New APPS: Art, Politics, Philosophy, Science

Interpersonal justification, argumentation, and epistemic autonomy

By Catarina Dutilh Novaes

I am currently supervising a MA thesis on interpersonal justification (by Sebastiano Lommi), and this is providing me with the opportunity to connect the dots between a number of topics and questions I’ve been interested in for years. In particular questions pertaining the epistemic value of deliberation, metaphors for argumentation, and the Enlightenment ideal of epistemic autonomy are all coming together. In this post I argue that the process whereby knowledge is shared through argumentation and exchange of reasons preserves the autonomy of the knowing subject to a greater extent than through testimony alone. Ultimately, the goal is to hit the sweet spot between preserving the autonomy of the knower while avoiding an overly individualistic picture of knowledge, i.e. one where the social dimension of knowledge is not sufficiently recognized.

The work of developmental psychologist Paul L. Harris (e.g. his book Trusting what you’re told) has been an important influence for my thinking on these matters. It is thanks to him that I got to see these issues through the lenses of Enlightenment ideals -- the exhortation to think for yourself -- which were a reaction to the then-prevailing model of (excessive) deference towards authority and testimony. Harris argues that the emphasis on the autonomy of the knowing subject thus conceived (as found in e.g. Kant, Rousseau, and centuries later in Piaget) swung the pendulum too far in the opposite direction, leading to a mistaken conception of knowledge and learning as essentially individual processes, disregarding how much we in fact learn from others.

In recent decades, the importance of taking into account the social aspects of knowledge became increasingly acknowledged in epistemology, leading to the emergence of the subfield of social epistemology. Arguably, the main focus of social epistemology until now has been on testimony, though there has also been some work on interpersonal justification, understood as "argument addressed to those who disagree with us, or to ourselves when we are of two minds" (Ralws) (see here for Goldman’s classic ‘Argumentation and interpersonal justification’, where he argues (mistakenly, in my opinion) that personal justification remains the primitive notion). While these may not be the two only processes whereby a person shares knowledge with others, for present purposes I take these to be paradigmatic cases.

Metaphors for argumentation

By Catarina Dutilh Novaes

It is well known that philosophers like to argue, and one of the things they like to argue about is arguing itself. Argumentation is frequently (and rightly, to my mind) taken to be a core feature of philosophical practice, and thus how to argue becomes a central topic for philosophical methodology. But many have claimed that the centrality of argumentation within philosophy is a weakness rather than a strength, deploring the excessively adversarial nature of argumentation in philosophy. Critics point out that philosophers are trained to find objections, counterexamples, rebuttals etc. to what their philosophical interlocutors say, who are tellingly described as one’s opponents. On this conception, argumentation is a duel between two opponents, and only one of them can win; blood will often ensue. Much of the criticism has been motivated by feminist concerns: aggressive, adversarial styles of argumentation are oppressive towards women and other disadvantaged groups, emphasizing competition (which is often presented to be an essentially ‘male’ feature) at the expense of cooperative, presumably more productive endeavors. Some of the authors having defended ideas along these lines are Janice Moulton and Andrea Nye (see here for a survey article by C. Hundleby).

A few years ago I became interested in how the presumed adversarial nature of philosophical argumentation affected not only the practice but also the outcome of philosophical investigation. It seemed to me that, while some of the feminist criticism definitely struck a cord if not with the theory at least with the practice of philosophy in some (well, many) quarters, the general critical stance that is characteristic of philosophical interactions was still an essential and epistemically valuable feature of the philosophical method. (Btw, it may be worth noting that this is not unique to philosophy; mathematics seems to proceed by ‘proofs and refutations’ (Lakatos), and in many if not all of the empirical and social sciences, objections and criticism are the bread-and-butter of the theorist.)

Formal Methods in Philosophy: a Brief Introduction (Part II)

By Catarina Dutilh Novaes

(Cross-posted at M-Phi) 

This is the second and final part of my 'brief introduction' to formal methods in philosophy to appear in the forthcoming Bloomsbury Philosophical Methodology Reader, being edited by Joachim Horvath. (Part I is here.) In this part I present in more detail the four papers included in the formal methods section, namely Tarski's 'On the concept of following logically', excerpts from Carnap's Logical Foundations of Probability, Hansson's 2000 'Formalization in philosophy', and a commissioned new piece by Michael Titelbaum focusing in particular (though not exclusively) on Bayesian epistemology. 

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Some of the pioneers in formal/mathematical approaches to philosophical questions had a number of interesting things to say on the issue of what counts as an adequate formalization, in particular Tarski and Carnap – hence the inclusion of pieces by each of them in the present volume. Indeed, both in his paper on truth and in his paper on logical consequence (in the 1930s), Tarski started out with an informal notion and then sought to develop an appropriate formal account of it. In the case of truth, the starting point was the correspondence conception of truth, which he claimed dated back to Aristotle. In the case of logical consequence, he was somewhat less precise and referred to the ‘common’ or ‘everyday’ notion of logical consequence. 

