Progress in Mathematics and Logic — III

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

Logic: the historical perspective

In the case of logic, too, a bit of a historical perspective will help making the case for how the field has, unquestionably, made progress, albeit differently from both the situations we have already explored in the cases of the natural sciences and of mathematics. To get us started, let us see if we can get clearer about what it is that defines the subject matter we are now tackling. According to Bird (1963) most authors agree that logic is concerned with the structure of propositions, independently of content, and Kneale (cited in Bird, p. 499) puts it this way: “[logic consists in] classifying and articulating the principles of formally valid inference” (which puts logic, at the least in part, in the business of carrying out normative analyses of human reasoning, as opposed to, say, the sort of descriptive picture we get from psychological studies of cognitive biases: Caverni et al. 1990; Pohl 2012). As we shall see in the next section, there actually are some exceptions to this general idea, but broadly speaking that is a good definition of the object of study of logicians.

Logic originated (and, to a point, evolved) pretty much independently in two places: Greece and India, and we will see that Indian logicians discovered (or invented) many of the same principles that are familiar to us from Greek logic, a pattern that we readily expect in the natural sciences, but that we have seen takes place also in mathematics. Nonetheless, Bird argues that progress in logic has not followed a linear trajectory, citing Bochenski (p. 494) as proposing a framework for understanding the history of (Western) logic that roughly speaking recognizes three “high points”: Aristotelian and Stoic ancient logic; Medieval Scholastic logic; and modern mathematical logic. I will get to some of the details below, but before proceeding let me follow King and Shapiro (1995) in providing a general panorama that I hope will be helpful to properly situate much of the rest of this chapter.

They begin with the ancient Greeks, crediting Aristotle both with the invention of syllogistics and of essential so-called “metalogical” theses: the Law of Bivalence, the Law of Noncontradiction, and the Principle of the Excluded Middle. After Aristotle and the Stoics, the first major innovator was Peter Abelard (12th century), most notably because of his formulation of relevance criteria for logical consequences. In the 14th century Jean Buridan, Albert of Saxony and William of Ockham were among those who helped develop supposition theory. We then jump to the 19th century, when Bolzano elaborated important new notions, such as those of analyticity and logical consequence. By the end of that century, according to King and Shapiro, we can distinguish a trifurcation of approaches in modern logic: (i) the algebraic school (exemplified by Boole and Venn), interested in developing calculi for reasoning, such as the one nowadays referred to as Boolean algebra, which has had countless applications in computer science; (ii) the logicist school (Frege and Russell, among others), aiming at codifying the general features of precise discourse within a single system. Famously, it was Russell (through the discovery of his namesake paradox) that managed to undermine Frege’s project of showing that arithmetic is a part of logic; and (iii) the mathematical school (e.g., Hilbert, Peano, Zermelo), attempting to axiomatize branches of mathematics, like geometry and set theory. Contemporary research in logic is a mixed blend of these three general tendencies, and has yielded plenty of fruits: work on meta-mathematics, culminating in Gödel’s incompleteness theorems; Alfred Tarski’s work on definitions of truths and logical consequence; and Alonzo Church’s demonstration that there is no algorithmic way to show that a given first-order formula is a logical truth, to mention just a few examples. Moreover, mathematical logic has provided the underlying structure for modern work in analytic philosophy, on which I will touch briefly below, including the contributions of authors like Davidson, Dummett, Quine, and Kripke.

With this broad vista in mind, let us take a closer look at three specific periods of development of logic, two in the Western world (ancient and medieval logic), the third one in India, so to better be able to appreciate the dynamics of progress in the field, as well as how they differ from what we have seen so far. Arguably, the beginning of logic took place once people started to think about the patterns of arguments, paying attention to their formal grammar (Bobzien 2006). Among the ancient Greeks, Zeno of Elea and Socrates used essentially what today are known as reductio arguments, though they took them for granted, without exploring their logical structure, while Eubulides originated a number of well known paradoxes, starting with the Liar and Sorites ones. Both the sophists and Plato were interested in logic and in fallacies of reasoning, and Plato makes an early distinction between syntax and semantics, i.e. between asking what a statement is and when that statement is true.

