Progress in Mathematics and Logic — IV

[for a brief explanation of this ongoing series, as well as a full table of contents, go here]

A panoply of logics

The striking thing about contemporary logic is that it is plural. Indeed, Logics (Nolt 1996) was the title of the book I used as a graduate student at the University of Tennessee, and that in itself was a surprise for me, since I had naively always thought of logic as a single, monolithic discipline. (But why, really? We have different ways of doing geometry and mathematics, and certainly a plethora of natural sciences!). The following brief look at modern logic should be enough to convince readers that the field is both vibrant and progressive, in the sense discussed above.

Consider, for instance, modal logic (Garson 2009), which deals with the behavior of sentences that include modal qualifiers, such as “necessarily,” “possibly,” and so on. It is actually a huge field within logic, as it includes deontic (“obligatory,” “permitted,” etc.: Hilpinen 1971), temporal (“always,” “never”), and doxastic (“believes that”) logics. Arguably the most familiar type of modal logic is the so-called K-logic (named after influential 20th century philosopher Saul Kripke), which is a compound logical system made of propositional logic, a necessitation rule (using the operator “it is necessary that”) and a distribution axiom. A stronger system, called M, is built by adding the axiom that whatever is necessary is the case, and this in turn yields an entire family of modal logics with different characteristics. Starting again with K-logic, one can also begin to build a deontic logic by adding an axiom that states that if x is obligatory, then x is permissible. Similarly, one can build temporal logics (with the details depending on one’s assumptions about the structure of time) and other types of modal logics. Indeed, Garson (2009) presents an elegant “map” of the relationships among a number of modal logics, demonstrating how they can all be built from K by adding different axioms. Furthermore, Lemmon and Scott (1977) have shown that there is a general parameter (G) from the specific values of which one can derive many (though not all) the axioms of modal logic. Garson’s connectivity map  and Lemmon and Scott’s G parameter are both, well, highly logical, and aesthetically very pleasing, at the least if you happen to have a developed aesthetic sensibility about logical matters. (Notice, once again, that these are all nice examples of non-teleonomic progress propelled by internally generated problems and consisting in exploring additional possibilities in a broad conceptual space that is evoked once one adds axioms or assumptions.)

A major issue with modal logic is that — unlike classical logic (Benthem 1983) — it is not possible to use truth tables to check the validity of an argument, for the simple reason that nobody has been able to come up with truth tables for expressions of the type “it is necessary that” and the like. The accepted solution to this problem (Garson 2009) is the use of possible worlds semantics (Copeland 2002), where truth values are assigned for each possible world in a given set W. Which implies that the same proposition may be true in world W1, say, but false in world W2. Of course one then has to specify whether W2, in this example, is correctly related to W1 and why. This is the sort of problem that has kept modal logicians occupied for some time, as you might imagine. Modal logic has a number of direct philosophical applications, as in the case of deontic logic (McNamara 2010), which deals with crucial notions in moral reasoning, such as permissible / impermissible, obligatory / optional, and so forth. Deontic logic has roots that go all the way back to the 14th century, although it became a formal branch of symbolic logic in the 20th century. As mentioned, deontic logic can be described in terms of Kripke-style possible worlds semantics, which allows formalized reasoning in metaethics (Fisher 2011).

