A note on logical pluralism

In this paper we are going to characterize the idea of logical pluralism according to Beall and Restall. In order to do that, this paper is divided into three sections. In the first section, we shall define briefly what Beall and Restall’s theory is about. In the second, we shall deal with some problems with the theory and some possible answers. Finally, three objections

I Logical pluralism is the thesis that there are at least two logics that are equa ly valid. But this could be understood in different ways. There are ways in which it is evident that there is pluralism. In contrast, there are other ways that are more questiona le. It is unquestiona le that the twentieth century has brought us many different logics: intuitionistic logic, relevant logic, many valued logics, etc. In this sense, it is obvious that there is a plurality of logics. But as Priest (2006, p. 195) ar ues, logical pluralism acquires significance and interest when we ta k about the ap lications of such logics in a ecific domain. In that case, pure logics, mathematical logic systems, become theories of that domain. In such a way, logics acquire the status of scientific theories and the pro lem of whether or not there is more than one valid logic for a certain domain turns into an important issue. Later we wi l see why this is interesting.
The version of logical pluralism that we are going to discuss here is that stated by Bea l and Re a l in "Logical Pluralism" (2000). According to them, there are at least two ways to understand deductive validity and these two ways are equa ly good (Bea l and Re a l, 2001, §1). Their pluralism is a pluralism about logical consequence. The main idea is that there are different formal ways to understand the pre-theoretical notion of logical consequence, (V), according to which a conclusion, A, follows from premises Σ, if and only if any case in which each premise in Σ is true is also a case in which A is true (Bea l and Re a l, 2000, p. 476). As there are at least two precisifications of such cases, there are at least two equa ly valid logics. One type of case could be a Tarskian first-order interpretation; another could be a possi le situation, or an inconsistent or incomplete interpretation.

II
At this point, let's analyse some objections that have been raised to this theory. A first objection ar ues that logical pluralism is an empty position since there are not many logicians that would adopt it. For instance, the law of the excluded mi dle is necessary for the classic logician, but it is not for the intuitionistjust as ex falso quo libet is valid for both of them, but it is rejected by the relevantist. As a l of them believe in their own notion of logical consequence, none would adopt a logical pluralism. Hence, we could conclude, logical pluralism is a position that would be maintained by few logicians. Bea l and Re a l (2006, p. 87-88) answer this objection by ar uing that this fact is not surprising. The reason why it is not surprising is because most intuitionists, relevantists, etc. are monists and consider their own logic as the only valid form. However, this a ect is not a necessary and essential feature of their theories. It could be said that relevantists develop their logic because they deny the classical notion of logical consequence, but as the authors claim, this is not necessary. One could develop a relevant logic with the motivation of formalising that othe notion of logical consequence according to which the premises must be relevant for the conclusion. Assuming and accepting this, there are thus other ways to understand it.
Another objection refers to our logical preferences. The objection says that if we prefer a particular logic, we are not actua ly defending a logical pluralism (Bea l and Re a l, 2006, p. 99). Bea l and Re a l reply that logical pluralism is not inconsistent with the idea of having fa ou ite logics. Their validity has nothing to do with individual preferences. This is why, since both logics are products of different ways of understanding the pre-theoretical notion of logical consequence, we keep on being pluralists.
Russe l (2013) ar ues that we can deduce two things from Bea l and Re a l's theses: (i) the meaning of the term 'case' is undefined; (ii) as it is undefined, we could find more than one precisification, which would make us pluralists. However, according to the author, none of them is unavoida le. As for the first thesis, Russe l maintain that philosophy of lan uage has showed us that we could find an ap roach that captures exactly what we mean by the notion of logical consequence. One possi le way, she su gests, could be making use of mathematical techniques. But this seems unsatisfactory. It is not clear how the concept of 'case' or 'logical consequence' in general could be analysed (even less with mathematical tools) for it to reach a completely defined meaning. The va ueness of the term and its possi le interpretations hap en to be a part of its meaning. Otherwise, we wi l have an artificial concept that is unlikely to reflect our pre-theoretical notion.
As for the second case, the author ar ues that even assuming that the term is indefinite, this does not necessarily mean that we are going to find more than one precisification of 'cases' . This objection may be valid in general, but it is not valid in this case, since Bea l and Re a l have already offered three such possi le ecifications.
The last objection begins by noting that in the principle (V) there is a universal quantifier. Regar less of the precisifications of cases, the conclusion must fo low from a l of them. So the true logic would be the intersection of a l systems accepted by (V). Bea l and Re a l (2006, p. 92) answer this objection by saying that the only inference that would rea ly survive the intersection is the law of identity. If A, then A. And it seems that to conclude that the law of identity is the only valid ar ument is somewhat implausi le and demotivating.
Priest ar ues that this might be the case for pure logics, but it seems unlikely when we ta k about ap lied logics, particularly in the canonical ap lication of logic: the analysis of human reasoning. Priest thus provides an example of an inference that would not fail in any case: "For example, any situation in which a conjunction holds, the conjunction holds, simply in virtue of the meaning of ^.
[…] As long as meanings are fixed, one can't vary them to dispose of valid inferences" (2006, p. 203). However, there are reasons to think like Bea l and Re a l in this re ect. Inferences are valid thanks to the meaning that has been assigned to the connective from each precisification of (V). In this manner, an intuitionist denies the law of the excluded mi dle by virtue of the meaning of the connectives ¬ and v. And for this reason it seems implausi le that his example may count as a valid inference for any logic. A similar response is offered by Russe l (2013).