Following Bobzien (2006; but see also: Kapp 1942; Kneale and Kneale 1962; Lear 1980), Aristotelian logic — certainly the turning point in the Western history of the field — shares elements in common with both predicate logic and set theory, though Diodorus Cronus, his student Philo, and later the Stoic Chrysippus developed a different approach to logic, which turns out to have similarities with the much later work of Frege. Aristotle’s Topics (which is part of his Organon, the target of the famous criticism by Francis Bacon, many centuries later) is the first complete treatise on logic, while his Sophistical Refutation is the first formal analysis of logical fallacies, building, presumably, on the early treatment by his mentor, Plato. Without question Aristotle’s most enduring contribution to logic, lasting well into the 19th century, was his detailed analysis of syllogisms — which was later commented upon and refined throughout the Middle Ages. And it was again Aristotle who invented modal logic, where the predicate holds actually, necessarily, possibly, contingently or impossibly.

Even before getting to the often unjustly under-appreciated (at least in popular lore) medieval logicians, the Aristotelian system was improved and refined by a number of people, including Theophrastus (who was a student of Aristotle) and Eudeumus, who pointed out that in modal logic the conclusion must have the same modal character of the weaker premise. These same authors also introduced what amounts to the forerunners of modern day modus ponens and modus tollens. The Stoics, such as Chrysippus of Soli, seem to have endorsed a type of deflationary view of truth, and were particularly interested in the “sayables” — i.e., in whatever underlies the meaning of everything we think. A subset of sayables is constituted by the so-called assertibles, which are characterized by truth values. The assertibles are the smallest expressions in a deductive system, and including them in one’s logic gives origin to a system of propositional logic in which arguments are composed of assertibles. The Stoics also developed a system of syllogisms, and they recognized that not all valid arguments are syllogisms. Their syllogistics, however, is different from Aristotle’s, and has more in common with modern day relevance logical systems. These two traditions in Greek logic, the so-called “Peripatetic” (i.e., Aristotelian) and Stoic were brought together by Galen in the 2nd Century, who made a first (and largely incomplete) attempt at synthesizing them. Stoic logic, however, pretty much disappeared from view by the 6th century CE, to eventually re-emerge only during the 20th century because of renewed interest in propositional logic.

Following Lagerlund (2010), the history of medieval logic is divided into “old logic” and “new logic,” separated by the figure of Abelard (12th century CE). This is because until Abelard’s time logicians only had access to parts of Aristotle’s works, which excluded the Prior Analytics, the book in which Aristotle developed his theory of syllogisms, although people knew about his theory from secondary sources. Aristotle’s theory was an impressive achievement, but was incomplete, and we owe its fuller development to the medieval logicians. In particular, Aristotle employed two methods to jointly prove validity in syllogistic theory (reductio ad impossibile and ekthesis [11]), and it was Alexander of Aphrodisias (c. 200 AD) who first showed that ekthesis was by itself sufficient for the purpose. The “old logic” period begins with Boethius (6th century CE), who provided a presentation of syllogistic theory that was clearer than Aristotle’s own (though anyone familiar with Aristotle’s work will testify that that particular bar is set a bit low). Boethius did make a few novel contributions, the main one of which was his introduction of the hypothetical syllogism, i.e., a syllogism in which one (or more) of the premises is a hypothetical, rather than a categorical sentence.

We then have to wait another six centuries — underscoring the fact that logic, more so even than mathematics, certainly doesn’t make progress in anything like a linear or steady fashion — for Abelard to clarify and improve on Boethius’ work and also to introduce the famous distinction between de dicto and de re modal sentences. The idea is that a sentence such as “Massimo necessarily has to write” can be interpreted in two ways: “Massimo writes necessarily” or “It is necessary that Massimo writes.” The difference is between a personal and an impersonal reading of the original sentence, and Abelard pointed out that the distinct meanings of modal sentences should be kept in mind, because attributes such as quantity and quality hold differently depending on the modal meaning. One of Abelard’s main contributions was the beginning of a theory of consequences, which later on during medieval times replaced syllogistic theory as the main interest of logicians.