Of particular interest, but for different reasons, are also the next two entries in our little catalog: many-valued and “fuzzy” logics. The term many-valued logic actually refers to a group of non-classical logics that does not restrict truth values to two possibilities (true or false; see Gottwald 2009). There are several types of many-valued logics, including perhaps most famously Łukasiewicz’s and Gödel’s. Some of these admit of a finite number of truth values, others of an infinite one. The Dunn/Belnap’s 4-valued system, for instance, has applications in both computer science and relevance logic. Multi-valued logic also presents aspects that reflect back on philosophical problems, as in the area of concepts of truth (Blackburn and Simmons 1999), or in the treatment of certain paradoxes (Beall 2003), like the heap and bald man ones. More practical applications are found in areas such as linguistics, hardware design and artificial intelligence (e.g., for the development of expert systems), as well as in mathematics. Fuzzy logic is nested within many-valued logics (Zadeh 1988; Hajek 2010), consisting of an approach that makes possible to analyze vagueness in both natural language (about degrees of beauty, age, etc.) and mathematics. The basic idea is that acceptable truth values range across the real interval [0,1], rather than lying only at the extremes (0 or 1) of that interval, as in classical logic. Fuzzy logic deals better than two-valued logic with the sort of problems raised by Sorites paradox, since these problems are generated in situations in which small/large, or many/few quantifiers are used, rather than simple binary choices. Therefore, fuzzy logic admits of things being almost true, rather than true, and it is for this reason that some authors have proposed that fuzzy logic can be thought of as a logic of vague notions.

A whole different way of thinking is afforded by so-called intuitionistic logic (Moschovakis 2010), which treats logic as a part of mathematics (as opposed to being foundational to it, as in the old Russell-Whitehead approach that we have discussed above), and — unlike mathematical Platonism (Linnebo 2011) — sees mathematical objects as mind-dependent constructs. Important steps in the development of intuitionistic logic were Gödel’s (who, ironically, was a mathematical Platonist!) proof (in 1933) that it is as consistent as classical logic, and Kripke’s formulation (in 1965) of a version of possible-worlds semantics that makes intuitionistic logic both complete and correct. Essentially, though, intuitionistic logic is Aristotelian logic without the (much contested) law of the excluded middle, which was developed for finite sets but was then extended without argument to the case of infinities.

Finally, a couple of words on what some consider the cutting edge, and some a wrong turn, in contemporary logic scholarship: paraconsistent and relevance logics. Paraconsistent logic is designed to deal with what in the context of classical logic are regarded as paradoxes (B. Brown 2002; Priest 2009). A paraconsistent logic is defined as one whose logical relations are not “explosive,” [12] with the classical candidate for treatment being the liar paradox: “This sentence is not true.” The paraconsistent approach is tightly connected to a general view known as dialethism (Priest et al. 2004), which is the idea that — contra popular wisdom — there are true contradictions, as oxymoronic as the phrase may sound. Paraconsistent logic, perhaps surprisingly, is not just of theoretical interest, as it turns out to have applications in automated reasoning: since computer databases will inevitably include inconsistent information (for instance because of error inputs), paraconsistent logic can be deployed to avoid wrong answers based on hidden contradictions. Similarly, paraconsistent logic can be deployed in the (not infrequent) cases in which people hold to inconsistent sets of beliefs, sometimes even rationally so (in the sense of instrumental rationality). More controversially, proponents of paraconsistent logic argue that it may allow us to bypass the constraints on arithmetic imposed by Gödel’s theorems. How does this work? One approach is known as “adaptive logic,” which begins with the idea that consistency is the norm, unless proven otherwise, or alternatively that consistency should be the first approach to a given sentence, with the alternative (inconsistency) being left as a last resort. That is to say, classical logic should be respected whenever possible. Paraconsistent logics can be generated using many-valued logic, as shown by Asenjo (1966), and the simplest way to do this is to allow a third truth value (besides true and false), referred to as “indeterminate” (i.e., neither true nor false).