These two conceptual starting points allowed Tarski to formulate what he described as ‘conditions of material adequacy’ for the formal accounts. He also formulated criteria of formal correctness, which pertain to the internal exactness of the formal theory. In the case of truth, the basic condition of material adequacy was the famous T-schema; in the case of logical consequence, the properties of necessary truth-preservation and of validity-preserving schematic substitution. Unsurprisingly, the formal theories he then went on to develop both passed the test of material adequacy he had formulated himself. But there is nothing particularly ad hoc about this, since the conceptual core of the notions he was after was presumably captured in these conditions, which thus could serve as conceptual ‘guides’ for the formulation of the formal theories. 

Formal Methods in Philosophy: a Brief Introduction (Part I)

By Catarina Dutilh Novaes

(Cross-posted in M-Phi)

There is a Bloomsbury Philosophical Methodology Reader in the making, being edited by Joachim Horvath (Cologne). Joachim asked me to edit the section on formal methods, which will contain four papers: Tarski's 'On the concept of following logically', excerpts from Carnap's Logical Foundations of Probability, Hansson's 2000 'Formalization in philosophy', and a commissioned new piece by Michael Titelbaum focusing in particular (though not exclusively) on Bayesian epistemology. It will also contain a brief introduction to the topic by me, which I will post in two installments. Here is part I: comments welcome!

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Since the inception of (Western) philosophy in ancient Greece, methods of regimentation and formalization, broadly understood, have been important items in the philosopher’s toolkit (Hodges 2009). The development of syllogistic logic by Aristotle and its extensive use in centuries of philosophical tradition as a formal tool for the analysis of arguments may be viewed as the first systematic application of formal methods to philosophical questions. In medieval times, philosophers and logicians relied extensively on logical tools other than syllogistic (which remained pervasive though) in their philosophical analyses (e.g. medieval theories of supposition, which come quite close to what is now known as formal semantics). But the level of sophistication and pervasiveness of formal tools in philosophy has increased significantly since the second half of the 19^th century. (Frege is probably the first name that comes to mind in this context.)

It is commonly held that reliance on formal methods is one of the hallmarks of analytic philosophy, in contrast with other philosophical traditions. Indeed, the birth of analytic philosophy at the turn of the 20^th century was marked in particular by Russell’s methodological decision to treat philosophical questions with the then-novel formal, logical tools developed for axiomatizations of mathematics (by Frege, Peano, Dedekind etc. – see (Awodey & Reck 2002) for an overview of these developments), for example in his influential ‘On denoting’ (1905). (Notice though that, from the start, there is an equally influential strand within analytic philosophy focusing on common sense and conceptual analysis, represented by Moore – see (Dutilh Novaes & Geerdink forthcoming).) This tradition was then continued by, among others, the philosophers of the Vienna Circle, who conceived of philosophical inquiry as closely related to the natural and exact sciences in terms of methods. Tarski, Carnap, Quine, Barcan Marcus, Kripke, and Putnam are some of those who have applied formal techniques to philosophical questions. Recently, there has been renewed interest in the use of formal, mathematical tools to treat philosophical questions, in particular with the use of probabilistic, Bayesian methods (e.g. formal epistemology). (See (Papineau 2012) for an overview of the main formal frameworks used for philosophical inquiry.)

Reductio arguments from a dialogical perspective: final considerations

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

 This is the final post in my series on reductio ad absurdum from a dialogical perspective. Here is Part I, here is Part II, here is Part III, here is Part IV, and here is Part V. I now return to the issues raised in the earlier posts equipped with the dialogical account of deduction, and of reductio ad absurdum in particular.

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 A general dialogical schema for reductio ad absurdum, following Proclus’ description but inspired by the Socratic elenchus, might look like this:

1. Interlocutor 1 commits to A (either prompted by a question from interlocutor 2, or spontaneously), which corresponds to assuming the initial hypothesis.
2. Interlocutor 2 leads the initial hypothesis to absurdity, typically by relying on additional discursive commitments of 1 (which may be elicited by 2 through questions).
3. Interlocutor 2 concludes ~A.

The main difference between the monological and the dialogical versions of a reductio is thus that in the latter there is a kind of division of labor that is absent from the former (as noted above). The agent making the initial assumption is not the same agent who will lead it to absurdity, and then conclude its contradictory. And so, the perceived pragmatic awkwardness of making an assumption precisely with the goal of ‘destroying’ it seems to vanish. Moreover, the adversarial component provides a compelling rationale for the general idea of ‘destroying’ the initial hypothesis; indeed, while the adversarial component is present in all deductive arguments (in particular given the requirement of necessary truth preservation, as argued above), it is even more pronounced in the case of reductio arguments, that is the procedure whereby someone’s discursive commitments are shown to be collectively incoherent since they lead to absurdity. There remains the question of why interlocutor 1 would want to engage in the dialogue at all, but presumably she simply wishes to voice a discursive commitment to A. From there on, the wheel begins to spin, mostly through 2’s actions.

Dialectical refutations and reductio ad absurdum

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

This is the fifth installment of my series of posts on reductio ad absurdum from a dialogical perspective. Here is Part I, here is Part II, here is Part III, and here is Part IV. In this post I discuss a closely related argumentative strategy, namely dialectical refutation, and argue that it can be viewed as a genealogical ancestor of reductio ad absurdum.

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Those familiar with Plato’s Socratic dialogues will undoubtedly recall the numerous instances in which Socrates, by means of questions, elicits a number of discursive commitments from his interlocutors, only to go on to show that, taken collectively, these commitments are incoherent. This is the procedure known as an elenchus, or dialectical refutation. 

