Abstract
In this paper we put forward and defend a view of the nature of logic that we call moderate anti-exceptionalism. In the first part of the paper we focus on the problem of genuine logical validity and consequence. We make use of examples from current debates to show that attempts to pinpoint the one and only authentic logic inevitably either yield irrefutable (and hence methodologically idle) theories or lead to dead ends. We then outline a thoroughly naturalist account of logical consequence as grounded in rules implicit in human linguistic practices (and thus immune to Quinean criticism of basing logic on explicit conventions). We insist that there are only two existing kinds of language: natural languages, and artificial languages that have been forged by us. There is thus no room for a "genuine" language (independent of us) and hence for "genuine" logic. We conclude that though logical theories are established—and are liable to criticism—in a similar fashion as those of the sciences, and in this sense logic is not exceptional, to fulfill its mission logic must lay a claim to normative authority over our argumentation and reasoning, which makes its methodology somewhat special. Logical theory is not meant to provide just an explanation, the standards it establishes serve also as a tool, providing for a reinforcement of our rational communication.
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Notes
Our understanding of naturalism is that of Quine (1969, p. 26): "Knowledge, mind, and meaning are part of the same world that they have to do with, and … they are to be studied in the same empirical spirit that animates natural science." Hence, we do not believe that studying logic necessitates any specific methods like a priori analysis or metaphysical inquiry.
When we speak about artificial languages, we don’t mean bare artificial languages consisting of mere lists of symbols plus formation rules, but rather languages with a "semantics"—be it specified in model-theoretic or proof-theoretic terms.
See Beall and Restall (2006) or the special issues of Synthese (Pluralistic Perspectives on Logic, 2020) and Inquiry (Logical Pluralism and Normativity, 2020).
What is the relationship between this plain variety of pluralism and the Beall–Restall variety? According to the latter, a single language can harbor a different version of consequence depending on the level of "counterfactual robustness" we choose. The version of pluralism inherent to our approach is less specific—it is compatible with their account, but we are not committed to this very version of pluralism.
In this article, we will follow the terminological convention we used in Peregrin and Svoboda (2017)—we will speak about (in)validity in case of forms of arguments and about (in)correctness in case of full-fledged arguments (arguments consisting of meaningful declarative sentences or of propositions expressed by such sentences.) How important the distinction is can be shown by means of a simple observation: Logical analysis can demonstrate that a certain argument is correct (because it exhibits a valid form) but it normally cannot demonstrate that it is incorrect—even if its form is invalid the argument can be correct (and hence it is wrong to call it "invalid"), as it can, for example, exhibit a valid form in a more fine-grained logical language. (See Svoboda & Peregrin, 2016.) Hence claims on validity (unlike claims on correctness) are, in effect, always relative (with respect to a given logical system).
The fact that we discuss an example marking the difference between these two concrete logics should not be seen as substantial. Historically, the disputes between CL and IL primarily concerned mathematics—the proponents of IL were not much interested in language outside of mathematics.
The term "language" is, as we all know, ambiguous. It is sometimes used for merely a syntactic structure, so that classical and intuitionistic logic can be said to share the same language. More often, it is used to cover also a semantic interpretation: a language with incidentally the same vocabulary and syntax as English but with different semantics would be considered a different language, and the same holds for artificial languages. We should also see a grave difference between natural languages used for communication and the artificial languages that derive their entitlement to being called "languages" merely from the fact that they have some features which are also characteristic of natural languages.
Not that negation in a natural language like English would be a transparent matter (see, e.g., Zeijlstra, 2007). But let us take a pass on this problem here. Let us assume, just for the sake of the present argument, that there is something that can be called negation in English; and similarly for other languages.
Where deciding which arguments do exhibit the form and which do not, of course, may often be far from a routine matter.
Some logicians claim that natural language does not have any logic (Glanzberg, 2015), some even use this to underpin the thesis of "logical nihilism" (Cotnoir, 2018). Our view is that what is properly called logic is reached via a theoretical reflection characterized in greater detail in Sect. 6. Given this, logic, strictly speaking, is not something to be found in natural language. It is, nevertheless, plausible to presume that natural languages do harbor a (proto)logic (or perhaps slightly different (proto)logics).