Yet another approach within the increasingly large family of paraconsistent logics it to adopt a form of relevance logic (Mares 2012), whereby one stipulates that the conclusion of a given instance of reasoning must be relevant to the premises, which is one way to block possible logical explosions. According to relevance logicians what generates apparent paradoxes is that the antecedent is irrelevant to the consequent, as in: “The moon is made of green cheese. Therefore, either it is raining in Ecuador now or it is not,” which is a valid inference in classical logic (I know, it takes a minute to get used to this). The typical objection to relevance logic is that logic is supposed to be about the form, not the content, of reasoning — as we have seen when briefly examining the history of both Western and Eastern logic — and that by invoking the notion of “relevance” (which is surprisingly hard to cash out, incidentally) one is shifting the focus at least in part to content. Mares (1997), however, suggests that a better way to think about relevance logic comes with the realization that a given world X (within the context of Kripe-style many worlds) contains informational links, such as laws of nature, causal principles, etc. It is these causal links, then, that are deployed by relevance logicians in order to flesh out the notion of relevance (in a given world or set of possible worlds) — perhaps reminiscent of Dharmakīrti’s invoking of causal relations to assure the truth of a first premiss, as we have discussed. Interestingly, some approaches to relevance logic (e.g., Priest 2008) can be deployed to describe the difference between a logic that applies to our world vs a logic that applies to a science fictional world (where, for instance, the laws of nature might be different), but relevance logic has a number of more practical applications too, including in mathematics, where it is used in attempts at developing mathematical approaches that are not set theoretical, as well as in the derivation of deontic logics, and in computer science (development of linear logic).

So, what do we get from the above historical and contemporary overviews of logic, with respect to how the field makes progress? The answer is, I maintain, significantly different from what we saw for the natural sciences in Chapter 4, but not too dissimilar from the one that emerged in the first part of this chapter in the case of mathematics. Logicians explore more and more areas of the conceptual space of their own discipline. Beginning, for instance, with classical two-valued logic it was only a matter of time before people started to consider three-valued and then multi-valued, and finally infinitely-valued (such as fuzzy) types of logical systems. Naturally enough, this progress was far from linear, with some historians of logic identifying three moments of vigorous activity during the history of Western logic, with the much maligned Middle Ages not devoid of interesting developments in the field. Other signs of progress can readily be seen in the expansion of the concerns of logicians, beginning with Aristotelian syllogisms or similar constructs (as in the parallel developments of the Stoics) and eventually exploding in the variety of contemporary approaches, including deontic, temporal, doxastic logics and the like. Even dialethism and the accompanying paraconsistent and relevance logics can be taken as further explorations of a broad territory that began to be mapped by the ancient Greeks: once you chew for a while (in this case, a very long while!) on the fact that classical logic can yield explosions and paradoxes, you might try somewhat radical alternatives, like biting the bullet and treating some paradoxes as “true,” or pushing for the need for a three-valued approach when considering paradoxes. As in the case of mathematics, there have been plenty of practical (i.e., empirical) applications of logic, which in themselves would justify the notion that the field has progressed. But in both mathematics and logic I don’t think the empirical aspect is quite as crucial as in the natural sciences. In science there is a good argument to be made that if theory loses contact with the empirical world (Baggott 2013) then it is essentially not science any longer. But in mathematics and logic that contact is entirely optional as far as any assessment of whether those fields have been making progress by the light of their own internal standards and how well they have tackled the problems that their own practitioners set out to resolve.

What about philosophy, then? It is finally to that field that I turn next, examining a number of examples of what I think clearly constitutes progress in philosophical inquiry. As we shall see, however, the terrain is more complex and perilous. More complex because while philosophy is very much a type of conceptual activity, sharing in this with mathematics and logic, it also depends on input from the empirical world, both in terms of commonsense and of scientific knowledge. More perilous because a relatively easy case could be made that a significant portion of philosophical meanderings aren’t really progressive, and some even smack of mystical nonsense. Unfortunately, this sort of pseudo-philosophy does appear in the philosophical literature, side by side with the serious stuff, and occasionally is even the result of the writings of the same authors! Nothing like that appears to be happening either in science or in mathematics and logic, which I believe is a major reason why philosophy keeps struggling to be taken seriously in the modern academic world.

Notes

[12] A logical explosion is a situation where everything follows from a contradiction, which is possible within the standard setup of classical logic.

References

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