The ultimate purpose of such a refutation may range from ridiculing the opponent to nobler didactic goals. The etymology of elenchus is related to shame, and indeed at least in some cases it seems that Socrates is out to shame the interlocutor by exposing the incoherence of their beliefs taken collectively (for example, so as to exhort them to positive action, as argued in (Brickhouse & Smith 1991)). However, as noted by Socrates himself in the Gorgias (470c7-10), refuting is also what friends do to each other, a process whereby someone rids a friend of nonsense. An elenchus can also have pedagogical purposes, in interactions between masters and pupils.

There has been much discussion in the secondary literature on what exactly an elenchus is, as well as on whether there is a sufficiently coherent core of properties for what counts as an elenchus, beyond a motley of vaguely related argumentative strategies deployed by Socrates (Carpenter & Polansky 2002). (A useful recent overview is (Wolfsdorf 2013); see also (Scott 2002).) For our purposes, it will be useful to take as our starting point the description of the ‘Socratic method’ in an influential article by G. Vlastos (1983) (a much shorter version of the same argument is to be found in (Vlastos 1982), and I'll be referring to the shorter version). Vlastos distinguishes two kinds of elenchi, the indirect elenchus and the standard elenchus:

A precis of the dialogical account of deduction

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

This is the fourth installment of my series of posts on reductio ad absurdum arguments from a dialogical perspective. Here is Part I, here is Part II, and here is Part III. In this post I offer a précis of the dialogical account of deduction which I have been developing over the last years, which will then allow me to return to the issue of reductio arguments equipped with a new perspective in the next installments. I have presented the basics of this conception in previous posts, but some details of the account have changed, and so it seems like a good idea to spell it out again.

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In this post, I present a brief account of the general dialogical conception of deduction that I endorse. Its relevance for the present purposes is to show that a dialogical conception of reductio ad absurdum arguments is not in any way ad-hoc; indeed, the claim is that this conception applies to deductive arguments in general, and thus a fortiori to reductio arguments. (But I will argue later on that the dialogical component is even more pronounced in reductio arguments than in other deductive arguments.)

Let us start with what can be described as functionalist questions pertaining to deductive arguments and deductive proofs. What is the point of deductive proofs? What are they good for? Why do mathematicians bother producing mathematical proofs at all? While these questions are typically ignored by mathematicians, they have been raised and addressed by so-called ‘maverick’ philosophers of mathematics, such as Hersh (1993) and Rav (1999). One promising vantage point to address these questions is the historical development of deductive proof in ancient Greek mathematics,[1] and on this topic the most authoritative study remains (Netz 1999). Netz emphasizes the importance of orality and dialogue for the emergence of classical, ‘Euclidean’ mathematics in ancient Greece:

Greek mathematics reflects the importance of persuasion. It reflects the role of orality, in the use of formulae, in the structure of proofs… But this orality is regimented into a written form, where vocabulary is limited, presentations follow a relatively rigid pattern… It is at once oral and written… (Netz 1999, 297/8)

Problems with reductio proofs: "jumping to conclusions"

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

This is the third installment of my series of posts on reductio ad absurdum arguments from a dialogical perspective. Here is Part I, and here is Part II. In this post I discuss issues pertaining specifically to the last step in a reductio argument, namely that of going from reaching absurdity to concluding the contradictory of the initial hypothesis.

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One worry we may have concerning reductio arguments is what could be described as ‘the culprit problem’. This is not a worry clearly formulated in the protocols previously described, but one which has been raised a number of times when I presented this material to different audiences. The basic problem is: we start with the initial assumption, which we intend to prove to be false, but along the way we avail ourselves to auxiliary hypotheses/premises. Now, it is the conjunction of all these premises and hypotheses that lead to absurdity, and it is not immediately clear whether we can single out one of them as the culprit to be rejected. For all we know, others may be to blame, and so there seems to be some arbitrariness involved in singling out one specific ingredient as responsible for things turning sour.

To be sure, in most practical cases this will not be a real concern; typically, the auxiliary premises we avail ourselves to are statements on which we have a high degree of epistemic confidence (for example, because they have been established by proofs that we recognize as correct). But it remains of philosophical significance that absurdity typically arises from the interaction between numerous elements, any of which can, in theory at least, be held to be responsible for the absurdity. A reductio argument, however, relies on the somewhat contentious assumption that we can isolate the culprit.

However, culprit considerations do not seem to be what motivates Fabio’s dramatic description of this last step as “an act of faith that I must do, a sacrifice I make”. Why is this step problematic then? Well, in first instance, what is established by leading the initial hypothesis to absurdity is that it is a bad idea to maintain this hypothesis (assuming that it can be reliably singled out as the culprit, e.g. if the auxiliary premises are beyond doubt). How does one go from it being a bad idea to maintain the hypothesis to it being a good idea to maintain its contradictory?

Problems with reductio proofs: assuming the impossible

By Catarina Dutilh Novaes

 (Cross-posted at M-Phi)

This is a series of posts with sections of the paper on reductio ad absurdum from a dialogical perspective that I am working on right now. This is Part II, here is Part I. In this post I discuss issues in connection with the first step in a reductio argument, that of assuming the impossible.

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We can think of a reductio ad absurdum as having three main components, following Proclus’ description:

(i) Assuming the initial hypothesis.