This is not, of course, to say that the correctness of any argument could be determined by a public poll. A lot of even minimally complex arguments may be correct despite the majority of speakers rejecting them or vice versa. But there are simple arguments the (in)correctness of which may be seen as constitutive of the meaning of the components they contain, and in their case there can be hardly any higher authority than the competent speakers (whose opinions, of course, need not have the same relevance—in case of some expressions the opinion of experts counts as more important than that of laypeople). There is, to be sure, no sharp boundary separating the "simple" from the "complex" arguments. But though logicians (or, for that matter, anybody else) can perhaps make us hold (A1) for (in)correct, this would not be a correction of a logical error, but rather a successful imposing of a specific meaning for "not" on speakers of English.
Already Aristotle intentionally disregarded, within his projects of syllogistics, the fact that terms like "all" and "some" were not used unanimously by his fellows and assigned them determinate meanings according to which "some" meant "at least one" and "all" entailed "some", thus treating the terms as logical constants. Modern logicians introduce artificial signs that serve as logical constants but the purpose is the same—regimentation.
It is symptomatic that Caret talks about the argument form "Necessarily, P. Therefore, P" as if it were an argument. The failure to distinguish between the two may cause confusion. A careful classical logician would probably refuse to assess the form as it contains an expression that doesn’t fit into the language of classical logic. However, when assessing an argument of the form she may conclude (seeing necessarily as an extralogical word) that it is correct but not logically correct.
In the sense of Quine’s (1986) famous dictum "change of logic, change of subject".
The difference between seeing formal languages of logic as self-contained structures and seeing them as such prisms or models of natural languages is discussed by Peregrin (2020).
Arguments, as we know, can be legitimately ascribed different logical forms, and the fact that one of the forms is invalid doesn't exclude that its other (typically more fine grained) form is valid.
We should notice that formal languages of logic are (unlike formalized ones, like that of Peano arithmetic) unable to express propositions at all (like the language of propositional logic) or only very specific, "trivial" propositions like ∀x(x = x) of classical predicate logic with identity. (Formulas like P ∧ Q or ∀x∃yR(x,y), with uninterpreted P, Q and R, do not express any specific propositions.) Thus, rather than languages they are mere language forms.
There are many ways of explicating the term "proposition", including such as "a class of possible worlds". Such propositions, then, do not have the kind of form we talk about. Here we focus on more traditionally conceived propositions viewed as structured entities that characteristically bear a given (unchangeable) truth-value. On the other hand, the term "proposition" is also used in a more mundane sense. Propositions in this sense are "that which two sentences in different languages must have in common in order to be correct translations each of the other" (Church, 1956, p. 25). We may call such propositions sentential propositions, while the propositions belonging to pure thought (pure reasoning) pure propositions. (We don’t mean to suggest that these accounts are the only ones associated with the term "proposition".) While it seems obvious that pure propositions as constant bearers of truth-values can’t be vague or unclear, sentential propositions can.
Note, however, that as the name "Socrates" certainly does not pick up a unique individual, the contrast with (A1) is not so sharp as it may prima facie seem.
Cf. Peregrin (2010). As adherents of naturalism we are, of course, very suspicious of explanations that involve supernatural perception or insight into a supernatural realm.
Russell (1919, pp. 169–170).
Thus Sider (2011, p. 115): "The status in contemporary philosophy of logical conventionalism and the related doctrine of 'truth by convention' is curious. On the one hand, few people self-identify as logical conventionalists. If pressed on why not, I suppose most would gesture at Quine’s famous critique in 'Truth by Convention'."
As Russell (1921) put it: "[w]e can hardly suppose a parliament of hitherto speechless elders meeting together and agreeing to call a cow a cow and a wolf a wolf" (p. 190).
In history of philosophy the term was used to bear a number of further meanings, and in today’s everyday discourse it is used in many ways and contexts, often to loosely suggest a kind of implicit rationality or coherency.