III
Fina ly, we wi l outline three objections. The first consists in defending the idea that Bea l and Re a l actua ly discuss theoretical pluralism, not ap lied pluralism. In this sense, it would not be an interesting pluralism. The second objection would consist in pointing out that, even assuming that the position defended by Bea l and Re a l was an ap lied pluralism, the canonical ap lication of logic is inaccurate. At this point we wi l attempt to distin uish between different domains of reasoning and defend the implausibility of a logical pluralism in them. A third objection, independent of the other two and hypothetical, would be to ar ue that the theoretical pluralism (not ap lied pluralism) defended by Bea l and Re a l is compati le with (and under some interpretations leads to) logical monism.
Let's start with the first objection. What Bea l and Resta l ar ue is that there are different, equa ly accepta le ways to understand and theorize the pre-theoretical notion of logical consequence. This therefore leads us to accept other formal systems. But this kind of pluralism would be a non-inter-esting pluralism because it simply creates different logics from different precisifications of cases. What we are e a lishing are pure logics, in the sense described by Graham Priest.
Thus, we have an immense number of pure logics (classical logic, modal logics: S4, S5, T, etc.), chara erised as we l-defined mathematical structures. Then we would have a subset of such logics consisting of those that adequately chara erize our pre-theoretical notion of logical consequenceat least three: classical logic, intuitionistic logic, and logic of relevance.
To ta k about a pluralism of this kind would simply mean that such logics are mathematical structures with the property of fitting with our pre-theoretical notion of logical consequence. This is indisputa le but not interesting.
Someone might object that Bea l and Re a l do not actua ly advocate for a theoretical pluralism, but for an ap lied pluralism. In this way, one could ar ue that as this is a pluralism about the notion of logical consequence, it has been already ecified that its scope is the analysis of human reasoning. Nevertheless, there are reasons to think otherwise, namely that Bea l and Re a l maintain a purely theoretical pluralism. Here we can find proof. In Chapter 8 of Logical Pluralism (2006) the authors a dress some objections to their theory, including the fo lowing.
What logic is the logical pluralist using when defending his theory? Like every reasoning, logical pluralism is ar ued following inference patterns. What then is the logic behind the pluralistic reasoning? According to the objection, this wi l lead to logical monism. The response of the authors is as fo lows: (Beall and Restall, 2006, p. 99).

