belief about the relation between NL and FOL is to introduce a harmful bias in the fields of logic and in the study of language. (12)

We call the previous belief--i.e., that NL has properties (i)-(iv)--an implicit foundational belief of the practice of giving truth-conditions to formulas in FOL by means of sentences in NL. This means that students, in practicing it, would take it for granted that NL has such properties. In this way, they come to believe that it is superfluous to reflect and question about the properties of NL.

It is important to make a brief digression about what it means for this belief to be an implicit foundational belief of TS. To this effect, we need to provide a very simple analytical matrix to situate this belief in a practice. The matrix has two axes: that the belief is implicit or not; and that it is foundational or not. Firstly, that the belief is implicit means that it is neither reflected upon nor presented when the practice is carried out. Secondly, that the belief is foundational means that a practitioner of the practice must assume that the belief is true without questioning or justification. Thus, the belief is an implicit foundational belief when the practice is carried out as if the hypothesis were true, and is never made explicit, not even for questioning or justification. The belief that NL has properties (i)-(iv) is an implicit foundational belief in the practice of using a TS in the explanation of SFOL.

Now, this belief can be an explicit foundational belief in a different practice. For instance, when in philosophical logic it is announced as a presupposition of logical analysis. (Compare the following statement by Borg and Lepore: "[t]o have any hope that a system of symbolic representation will work, we need to assume that natural language sentences actually possess some kind of formal structure on the basis of which we can project out the explanations of the inferential properties of novel linguistic items" (Borg and Lepore, 2002: 99, emphasis in the original).) Thus, the implicit or explicit character of this foundational belief depends upon the practice--it changes from implicit in TS to explicit in logical analysis.

But the belief in this hypothesis can also cease to be foundational when we move on to the methodological reflections of logical analysis, such as those of Carnap's and Quine's about 'explication' (Carnap, 1947, 1962; Quine, 1964), in which logical tools are not synonymous with some natural language expressions, but replace them. To sum up, the implicit or explicit character, as well as the foundational or nonfoundational character of the belief in the logical-core-of-language hypothesis, depends upon the practice.

To bring this subsection to a close, we want to emphasize again that the gist of our critique is not that, in teaching logic by means of TS, we are teaching too simple a tool for the task of translating NL into FOL. This, of course, could be solved by teaching more powerful tools. Rather, the problem is that, in teaching logic by means of TS, we are introducing a bias in our understanding of NL. In the next subsection we will argue that this bias has the problem of hiding the difference between two kinds of study.

4.2 The hypothesis is confused

There is an aspect to this hypothesis that is even more problematic, namely, that it confuses two different kinds of study: the study of SFOL and that of SNL. The difference that we want to emphasize can be brought out by means of the following analogy with other kinds of studies. Consider the study of the Cathedral of Cologne and the study of the emergence of prices in a free market. To confuse these two studies would be like to suppose that a cathedral can be understood as the result of either an action of nature or as the complex but anarchic confluence of many individual actions, on the one hand; or that the emergence of prices in a free market can be studied by means of criteria such as the cultural influences, the technical abilities and the purpose of a designer, on the other hand. Mutatis mutandis, the study of SFOL is like the study of the Cathedral of Cologne, inasmuch as SFOL is an object-of-design; the study of SNL is like the study of the emergence of prices in a free market, inasmuch as it is complex and anarchic an object. The confusion produced by the use of TS under the logical-core-of-language hypothesis is that the study of natural language is confused with the study of logic. This confusion makes the study of SNL seem akin to the study of the artifacts of Model Theory, which are used in the construction of SFOL. Such artifacts, which come from Set Theory, were originally designed to address worries about the foundations of mathematics. But as it is known by any architect, it is one thing to design the foundations of something, and quite another to discover the foundations of something. SFOL is an artifact designed with a purpose, which is absent in natural language--natural language per se does not have a purpose, even if it can be used with one. There is no way around the conclusion that the studies of SFOL and SNL are of a different kind, since their objects of study are of a different nature.