Again following Lagerlund (2010; see also: Lagerlund 2000; Zupko, 2003; Dutilh-Novaes 2008), Richard of Campsall (14th century CE) provided a complex rendition of syllogistic theory, which ended up implicitly showing that a consistent interpretation of Aristotle’s Prior Analytics is actually not possible. This in turn led William of Ockham to take the extraordinary step of simply abandoning Aristotle’s approach to seek a more systematic account of syllogistic theory. But it was John Buridan who showed that syllogistic theory is in fact a special case of a more general theory of consequences. His work achieved the most complete account of modal logic available at the time, and by implication the most powerful system of logic devised within the Western canon before the modern era, with some modern commentators even arguing that Buridan’s system was developed by thinking in terms of a (very contemporary) “possible worlds” model — though recall the warning given above about unwisely lapsing into a heritage view of history.

The reader may have noticed that I have used several times the term “development” to describe the above sequence of discoveries (or inventions) in Western logic. That term seems to me to best capture the sense in which logic makes progress: not by accumulating truths about the world (a la natural science), but — similar to mathematics — by becoming more and more explanatory in response to largely internally generated problems. This is, again, not teleonomic progress toward an end goal, or overarching theory of everything, but expansion by multiplicative exploration of novel areas of conceptual space — areas that, to use Smolin’s terminology introduced at the beginning of the book, are progressively “evoked” once logicians adopt new axioms and build new systems. This will appear even more obviously to be the case below, once we get to the panoply of contemporary logics (plural).

Our third glance at some of the details of the history of logic turns east, toward the parallel development of the field in the Indian tradition (for an overview see: Matilal 1998; Ganeri 2001; Gillon 2011; see also Chapter 2). Here the early Buddhist literature already mentions debates and other forms of public deliberation in ancient India, suggesting an early interest in the art of reasoning similar to the one developed independently in ancient Greece. Buddhist writers from the 3rd century BCE were aware that the form of an argument is crucial to the argument being a good one. Moreover, we have texts that document Indian logicians’ familiarity with a number of forms of reasoning that are equivalent to several of those developed in the West, including modus tollens, modus ponens, and reductio ad absurdum. The principle of non-contradiction was not studied explicitly, but was often implicitly invoked (e.g., by the Buddhist philosopher Nāgārjuna in the 2nd century CE). At about the same time Gautama wrote Nyāya-sūtra (Aphorisms on Logic), an early treatise on inference and logic, and a bit later (6th century CE) Bhartṛhari formulated a version of the principle of the excluded middle.

Even before Bhartṛhari, Vātsyāyana (5th century CE) rejected similarity and dissimilarity (i.e., arguments from analogy) as underlying syllogisms, proposing instead a view of syllogism that invokes the concept of causation. While this was far from a complete account, it demonstrates a very clear understanding of the problem of syllogistic soundness. Dignāga (c. 5th–6th century CE) then made the connection between inference and argument, treating them as two aspects of the same reasoning process, although according to Gillon (2011) he seemed to be confused about the role of examples in syllogisms (he thought they were necessary, and they are not, they are just illustrative). That confusion apparently stemmed from a deeper one: failing to make a sharp distinction between validity and persuasiveness (examples are crucial to the latter — as any good lawyer will tell you — but irrelevant to the former).

One of Dignāga’s students, Īśvarasena, seems to have been the first to recognize the problem of induction (which was not formulated explicitly in the West until Hume in the 18th century!), though his solution of it (invoking non-perception of the first premiss) was inadequate (as pretty much any other solution proposed thus far, I might add). One of his students, Dharmakīrti (7th century CE), in turn tried his hand at solving the problem by arguing that the truth of the first premiss is guaranteed by causal relations or by identity. The story could continue with a number of other convergences and parallels with the history of Western logic, but I think that the examples given above clearly affirm that logical principles were arrived at in both India and the West, and that the field made progress — in the sense clarified above — for centuries before the modern era, albeit of course at different paces between East and West, and with partially different focus and approaches. Let’s now turn from history to examine what contemporary logic as a field of scholarship looks like. It will be immediately clear that the telltale signs are the same: diversification and non-teleonomic progress interpreted as continuously expanding explanatory power with regard to internally generated problems.