As anyone who applies formal logics knows, the fit between deductive validity and analysis of actual reasoning is not always an easy one. […] The pluralist claim is that, given a body of informal reasoning […], you can use different consequence relations in order to analyse the reasoning. As to which relation we wish our own reasoning to be evaluated by, we are happy to say: any and all (admissible) ones! Our arguments might be valid by some and invalid by others, good in some senses and bad in others
We can te l from the first sentence that deductive validity does not exactly fit with the analysis of the actual reasoning. In a dition, in their theory they eak of deductive validity. Therefore, we can conclude that their logical pluralism is a purely theoretical pluralism.
But let's sup ose that Bea l and Re a l ta k about ap lied pluralism. In that case, we would pass to the second objection: the one of the various areas of ap lication. The scope of application, we remember, is the canonical analysis of human reasoning (Priest, 2006, p. 196). A pluralist position on this ap lication (fo lowing the tenets of Bea l and Re a l) would say that at least two precisifications of cases are compati le with our notion of logical consequence, and would be equa ly valid in order to analyse human reasoning.
As a matter of fact, there is a variety of types of discourse, and in order to formalize them, we need different logics, focusing on the ecial features of each type of discourse. For instance, modal logic is more ap ropriate than classical logic in order to eak of necessity and possibility. And classical logic may be more ap ropriate than logic of relevance in certain situations. For instance, it seems that the principle of explosion is used in our ordinary lan uage. It does not seem absurd to say that if a contradiction is true, then we can conclude anything. "What do you mean? If it is true that you were in Barcelona and in Madrid at the same time, then I am a Martian" . What this is trying to show is that the way we understand the canonical ap lication of logic is wrong and that human reasoning cannot be understood as a single field. Instead, we need to ecify the types of eech that are in it. The canonical ap lication should not be understood as the analysis of reasoning in order to see if a conclusion, A, fo lows from a set of premises, Σ. What we must take into account is that the content of these premises and their context is relevant to see what logic we should use. Therefore, the main claim at this point is that there is no canonical ap lication, namely, human reasoning, but there are many ap lications depending on the content of the discourse that we are treating.
Thus, different discourses may require different logics. For instance, modal discourse would require a different logical analysis to temporal discourse. In this sense, it is clear that there is a plurality of equa ly valid logics, since they simply try to analyse different areas of ap lication. However, this would not be an interesting but an obvious pluralism. We can remind ourselves of Bea l and Re a l at this point. They remarked that their ar uments could be valid in one logic but invalid according to another. They ar ue that the logic that we should ap ly wi l depend on the type of verification we need for the task we are performing 2 .
Instead, a type of logical pluralism that might be interesting, as we anticipated at the beginning, would fo low if in a particular type of eech there was more than one equa ly valid logic. For example, focusing on necessity and possibility, we might ask whether there is more than one equa ly valid modal logic. As Priest (2006, p. 195) says, in its scope, logic is no longer a pure mathematical structure that does not ta k about the world as an explanatory theory of its field. However, it seems implausi le that we could be ta king about logical pluralism when we ta k about enclosed fields of reasoning, since we understand this logic as an empirical theory that attempts to account for a universe of discourse. In this way, as in other sciences, we do not accept various theories as valid in the same sense. We try to see which one is the best under dif-ferent criteria (for its simplicity, its lack of a hoc postulates, etc.). For instance, in the realm of necessity and contingency, a modal theory wi l be the best theory that fits in that particular realm. Ties could be possi le, but momentary. In the end, for one criteria or another, we would end up choosing one theory (logic) over another.
Fina ly, we sha l turn to the third objection. Here we wi l focus on the possibility of understanding the relation of logical consequence as we could understand negation or conditional if our lan uage use was ambi uous. If our use of negation is ambi uous such that, for instance, sometimes we use it with the meaning of classical negation and other times with the meaning of the intuitionistic negation, then we actua ly have two legitimate meanings of negation that therefore should be treated differently. We should have two signs in formal lan uage to represent both meanings, as we have different symbols for the conditional depending on its truth conditions.
In such a way, ¬ 1 ¬ 1 p |-p, meaning ¬ 1 as the intuitionistic negation, would be an invalid inference. While ¬ 2 ¬ 2 p |-p, meaning ¬ 2 as the classical negation, would be a valid inference. As we can see, we are not changing between logics and ar uing that in classical logic stating that ¬¬p |-p is a valid inference while in intuitionistic logic it is not. We are just treating two connectives with different truth conditions in a different way within the same logic (Priest, 2006, p. 198-199). Now, we have two possibilities: either our pre-theoretical notion of logical consequence relation is ambi uous, or it is not. If not, then we would conclude that there is only one way to understand it, one precisification of cases; therefore, we would maintain logical monism. However, if it is ambi uous, then this notion could be instantiated in different ways. An example of this is offered by Bea l and Re a l.
Nevertheless, another possi le way of putting this ambiuity is the fo lowing. If there is more than one precisification of cases, because of the ambi uity of the notion of logical consequence, then we can keep both the truth of, for example, |-1 p v ¬ p, and the falsity of |-2 p v ¬ p, meaning |-1 as the relation of logical consequence maintained by classical logicians, and |-2 as the relation of logical consequence maintained by intuitionist logicians. Since there is no single way of understanding the relation of logical consequence, we should act in the same way as with conditional or negation and e a lish that there are at least two types, with their own logical symbols. Thus, we are not saying that p ^ ¬ p is valid and invalid at the same time, 3 but that it is valid for a type of logical consequence and invalid for another.
What, then, is the difference from the logical pluralism developed by Bea l and Re a l? Bea l and Re a l's logical pluralism assumes the ambi uity of the notion of logical consequence, but solves it by validating different and mutua ly incompati le logics. According to the hypothesis given above, it would be possi le to create a new logic that, providing different signs for different ways of understanding this concept, could be a consistent theory.
However, we should not conclude that a l kinds of logical pluralism could entail monism. This objection works against the idea of a pluralism based on the breadth of possibilities that arise from a pre-theoretical conception of logical consequence. Thus, there could be other ways of understanding logical pluralism.