Now, even if SNL and SFOL were structurally identical--and thus the hypothesis were true-, the discovery of such a truth requires two different studies: the empirical study of natural language and the mathematical study of Model Theory. Only on the basis of the results of both studies can we compare SNL and SFOL in order to assert that they are indeed identical. Thereby, both studies need to be of a different kind even regardless our assumption of the structural identity of their object of study. To confuse both kinds of study in this case is like to take counterfeit money as if it were authentic, just because we cannot distinguish a good counterfeit--which imitates every detail--from the legitimate note. But without knowing from where they come from, we cannot distinguish them with certainty. In like manner, the two semantics demand a different kind of study, in virtue of their coming from different places, namely, everyday life vs. the design made by philosophers and mathematicians. We can even say that their coming from different places makes them objects of different nature. To bring this line of argument to a close, suppose further that there is indeed a consolidated body of knowledge about natural language, and that this indicates that SNL coincides with SFOL. Even in this case we maintain that the two studies are of a different kind. For any advance in our knowledge of SFOL, say a new theorem that is proved, is not by itself an advance in our knowledge of SNL. We would need to map these results from one study to the other, and to this effect we require two different kinds of studies.

TS is a practice in which the logical-core-of-language hypothesis features as an implicit foundational belief. It assumes, for instance, that 'Socrates is human' and 'I(s) [member of] I(H)' have the same structure, and does so unreflectively and without justification. It comes as no surprise that the comparison between the empirical study of natural language and the mathematical study of Model Theory is conspicuously absent from the textbooks that promote TS. They promote a practice that confuses the empirical study of the semantics of 'Socrates is human' and the theoretical study of 'I(s) [member of] I(H)'. This a methodological confusion that should not be disseminated.

***

We have argued that the practice of teaching the truth-conditions of first-order logic by means of natural language--what we have called translational semantics--has detrimental consequences for understanding the semantics of first-order logic. In fact, it turns out to be necessary to restrict the constructions in natural language that can be used in the formulation of truth-conditions, in order not to invalidate logical laws. But even if this restriction were feasible, which we argue is not, the clauses that connect formal language with natural language (say 'Hs' is true iff Socrates is human) do not explain the semantics of first-order logic. In other words, the practice of translational semantics is not an explanation of the semantics of first-order logic. Even if some of its practitioners believe that there are advantages in the use of translational semantics, that such advantages overweight the problems it causes is something that should be actively defended, and not just taken for granted as part of the way this business usually goes. More seriously, an implicit foundational hypothesis of this practice is that (parts of) natural language and firstorder logic share a structure. This hypothesis is unjustified and controversial, and it incites the misconception that, when we study the semantics of first-order logic, we are unearthing it from the semantics of natural language--this a methodological confusion that should not be disseminated.

Acknowledgements

We would like to thank the anonymous referees for their useful feedback. We would also want to thank Graham Priest, Andres Paez, Tomas Barrero Carlos Cardona, Fabio Fang and Juan Raul Loaiza for useful comments on earlier versions of this text.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

NOTES

(1.) We wrote several alternatives for the relation between symbols in first-order logic and expressions in natural language ('is translated as/means/stands for') because there is no agreed upon concept to use when it comes to the connection between natural language and formal language. There are other options, including 'represents', 'formalizes', 'models', 'is interpreted by', etc. In the sequel we will mostly use 'represents' and 'is interpreted by', but the reader should bear in mind that there are several alternatives that could be used, and that there is no agreed upon concept for this relation. Note in passing that there is no noticeable effort in these textbooks to distinguish the different ways of relating the constructions in natural language with the formal constructions of logic. For instance, one thing is to claim that a sentence x is translated for a sentence y, and another is to claim that y interprets x, or that y represents x, or that y is a formalization of x.

(2.) It is worth mentioning that the characteristics of this transformation are obscure, for what is the relation between the input of the process (i.e. a natural language sentence) and its output (a first-order formula)? What is changed and what preserved by this process? Quine is not explicit about this issue and only mentions in passing that this is an abstraction process.

(3.) This makes this presentation found in the first five editions of The Logic Book a paradigmatic example of a TS, even more than Carnap's semantical systems. Carnap does not define the metatheoretical concepts of logic by means of his rules of truth, but by means of his state descriptions. Nonetheless, Carnap uses his rules of truth to explicate concepts such as contingent truth.

(4.) Compare (2) with the truth-conditions obtained from a Carnapian semantical system.

(5.) Despite the fact that there is no conventionally accepted standard rule-based procedure, there is a de facto agreement on the outline of such a process, which mostly consists in convenient replacements that end up in a logical construction aptly suited for drawing the inferential relations of relevance for the initial argument in natural language.

(6.) We are aware that some philosophers would disagree with calling an argument in NL valid, because they would prefer to restrict the use of such a term to the formal languages used in logic. We agree with this qualification, since the discussion whether the entailment that we find in NL is a genuine logical consequence relation is still open (cf. Glanzberg, 2015).

(7.) This would be the case if we thought that 'cat' was a natural kind.