[11] On the not exactly crystalline topic of ekthesis, see: Smith, 1982.


Bird, O. (1963) The history of logic. The Review of Metaphysics 16:491-502.

Bobzien, S. (2006) Ancient logic. Stanford Encyclopedia of Philosophy (accessed on 2 August 2013).

Caverni, J.-P., Fabre, J.-M., and Gonzalez, M. (1990) Cognitive Biases. Elsevier Science.

Dutilh-Novaes, C. (2008) Logic in the 14th Century after Ockham. In: D. Gabbay and J. Woods (eds.), Handbook of the History of Logic, vol. 2, Medieval and Renaissance Logic, North Holland.

Ganeri, J. (ed.) (2001) Indian Logic: A Reader. Curzon.

Gillon, B. (2011) Logic in classical Indian philosophy. Stanford Encyclopedia of Philosophy (accessed on 20 July 2012).

Kapp, E., (1942) Greek Foundations of Traditional Logic. Columbia University Press.

King, P. and Shapiro, S. (1995) The history of logic. In: T. Honderich (ed.), The Oxford Companion of Philosophy, Oxford University Press, pp. 496-500.

Kneale, M. and Kneale, W. (1962) The Development of Logic. Clarendon Press.

Lagerlund, H. (2000). Modal Syllogistics in the Middle Ages. Brill.

Lagerlund, H. (2010) Medieval theories of the syllogism. Stanford Encyclopedia of Philosophy accessed on 2 August 2013.

Lear, J. (1980) Aristotle and Logical Theory. Cambridge University Press.

Matilal, B.K. (1998) The Character of Indian Logic. State University of New York Press.

Pohl, R.F. (2012) Cognitive Illusions: A Handbook on Fallacies and Biases in Thinking, Judgement and Memory. Psychology Press.

Smith, R. (1982) What is Aristotelian Ecthesis? History and Philosophy of Logic 3:113–127.

Zupko, J. (2003) John Buridan: Portrait of a Fourteenth-Century Arts Master, Publications in Medieval Studies, University of Notre Dame Press.


16 thoughts on “Progress in Mathematics and Logic — III

  1. Hi Massimo,

    “– tough recall the warning given above about unwisely incurring into a heritage view of history.”

    Typo: “tough” => “though”. “Incurring into” is also not quite the right words. Maybe “lapsing into” or “adopting”.

    Liked by 1 person


    Not on today’s topic, but relevant to your book and why popular science books by scientist are often so bad. Using them as to represent science borders on a ‘straw man’ argument, accept of course you didn’t have to assemble the ‘straw man’ yourself, but found him premade by himself.

    Mathematicians don’t seem as bad, but maybe that’s because I can’t see when they’re BS’ing as easily.

    Why in the world is fake new more accurate than so called news? On physics I’ve found only the ‘Economist’ is reasonably accurate.


  3. “The great mystery of mathematics is its lack of mystery! So says this article and it’s author claims that every mathematician he has ever spoken with agrees.”

    Here’s a math mystery:

    Though I gather they have some idea why this occurs, it is weird.

    It would be even stranger in base 2. The last digit of a prime is of course always 1, but What if 0 was followed by 0 more often than 1 or something like that. As far I can see they haven’t looked into that or any base other than 10.Giben a binary table of primes, I could do that myself, though the why would be beyond me.


  4. From I: and very much in contrast to the general neglect of the history of science that scientists often display.

    This reminds me of how I forgot how, Bohr preceded de Broglie, because it easier to teach the Bohr atom if you start with de Broglie, In fact Bohr has to make some brilliant leaps that are hard to follow which seem to logically follow once you have matter as wave.

    The book [a] I 1st learned Bohr Atom does in fact get the historical order right and then proceeds (after due warning) to explain it the other way ’round.

    I was always better with concepts than dates, but this was a pretty extreme example of that.

    [a] Weidner and Sells, ‘Elementary Modern Physics’

    I have my copy some place. Likely a first addition. There’s no mention of quarks or strings in it.