(ii) Leading the hypothesis to absurdity.

(iii) Concluding the contradictory of the initial hypothesis.

I discuss two problems pertaining to (i) in this post, and two problems pertaining to (iii) in the next post. (ii) is not itself unproblematic, and we have seen for example that Maria worries whether the ‘usual’ rules for reasoning still apply once we’ve entered the impossible world established by (i). Moreover, the problematic status of (i) arises to a great extent from its perceived pragmatic conflict with (ii). But the focus will be on issues arising in connection with (i) and (iii).

A reductio proof starts with the assumption of precisely that which we want to prove is impossible (or false). As we’ve seen, this seems to create a feeling of cognitive dissonance in (some) reasoners: “I do not know what is true and what I pretend [to be] true.” (Maria) This may seem surprising at first sight: don’t we all regularly reason on the basis of false propositions, such as in counterfactual reasoning? (“If I had eaten a proper meal earlier today, I wouldn’t be so damn hungry now!”) However, as a matter of fact, there is considerable empirical evidence suggesting that dissociating one’s beliefs from reasoning is a very complex task, cognitively speaking (to ‘pretend that something is true’, in Maria’s terms). The belief bias literature, for example, has amply demonstrated the effect of belief on reasoning, even when participants are told to focus only on the connections between premises and conclusions. Moreover, empirical studies of reasoning behavior among adults with low to no schooling show their reluctance to reason with premises of which they have no knowledge (Harris 2000; Dutilh Novaes 2013). From this perspective, reasoning on the basis of hypotheses or suppositions may well be something that requires some sort of training (e.g. schooling) to be mastered.

Problems with reductio proofs: cognitive aspects

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

As some readers may recall, I ran a couple of posts on reductio proofs from a dialogical perspective quite some time ago (here and here). I am now *finally* writing the paper where I systematize the account. In the coming days I'll be posting sections of the paper; as always, feedback is most welcome! The first part will focus on what seem to be the cognitive challenges that reasoners face when formulating reductio arguments.

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For philosophers and mathematicians having been suitably ‘indoctrinated’ in the relevant methodologies, the issues pertaining to reductio ad absurdum arguments may not become immediately apparent, given their familiarity with the technique. And so, to get a sense of what is problematic about these arguments, let us start with a somewhat dramatic but in fact quite accurate account of what we could describe as the ‘phenomenology’ of producing a reductio argument, in the words of math education researcher U. Leron:

We begin the proof with a declaration that we are about to enter a false, impossible world, and all our subsequent efforts are directed towards ‘destroying’ this world, proving it is indeed false and impossible. (Leron 1985, 323)

In other words, we are first required to postulate this impossible world (which we know to be impossible, given that our very goal is to refute the initial hypothesis), and then required to show that this impossible world is indeed impossible. The first step already raises a number of issues (to be discussed shortly), but the tension between the two main steps (postulating a world, as it were, and then proceeding towards destroying it) is perhaps even more striking. As it so happens, these are not the only two issues that arise once one starts digging deeper. 

To obtain a better grasp of the puzzling nature of reductio arguments, let us start with a discussion of why these arguments appear to be cognitively demanding – that is, if we are to believe findings in the math education literature as well as anecdotal evidence (e.g. of those with experience teaching the technique to students). This will offer a suitable framework to formulate further issues later on.

Aristotle's definition of the syllogism -- a dialogical interpretation

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

(I am currently finishing a paper on the definition of the syllogism according to Aristotle, Ockham, and Buridan. I post below the section where I present a dialogical interpretation of Aristotle's definition.)

Aristotle’s definition of ‘syllogismos’ in Prior Analytics (APri)  24b18-22 is among one of the most commented-upon passages of the Aristotelian corpus, by ancient as well as (Arabic and Latin) medieval commentators. He offers very similar definitions of syllogismos in the Topics, Sophistical Refutations, and the Rhetoric, but the one in APri is the one having received most attention from commentators. In the recent Striker (2009) translation, it goes like this (emphasis added):

A ‘syllogismos’ is an argument (logos) in which, (i) certain things being posited (tethentôn), (ii) something other than what was laid down (keimenôn) (iii) results by necessity (eks anagkês sumbainei)(iv) because these things are so. By ‘because these things are so’ I mean that it results through these, and by ‘resulting through these’ I mean that no term is required from outside for the necessity to come about.

It became customary among commentators to take ‘syllogismos’ as belonging to the genus ‘logos’ (discourse, argument), and as characterized by four (sometimes five) differentiae:

Review of Williamson's Tetralogue

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

I've been asked to write a review of Williamson's brand new book Tetralogue for the Times Higher Education. Here is what I've come up with so far. Comments are very welcome, as I still have some time before submitting the final version. (For more background on the book, here is a short video where Williamson explains the project.)

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Disagreement in debates and discussions is an interesting phenomenon. On the one hand, having to justify your views and opinions vis-à-vis those who disagree with you is perhaps one of the best ways to induce a critical reevaluation of these views. On the other hand, it is far from clear that a clash of views will eventually lead to a consensus where the parties come to hold better views than the ones they held before. This is one of the promises of rational discourse, but one that is all too often not kept. What to do in situations of discursive deadlock?