Priest (2014) presents a similar threefold division of logic. He reiterates the medieval division talking about logica docens (which is basically our sense (2)) and logica utens (which is our (1)); and adds his own invention, logica ens, which concerns "what is actually valid: what really follows from what". Needless to say, we have no room for this sense in our theory: our sense (3) is something utterly different.
It is even possible to consider the phrase "implicit convention" as a case of contradictio in adjecto. As Quine asks: "What is convention when there can be no thought of convening?" (Quine, 2008, p. xi). We might perhaps say that logical truths of natural logics are something like "covertly conventional", but this would not help elucidate the situation in any way. Logical truths of formal or formalized languages are ("overtly") conventional in the sense that they are established by definitions. But we are convinced that the space within which logicians can sensibly introduce their conventions is quite limited (similarly as is limited the space for explicit conventions posited by linguists dealing with grammar, morphology, phonology, etc.). Cf. Warren (2020, Chapter 7).
We should perhaps stress once again that only theories which are tried and true deserve the status of logical theory. For details cf. Peregrin and Svoboda (2017).
Logic, of course, does not target argumentation in its entirety—it restricts itself to cases where logical vocabulary plays the crucial role. And despite the fact that it targets de facto argumentation, it generally operates on a more abstract level than disciplines such as critical thinking or the theory of argumentation.
It is worth pointing out that this is a simplification—there is no single natural logic. Languages can to some extent differ in their logical build-up. There are, for example, logically relevant differences between languages that employ definite and indefinite articles and those which don’t; or between the ways languages treat quantification (Bach et al., 2013). The differences are not so serious to make their sentences mutually untranslatable, but they affect the perception of the (natural) logical structure of sentences and they also may influence the selection of issues to be addressed.
For a discussion on intricacy of economic predictions see McCloskey (1998).
This, of course, doesn’t exclude the relevant mathematical structures from the purview of logic—and hence seeing them as a legitimate object of its study. We, however, should appreciate the importance of the question as to what makes some mathematical structures (rather than others) interesting for logic. The answer, we think, is that they are capable of functioning as useful models of overt (publically accessible) human reasoning.
The view that formal logic essentially offers models of reasoning is put forward, e.g., by Shapiro (2001).
By saying this we don’t want to suggest that all mental processes we tend to associate with reasoning have the character of a kind of "inner talk". We only insist that insofar as we see reasoning as a process of moving from premises to conclusions, both the premises and the conclusions have to assume a kind of linguistic shape.
It is worth noting that here the term "logic" is used in yet another way than were the three understandings that we distinguished above. This logic is neither a natural phenomenon nor a theory. Perhaps we should also have distinguished using the term as a name of a supernatural phenomenon (after all, that such logic exists was rather commonly assumed—or even taken for granted—during the history of European logic) but we, being naturalists, are reluctant to admit such phenomena.
Admittedly, what counts as belonging to natural language develops. Some originally logical constructs have been integrated into the vocabularies of natural languages. Phrases like "material implication" or "exclusive disjunction" are comprehensible to a majority of educated English speakers.
There is little doubt that we can (and do) have knowledge that concerns internal structures of individual logical theories as well as knowledge concerning their mutual relationships. In our view, however, we cannot have real knowledge of which logical theory (or theories) are the "right" ones; we, nevertheless, do have some methodological mechanisms that eliminate theories which do not deserve the label "logical". For discussion of criteria that underpin such mechanisms see Peregrin and Svoboda (2017).
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Funding
Work on this paper was supported by Grant No. 20-18675S of the Czech Science Foundation. The authors are grateful to Georg Brun, Ulf Hlobil, Vít Punčochář, and two reviewers of this journal for valuable critical comments.
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This article belongs to the topical collection on Anti-Exceptionalism about Logic, edited by Ben Martin, Maria Paola Sforza Fogliani, and Filippo Ferrari.
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Peregrin, J., Svoboda, V. Moderate anti-exceptionalism and earthborn logic. Synthese 199, 8781–8806 (2021). https://doi.org/10.1007/s11229-021-03182-9
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DOI: https://doi.org/10.1007/s11229-021-03182-9