  5. From I; The second approach, geometric analysis (Mahoney 1968), led to both a preoccupation with general problems in geometry and to the solution of specific issues, such as trisecting the angle.

    This gives the impression that the ‘problem of trisecting the angle’ can be solved. It can’t, so the ‘solution’ is that it was proved that it can’t.

    I think you might want to re-word that slightly.

    We had a lot of fun trying to trisect angles in high school geometry literally with compass and straight edge. A binary search gives an approximation –as close as you care to pursue and, thus, one encounters infinity in the real world (so to speak).


  6. Consequently, students get a misleading view of a series of finished products with no sense of why the questions were raised to begin with, or of what struggle ensued in the attempt to answer them.

    This certainly happened to me.

    I do think you should remove ‘monotonic’ from your description of progress of science.


  7. Rotating Reference Frames

    Another confused discussion in ‘Dreams’ – there’s actually some good stuff too, but it’s less relevant to the issues of this book.

    Weinberg:[The philosopher] Mach’s hint was picked up by Einstein and made concrete in his general theory of relativity. In general relativity there is indeed an influence exerted by the distant stars that creates the phenomenon of centrifugal force in a spinning merry-go-round: it is the force of gravity.

    Weinberg, Steven. Dreams of a Final Theory: The Scientist’s Search for the Ultimate Laws of Nature (pp. 143-144). Knopf Doubleday Publishing Group. Kindle Edition.

    I don’t think this makes much sense. It borders celestial spheres moving at speed greater than light.

    Weinberg: The equations of general relativity, unlike those of Newtonian mechanics, are precisely the same in the merry-go-round laboratory and conventional laboratories;

    Weinberg, Steven. Dreams of a Final Theory: The Scientist’s Search for the Ultimate Laws of Nature (p. 144). Knopf Doubleday Publishing Group. Kindle Edition.

    This part I don’t doubt the laws of physics found on a rotating platform would be the same. Indeed, Newton’s laws are not different on rotating platform and while scientist on the platform could using GR could formulate those laws in a rotating coordinate system such that the same equations apply, they might still find it simpler to formulate them in an inertial frame in which they happen to be rotating.

    Newton laws can’t be formulated the same, but still using them in the an inertial frame those in a rotating frame can still calculate the how a ball will move in their frame.

    Come to think I we are on a rotating platform. We don’t worry about Alpha Centauri moving 27 light-years in one day.

    Weinberg: …and the centrifugal force felt on a merry-go-round would allow us to distinguish between the merry-go-round and more conventional laboratories

    Weinberg, Steven. Dreams of a Final Theory: The Scientist’s Search for the Ultimate Laws of Nature (p. 144). Knopf Doubleday Publishing Group. Kindle Edition.

    But, of course, we can distinguish between being on a merry-go-round based lab and a ‘conventional’ one. Just try playing baseball or catch [a]. The laws of physics are the same, as formulated in GR they even look the same; the situation is not.

    Things are complicated to solve in a rotating a coordinate system.

    There are apparently claims in GR that rotating the universe and rotating the merry-go-round are equivalent:

    If you spin a bucket of water, the surface of the water deforms because as Mach said it is rotating with respect to the frame of the distant fixed stars, then by relativity, we should be able to keep the bucked fixed and rotate the universe with the same angular velocity, and the water should still deform even though the bucket is not ‘actually’ rotating. Evidently, in the 1960’s, theorists were able to give a partial answer to whether these two experiments gave the same outcome, and showed that the two ‘experiments’ in general relativity would give equal outcomes.

    I guess Alfa Centauri is locally moving less than c and the space round it is moving. Something like that; something like the expansion of space.

    I think they must be only considering the only a static situation. If the merry-go-round accelerates its rate of rotation, the locals would feel the acceleration. After a while the merry-go-rounders could ask the star people, if they’d felt anything at the time of the acceleration. They couldn’t have.

    The situation is not symmetric. My guess is that these equivalence calculations don’t take the ‘when you spin the bucket part,’ but assume it’s been spinning forever. If you by some miracle of coordinated rocket fire, you accelerated the universe, the folks on the merry-go-round wouldn’t know about it for quite a while.