Timothy Williamson’s Tetralogue is precisely an investigation on the merits and limits of rational debate. Four people holding very different views sit across each other in a train and discuss a wide range of topics, such as the existence of witchcraft, the superiority and falibilism of scientific reasoning, whether anyone can ever be sure to really know anything, what it means for a statement to be true, and many others. As one of the most influential philosophers currently in activity, Williamson is well placed to give the reader an overview of some of the main debates in recent philosophy, as his characters debate their views.

In memoriam: Ivor Grattan-Guinness

The great historian of logic and mathematics Ivor Grattan-Guinness passed away about a month ago, aged 73. I only heard it yesterday, when Stephen Read posted a link to the Guardian obituary on Facebook. From the obituary: 

He rescued the moribund journal Annals of Science, founded the journal History and Philosophy of Logic, and was on the board of Historia Mathematica from its inception. A member of the council of the Society for Psychical Research, he wrote Psychical Research: A Guide to Its History (1982). In 1971 the British Society for the History of Mathematics was founded: Ivor served as its president (1986-88) and instituted a formal constitution.

Indeed, many of us owe him eternal gratitude for founding the journal History and Philosophy of Logic, which continues to be the main journal for studies combining historical and philosophical perspectives on logic. Ivor's work and scholarship spans over an impressive range of topics and areas, and is bound to continue to influence many generations of scholars to come. It is a great loss.

The upside down world paradox

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

As most kids (I suspect), my daughters sometimes play ‘upside down world’, especially when I ask them something to which they should say ‘yes’, but instead they say ‘no’ and immediately regret it: ‘Upside down world!’ The upside down world game basically functions as a truth-value flipping operator: if you say yes, you mean no, and if you say no, you mean yes.

My younger daughter recently came across the upside down world paradox: if someone asks you ‘are you playing upside down world?’, all kinds of weird things happen to each of the answers you may give. If you are not playing upside down world, you will say no; but if you are playing upside down world you will also say no. So the ‘no’ answer underdetermines its truth-value, a bit like the no-no paradox. Now for the ‘yes’ answer: if you are playing upside down world and say ‘yes’, then that means ‘no’, and so you are not playing the game after all if you are speaking truthfully. But then your ‘yes’ was a genuine yes in the first place, and so you are playing the game and said yes, which takes us back to the beginning. (In other words, 'no' is the only coherent answer, but it still doesn't say anything about whether you are actually playing the game or not.)

 

Review of T. Parsons' Articulating Medieval Logic

By Catarina Dutilh Novaes

(Cross-posted at M-Phi)

I was asked to write a review of Terry Parsons' Articulating Medieval Logic for the Australasian Journal of Philosophy. This is what I've come up with so far. Comments welcome!

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 Scholars working on (Latin) medieval logic can be viewed as populating a spectrum. At one extremity are those who adopt a purely historical and textual approach to the material: they are the ones who produce the invaluable modern editions of important texts, without which the field would to a great extent simply not exist; they also typically seek to place the doctrines presented in the texts in a broader historical context. At the other extremity are those who study the medieval theories first and foremost from the point of view of modern philosophical and logical concerns; various techniques of formalization are then employed to ‘translate’ the medieval theories into something more intelligible to the modern non-historian philosopher. Between the two extremes one encounters a variety of positions. (Notice that one and the same scholar can at times wear the historian’s hat, and at other times the systematic philosopher’s hat.) For those adopting one of the many intermediary positions, life can be hard at times: when trying to combine the two paradigms, these scholars sometimes end up displeasing everyone (speaking from personal experience).

 

Terence Parsons’ Articulating Medieval Logic occupies one of these intermediate positions, but very close to the second extremity; indeed, it represents the daring attempt to combine the author’s expertise in natural language semantics, linguistics, and modern philosophy with his interest in medieval logical theories (which arose in particular from his decade-long collaboration with Calvin Normore, to whom the book is dedicated). For scholars of Latin medieval logic, the fact that such a distinguished expert in contemporary philosophy and linguistics became interested in these medieval theories only confirms what we’ve known all along: medieval logical theories have intrinsic systematic interest; they are not only curious museum pieces.

 

Despite not being the first to employ modern logical techniques to analyze medieval theories, Parsons' approach is quite unique (one might even say idiosyncratic). It seems fair to say that nobody has ever before attempted to achieve what he wants to achieve with this book. A passage from the book’s Introduction is quite revealing with respect to its goals:

 

What’s the big deal with consistency?

(Cross-posted at M-Phi)

It is no news to anyone that the concept of consistency is a hotly debated topic in philosophy of logic and epistemology (as well as elsewhere). Indeed, a number of philosophers throughout history have defended the view that consistency, in particular in the form of the principle of non-contradiction (PNC), is the most fundamental principle governing human rationality – so much so that rational debate about PNC itself wouldn’t even be possible, as famously stated by David Lewis. It is also the presumed privileged status of consistency that seems to motivate the philosophical obsession with paradoxes across time; to be caught entertaining inconsistent beliefs/concepts is really bad, so blocking the emergence of paradoxes is top-priority. Moreover, in classical as well as other logical systems, inconsistency entails triviality, and that of course amounts to complete disaster.