(8.) And if we favored a causal-historical theory of reference, it would not even be clear what the use of a name would give us, since it would be based on a process of 'reference borrowing'.

(9.) On a more technical note, it could be argued that another reason why (13) cannot be used to explain (14) is that the former provides absolute truth-conditions whereas the latter establishes truth-conditions that are relative to a model (cf. Glanzberg 2015, in particular [section]II.1.1; Higginbotham 1988; Lepore 1983). Such a critique can be addressed to some versions of TS, such as, arguably, Carnap's and Quine's (cf. [section]1 above). However, there are versions of TS, such as the one we find in Bergmann et al. (2013) and in Paez (2007), that emphasize that the specification of truth-conditions must begin with a universe of discourse. Thus, such versions of TS provide truth-conditions relative to a model and, therefore, we cannot use the difference between an absolute and a relative semantics in (13) and (14) in order to criticize them.

(10.) In this section we have criticized the idea that the use of sentences in NL explains the semantics of SFOL. It is a presupposition of this idea that the student can rely on his or her own understanding of NL in the task of interpreting SFOL. Thus, sentences in NL are supposed to be endowed with unambiguous meanings that can be fruitfully exploited. But sentences in NL stand in need of interpretation, and recent research has shown that, even in mathematical environments, students can interpret sentences in different ways. Some interpretations hinder and some facilitate an appropriate use of NL sentences in mathematical proofs (cf. Dawkins, 2017; Dawkins and Cook, 2017). For instance, the evaluation of the truth value of the sentence 'Given any quadrilateral, it is a square or it is not a rectangle', is influenced by the way in which students represent the predicates 'quadrilateral', 'square', 'rectangle'. On the one hand, interpretations based on exemplars (or prototypes) hinder the proper normative analysis of the statement; on the other hand, interpretations relying on setbased meanings expedite the normative analysis of the truth value (Dawkins, 2017).

Dawkins and Cook (2017) suggest that set-based meanings emerge from the students' engagement in proof-oriented practices in mathematics. This is an interesting line of research, and one that could improve the teaching of the semantics of FOL, but in this text it should remain as a suggestion for future work.

(11.) We do not discard that some philosophers may have a hidden agenda that explicitly adheres to such a hypothesis. However, our argument does not depend on the intentions of the practitioners of this practice, but on the presuppositions of this practice itself.

(12.) We would like to quote Steward Shapiro who, in some recent reflections about the role of mathematics in philosophy, says that: "We philosophers were bequeathed a powerful batch of tools: model theory, proof theory, set theory, recursion theory, you name it. It is natural for us to hold that everything we encounter will yield to those tools, and be enlightened thereby" (Shapiro, 2016: 5). We think that improper practices such as TS are responsible for the fact that philosophers feel that it is natural to think in this way. But this is not what a good education in philosophy should provide. Shapiro equals this wrong way of thinking with a "child who is given a hammer for her birthday. As she goes around the house, she discovers, to her delight (and her parents consternation), that everything she encounters is in need of hammering. [...] Perhaps we should keep in mind that there may be important mismatches between mathematical languages, especially how they are regimented in logic, and the rough and ready world of ordinary thought or even scientific thought. It may be that things are not as determinate and clear as the mathematics for which our cherished tools were developed" (idem). Nevertheless, our point has wider scope than Shapiro's, since our worries go beyond the mismatch between mathematics and natural language. We also consider the confusion between two different kinds of inquiries about two different kinds of realms and the nature of their constituent elements (cf. [section]4.2 below).

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EDGAR ANDRADE-LOTERO

edgar.andrade@urosario.edu.co

Department of Applied Mathematics and Computer Science, Universidad del Rosario, Bogota, Colombia

JULIAN M. ORTIZ-DUQUE

ortizd.julianur@gmail.com

Independent Scholar, Colombia

How to cite: Andrade-Lotero, Edgar, and Julian M. Ortiz-Duque (2019). "A Misconception of Logic in Its Diffusion across Non-Mathematical Environments: The Embarrassing Mistake of Relying on Natural Language," Linguistic and Philosophical Investigations 18: 77-96.

Received 28 November 2017 * Received in revised form 7 February 2018

Accepted 8 February 2018 * Available online 27 February 2018

doi:10.22381/LPI1820194

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Author: | Andrade-Lotero, Edgar; Ortiz-Duque, Julian M. |
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Publication: | Linguistic and Philosophical Investigations |

Geographic Code: | 4EUUK |

Date: | Jan 1, 2019 |

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