    The point that you can tell if you’re in a rotating frame or not is unperturbed whether you take rotating universe as equivalent or not. The difference in the force in a rotating environment allows you do tell whether you and/or the universe is rotating whether you call it centrifugal or call it gravity.

    Weinberg’s motivation for this discussion seems to be to show GR must be the way it is – well not claim anything quite that strong – but in that direction.

    GR is impressively constrained once you put in equivalence principle and invariance of the formulation in all reference frames (not just inertial). This discussion doesn’t help his case though.

    It is impressive that such a mathy kind of invariance can produce such a predictive theory. Same thing applies to ‘gauge principles’ in which a rather abstract principles leads to predictive theories.

    I don’t know if he’s aiming at a ‘Final Theory’ that has to be what it is. That doesn’t seem that plausible and with string theory we’re moving away from it.

    [a] There used to be a ride at the Santa Cruz beach boardwalk where you could try to play catch on a rotating platform. It was almost impossible, but you could learn to aim off center and at as constant as speed as possible, so your partner and the ball would arrive at the same point at the same time. I never mastered it.


  8. ” but — similar to mathematics — by becoming more and more explanatory in response to largely internally generated problems. This is, again, not teleonomic progress toward an end goal, or overarching theory of everything, but expansion by multiplicative exploration of novel areas of conceptual space — areas that, to use Smolin’s terminology introduced at the beginning of the book, are progressively “evoked” once logicians adopt new axioms and build new systems.”

    “expansion by multiplicative exploration of novel areas of conceptual space”

    “explanatory in response to largely internally generated problems.”

    It would seem to be a process of expansion and consolidation. The result then becomes the basis of further expansion.

    The result is the heritage, while the process is the history.

    Then there is feedback. Having to go back and fill in the gaps.

    “Zeno of Elea and Socrates used essentially what today are known as reductio arguments, though they took them for granted, without exploring their logical structure, while Eubulides originated a number of well known paradoxes, starting with the Liar and Sorites ones. Both the sophists and Plato were interested in logic and in fallacies of reasoning,”

    “A subset of sayables is constituted by the so-called assertibles, which are characterized by truth values. The assertibles are the smallest expressions in a deductive system, and including them in one’s logic gives origin to a system of propositional logic in which arguments are composed of assertibles.”

    “But it was John Buridan who showed that syllogistic theory is in fact a special case of a more general theory of consequences.”

    It does seem analogous to an biological process, where causal effects/consequences invariably create further niches, which then have to be filled in. So it is not so much a linear “progression,” but increasing complexity to both fill and expand in and on the original space. With every effort presumably going to seek out conceptual efficiencies, which in turn creates more spaces/niches to fill.

    The search for answers becomes the source of questions.

    Turning noise into complexity.

    If it seems like its going in circles……


  9. Rotating reference “mollusc”, not frame, to use Einstein’s cute terminology, but I think that is the key, as I am slowly starting to understand, stars are not whilrling round us at many times the speed of light. But the photons and so on from them that have almost reached us are. Or something like that.


  10. So with a mollusc, as opposed to a frame we are not rotating the universe with respect to the bucket, we are rotating a gravitational field with respect to the bucket. Although I could have that completely wrong.


  11. Arthur,

    Though I should leave the topic alone, if a clock slows in a moving frame, as space and time dilate, wouldn’t that imply there is an inertial frame to the vacuum and the frame with the fastest clock/least dilated time and space, would be closest to it?

    Not that it could be exactly measured, given there might always be a faster clock/frame, the incremental increase would imply it is quite close. Like getting close to absolute zero, with temperature.


  12. Such that it is the basis of the speed of light and other frames are moving relative to it, not so much each other?


  13. Massimo,

    Since the topics are serial, can we respond to the prior thread? If not delete;


    “Wrong. Causal and epistemic priority are entirely separate and can come apart. This is a pretty basic idea in philosophy.”

    I was simply trying to say there is no objective, ontological, or physical past. Yet events do have to occur, in order for there to be any knowledge or residue of them.
    No, the sequence of events may not be causal, but I didn’t say that.


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