 

Since the advent of dialetheism, and in particular under the powerful assaults of karateka Graham Priest, PNC has been under pressure. Priest is right to point out that there are very few arguments in favor of the principle of non-contradiction in the history of philosophy, and many of them are in fact rather unconvincing. According to him, this holds in particular of Aristotle’s elenctic argument in Metaphysics gamma. (I agree with him that the argument there does not go through, but we disagree on its exact structure. At any rate, it is worth noticing that, unlike David Lewis, Aristotle did think it was possible to debate with the opponent of PNC about PNC itself.) But despite the best efforts of dialetheists, the principle of non-contradiction and consistency are still widely viewed as cornerstones of the very concept of rationality.

 

However, in the spirit of my genealogical approach to philosophical issues, I believe that an important question to be asked is: What’s the big deal with consistency in the first place? What does it do for us? Why do we want consistency so badly to start with? When and why did we start thinking that consistency was a good norm to be had for rational discourse? And this of course takes me back to the Greeks, and in particular the Greeks before Aristotle.

 

Did Russell's Paradox entail that set theory is incomplete (if axiomatized)?

Tonight I was fondly recalling Michael Hand and Jonathan Kvanvig's old paper on Tennant's solution to Fitch's Paradox (it's a beautiful read) and a weird thought occured to me.

Hand and Kvanvig argue that Tennant's solution would be analogous to a set theorist responding to Russell's Paradox by proposing naive set theory with Frege's comprehension axiom restricted to instances that don't allow one to prove absurdity from that instance and the other axioms (call this theory N'). For Hand and Kvanvig this is a reductio of Tennant, and in my response* I argued that Tennant's solution was not actually analogous to N'.

If I remember right, Hand and Kvanvig argue that N' is bad because it doesn't illuminate the nature of sets in the way we properly expect of solutions to paradoxes. But they don't go into the logical properties of N' at all, and tonight I'm thinking that this is actually an important question in its own right. Let's just consider consistency, completeness, and axiomatizability.

Preferential logics, supraclassicality, and human reasoning

(Cross-posted at M-Phi)

 

Some time ago, I wrote a blog post defending the idea that a particular family of non-monotonic logics, called preferential logics, offered the resources to explain a number of empirical findings about human reasoning, as experimentally established. (To be clear: I am here adopting a purely descriptive perspective and leaving thorny normative questions aside. Naturally, formal models of rationality also typically include normative claims about human cognition.)  

 

In particular, I claimed that preferential logics could explain what is known as the modus ponens-modus tollens asymmetry, i.e. the fact that in experiments, participants will readily reason following the modus ponens principle, but tend to ‘fail’ quite miserably with modus tollens reasoning – even though these are equivalent according to classical as well as many non-classical logics. I also defended (e.g. at a number of talks, including one at the Munich Center for Mathematical Philosophy which is immortalized in video here and here) that preferential logics could be applied to another well-known, robust psychological phenomenon, namely what is known as belief bias. Belief bias is the tendency that human reasoners seem to have to let the believability of a conclusion guide both their evaluation and production of arguments, rather than the validity of the argument as such.

 

Well, I am now officially taking most of it back (and mostly thanks to working on these issues with my student Herman Veluwenkamp).

Williamson on David Lewis

This summer I'm trying to get a little bit up to speed on modality issues by doing an independent study with some students.* I've started looking ahead to Williamson's recent magnum opus and this little bit of the preface weirded me out:

Since cosmological theories in physics are naturally understood as embodying no restriction of their purview to exclude Lewis's multiple spatiotemporal systems, many of which are supposed to violate their laws, his cosmology is inconsistent with physicists', and so in competition with them as a theory of total spatiotemporal reality. On such matters, physicists may be felt to speak with more authority than metaphysicians. The effect of Lewis's influential and ingenious system-building was to keep centre stage a view that imposed Quine's puritan standards on modality long after Quine's own eliminativist application of those standards have been marginalized (Williamson 2013, xii)

I don't get this at all.

The connection between Lewisian Genuine Realism and Quine's eliminativism is a promissory note that I assume he'll cash in later, but the first bit just makes no sense to me. In On the Plurality of Worlds, Lewis explicitly says that the nomologically possible worlds will be a subset of all possible worlds and he discusses physically impossible forms of space time in this context. He has to do this, since possible worlds are individuated by the space-time which each world shares with itself. But nowhere does he make claims about which class of worlds will be the nomologically possible ones.

"Mathematicians are supposed to prove things *in* arithmetic, not *about* arithmetic!"

Hitler does not like Gödel's theorem one bit. Perhaps surprisingly, he displays a sophisticated understanding of the implications and presuppositions of the theorem. (In other words, there's some very solid philosophy of logic in the background -- I think I could teach a whole course only on the material presupposed here.)

(Courtesy of Diego Tajer, talented young logician from Buenos Aires, giving continuation to the best Monty Python tradition!)

 

Is there a recieved view about Evans' argument against ontic vagueness?

My co-writer* Joshua Heller is currently working on a project connections between vagueness literature, literature on semantic underdetermination, and new work on metaphysical indeterminacy.**

One thing we're both interested in exploring the next few weeks is the extent to which Evans' argument against ontic vagueness applies to either semantic underdetermination or metaphysical indeterminacy. But I'm about ten years out of date on the vagueness literature. The last time I dipped my toe in this, it seemed like everyone was trying to save supervaulationism from Williamson's criticisms about wide and narrow entailment and from the charge that it has no advantages over three valued systems with respect to modelling higher order vagueness. I didn't think there was any consensus on Evans' argument
then.

Is there now anything approaching a consensus among people working on vagueness about Evans' argument? If so, what should I read? Have any of the new people working on metaphysical indeterminacy or semantic underdetermination said anything interesting about Evans' argument?

Is the Barwise/Etchemendy system unsound?

I might be being an idiot here, but I can't for the life of me figure out how the following inference is disallowed in the Barker-Plummer/Barwise/Etchemendy textbook:

1. Ex(Rax)

2. ExEx (Rxx)   1, E introduction

This is clearly invalid, because there are anti-symmetric relations (note that if it were valid one can prove ExEx (Rxx) from ExEy (Rxy), and that ExEx(Rxx) has the same truth conditions as Ex(Rxx)).

Intuitively, Existential Introduction should be restricted so that one cannot replace a name or eigenvariable with a variable that is already bound in the sentence. But I can't find this restriction in Barker-Plummer/Barwise/Etchemendy. My friend couldn't find it in the old red Mates book either, so we think it's not unlikely that we're both missing something obvious.

I turned to the soundness proof in Barker-Plummer/Barwise/Etchemendy and they leave the case of Existential Introduction as an excercise to the reader. It has a little star next to it showing that it is a difficult problem. I'm wondering if it's impossible. But, again, it's more likely that I'm missing something, so if anyone who teaches from the book could take a look my introductory logic students would appreciate it.

reflections on beautiful souls, hypocrites, undecidability results, and the structure of normativity

In much of the philosophy of language and mind coming out of the late Wittgenstein and/or early Heidegger, a distinction is made between merely following a norm versus also being able to correctly assess whether others are following that norm. Note that the Brandom of "Dasein, the Being that Thematizes" (in Tales of the Mighty Dead) and the Mark Okrent of "On Layer Cakes" both mark this distinction, though they disagree on whether the latter ability requires language. Okrent (whose objects that Brandom's view entails that human aphaisics and non-linguistic deaf adults have no minds) writes:

Because all tool use is embedded in a context of instrumental rationality, there is more to using a hammer correctly than using it as others do. Sometimes it is possible to use a hammer better than the others do, even if no one else has ever done it in that way, and no one else recognizes that one is doing so, because the norm that defines this use as ‘better’ is independent of what is actually recognized within the community. That norm is the norm of instrumental rationality: it is good to do that which would achieve one’s ends most completely and most efficiently, were anyone to do it in that way. For the same reason, it is sometimes possible for a member of a society to improve a hammer, or repair it, by giving it a structure that no hammer has previously had in that society.

A plea for descriptive metaphilosophy as philosophy

I am currently supervising a student writing a paper on Wittgenstein’s notion of therapy as a metaphilosophical concept. The paper relies centrally on a very useful distinction discussed in N. Rescher’s 1985 book The Strife of Systems (though I do not know whether it was introduced there for the first time), namely the distinction between prescriptive vs. descriptive metaphilosophy (the topic of chap. 14 of the book).

The descriptive issue of how philosophy has been done is one of factual inquiry largely to be handled in terms of the history of the field. But the normative issue of how philosophy should be done – or significant questions, adequate solutions, and good arguments – is something very different. (Rescher 1985, 261) 

Rescher goes on to argue that descriptive metaphilosophy is not part of philosophy at all; it is a branch of factual inquiry, namely the history of philosophy and perhaps its sociology. Prescriptive metaphilosophy, by contrast, is real philosophy: methodological claims on how philosophy should be done are themselves philosophical claims. (Full disclosure: I haven’t read the whole chapter, only what google books allows me to see…) Rescher’s position as described here seems to be quite widespread, encapsulating the ‘disdain’ with which not only descriptive metaphilosophy, but also the history of philosophy in general, is often viewed by ‘real philosophers’. And yet, this position seems to me to be fundamentally wrong (and this is also the claim that my student is defending in his paper). 

(Notice that to discuss the status of descriptive metaphilosophy as philosophy, we need to go meta-metaphilosophical! It’s turtles all the way up, or down, depending on how you look at it.)

Does the inferentialist have to worry about Kaplan-Russell type paradoxes?

)

Genuine Realists about modality typically understand propositional content to be a function of the set of worlds where that proposition is true (the set of worlds might include impossible ones). Actualist Realists take the dependence to go in the other direction, taking a world to be a function of the set of propositions true at that world. Since this function is almost always identity,* let's treat it as such in what follows.

Kaplan established a cardinality paradox against Genuine Realism analogous to an earlier paradox about the set of all propositions put forward by Russell. Russell's paradox** is now taken analogically to present a problem for Actual Realists.

Here's how Kaplan's paradox goes. Assume the set of all possible worlds has the cardinality K. Then, by Cantor's Theorem, the powerset of the set of possible worlds has a greater cardinality. But if a proposition is a set of worlds, then the cardinality of the set of propositions is greater than the cardinality of the set of worlds. O.K. so far. But let's consider for each proposition a world where one being is thinking that proposition.**** But then the set of worlds has at least the cardinality as that of the set of propositions. Contradiction.

Axiomatizations of arithmetic and the first-order/second-order divide

I'm currently running a series of posts at M-Phi with sections of a paper I'm working on, 'Axiomatizations of arithmetic and the first-order/second-order divide', which may be of interest to at least some of the NewAPPS readership. It focuses on the idea that, when it comes to axiomatizing arithmetic, descriptive power and deductive power cannot be combined: axiomatizations that are categorical (using a highly expressive logical language, typically second-order logic) will typically be intractable, whereas axiomatizations with deductively better-behaved underlying logics (typically, first-order logic) will not be categorical -- i.e. will be true of models other than the intended model of the series of the natural numbers. Based on a distinction proposed by Hintikka between the descriptive use and the deductive use of logic in the foundations of mathematics, I discuss what the impossibility of having our arithmetical cake and eating it (i.e. of combining deductive power with expressive power to characterize arithmetic with logical tools) means for the first-order logic vs. second-order logic debate.

Part I is here, Part II here, and Part III here. I still hope to post Part IV tomorrow, and then the final Part V will have to wait for a while.

Critical Reasoning Materials with a Dialectical Focus

Over on Facebook, Bijan Parsia asked a really great question.  

[... are there] any critical reasoning courses/textbooks out there that focus at the dialectical (or beyond) level rather than at the argument level. My recollection is that they are very focused at the individual argument level with an unhealthy focus on fallacies rather than thinking very much about overall cognitive strategies (esp. in group settings) or other goals than the cognitive. I recall getting a lot of that from phil of science classes and pedagogy and (interestingly) online dissuasion analysis (see the "poisonous people" video floating about, or even troll bestiaries), but not so much from critical reasoning (which often was shoehorned into a symbolic logic class).

While I haven't taught critical reasoning in a few years, I also can't recall having run across anything like what Bijan is looking for here. But I don't think it's difficult to see why materials of the sort would be of great value. In fact, I can see how they would be very helpful not just in the 'critical reasoning' context, but more broadly as part of the kind of instruction might give in philosophical process in a lot of our classes. 

And with that, I throw the question out to the rest of you. Do you know of materials of this sort? Have you developed something of your own that you'd like to share?

(Possibly Boneheaded) Question about Axiom of Choice, Soundness/Completeness, and Pi 2 Formulas

I'm not a logician. Nor do I play one on T.V. So please be patient if I'm messing up something basic in what follows. An explanation of what I'm messing up and/or some relevant citations would be pretty helpful too.

Gregory Moore's masterful Zermelo's Axiom of Choice: Its Origins, Development, and Influence contains the following passage:

Vestiges of the first state - choosing an unspecified element from a single set - can be found in Euclid's Elements, if not earlier. Such choices formed the basis of the ancient method of proving a generalization by considering an arbitrary but definite object, and then executing the argument for that object. This first stage also included the arbitrary choice of an element form each of finitely many sets. It is important to understand that the Axiom was not needed for an arbitrary choice from a single set, even if the set contained infinitely many elements. For in a formal system a single arbitrary choice can be eliminated through the use of universal generalization or similar rule of inference. By induction on the natural numbers, such a procedure can be extended to any finite family of sets.

I don't get this at all.

Two conceptions of criteria of adequacy for formalization: Tarski and Carnap

(Cross-posted at M-Phi)

Formal/mathematical philosophy is a well-established approach within philosophical inquiry, having its friends as well as its foes. Now, even though I am very much a formal-approaches-enthusiast, I believe that fundamental methodological questions tend not to receive as much attention as they deserve within this tradition. In particular, a key question which is unfortunately not asked often enough is: what counts as a ‘good’ formalization? How do we know that a given proposed formalization is adequate, so that the insights provided by it are indeed insights about the target phenomenon in question? In recent years, the question of what counts as adequate formalization seems to be for the most part a ‘Swiss obsession’, with the thought-provoking work of Georg Brun, and Michael Baumgartner & Timm Lampert. But even these authors seem to me to restrict the question to a limited notion of formalization, as translation of pieces of natural language into some formalism. (I argued in chapter 3 of my book Formal Languages in Logic that this is not the best way to think about formalization.)

However, some of the pioneers in formal/mathematical approaches to philosophical questions did pay at least some attention to the issue of what counts as an adequate formalization. In this post, I want to discuss how Tarski and Carnap approached the issue, hoping to convince more ‘formal philosophers’ to go back to these questions. (I also find the ‘squeezing argument’ framework developed by Kreisel particularly illuminating, but will leave it out for now, for reasons of space.)

My Dad went to LSU, and all I got was this lousy pee-cup

A few days ago, while trying to open the interwebs thingy to allow me to start entering my grades, I was prevented from doing so by a pop-up menu that referenced LSU's Policy Statement 67. The text included unsubstantiated and highly dubious claims such as that most workplace problems are the result of drugs and alcohol abuse by workers. And this was only a few weeks after all of the chairs at LSU had to provide verification that every single faculty member had read a hysterical message from our staff and administrative overlords that justified expanding the extension of pee-tested employees at LSU to now include faculty. The wretched communiqué justified pee-testing faculty because of new evidence showing that marijuana is harmful to 13 year olds.*

Anyhow, when I scrolled to the bottom of the popup, I had to click a button saying not only that I read the document but also that I "agreed" with it. 

I honestly don't get this. Are my beliefs a condition of employment at LSU? There was no button that said I read it but didn't agree with it.