New APPS: Art, Politics, Philosophy, Science

Wittgenstein on the barest essentials

(Cross-posted at M-Phi)

( From the graphic novel Logicomix, taken from this blog post by Richard Zach.)

“He doesn’t want to prove this or that, but to find out how things really are.” This is how Russell describes Wittgenstein in a letter to Lady Ottoline Morrell (as reported in M. Potter’s wonderful book Wittgenstein's Notes on Logic, p. 50 – see my critical note on the book). This may well be the most accurate characterization of Wittgenstein’s approach to philosophy in general, in fact a fitting description of the different phases Wittgenstein went through. Indeed, if there is a common denominator to the first, second, intermediate etc. Wittgensteins, it is the fundamental nature of the questions he asked: different answers, but similar questions throughout. So instead of proving ‘this or that’, for example, he asks what a proof is in the first place.

Aristotle's notion of dialectic as a model for philosophical debate and inquiry

This week, we’ve had a new round of discussions on the ‘combative’ nature of philosophy as currently practiced and its implications, prompted by a remark in a column by Jonathan Wolff on the scarcity of women in the profession. (Recall the last wave of such discussions, then prompted by Rebecca Kukla’s 3AM interview.) Brian Leiter retorted that there’s nothing wrong with combativeness in philosophy (“Insofar as truth is at stake, combat seems the right posture!”). Chris Bertram in turn remarked that this is the case only if “there’s some good reason to believe that combat leads to truth more reliably than some alternative, more co-operative approach”, which he (apparently) does not think there is. Our own John Protevi pointed out the possible effects of individualized grading for the establishment of a competitive culture.

As I argued in a previous post on the topic some months ago, I am of the opinion that adversariality can have a productive, positive effect for philosophical inquiry, but not just any adversariality/combativeness. (In that post, I placed the discussion against the background of gender considerations; I will not do so here, even though there are obvious gender-related implications to be explored.) In fact, what I defend is a form of adversariality which combines adversariality/opposition with a form of cooperation.

On a Beautiful Review

This is a beautiful review. It is clear on technical issues; it is very critical, albeit respectful. It is informative to experts and non-experts alike; the formal apparatus is used to provide clarity not to create an esoteric, gated garden. It calls attention to unexplored alternative positions, and does so not just to keep scholarly score, but especially in order to illuminate the philosophical possibility space. It also contains a touch of wicked humor. (I return to that below.)

Note that Takashi Yagisawa (the reviewer) does not offer a detailed summary of the book; it is, thus, not balanced. Readers have to trust his judgment that he has focused on the central issues that are relevant to the community. Only competent readers of the whole book can decide, thus, if the review is fair. For some, the lack of summary may be a fatal flaw. Such people think that the main duty of a review is to tell people what's in a book. While that is important (which is why judicious summaries are often part of great reviews), it need not trump other considerations of the sort mentioned in the first paragraph. 

A dialogical analysis of structural rules - Part II

(Cross-posted at M-Phi)

 

Two weeks ago, I wrote a post proposing a dialogical perspective on structural rules. In fact, at that point I offered an analysis of only one structural rule, namely left-weakening, and promised that I would come back for more. In this post, I will discuss contraction and exchange (for both, I again restrict myself to the left cases). (I will assume that readers are familiar with the basic principles of my dialogical approach to deductive proofs, as recapped in my previous post on structural rules.)

 

Contraction, in particular, is very significant, given the recent popularity of restriction on contraction as a way to block the derivation of paradoxes such as the Liar and Curry. What does contraction mean, generally speaking? Contraction is the rule according to which two or more copies of a given formula in a sequent can be collapsed into each other (contracted); in other words, the idea is that the number of copies should not matter for the derivation of the conclusion:

 

What female orgasm and deductive reasoning have in common: they are not adaptations

Evolutionary accounts of deductive reasoning have been enjoying a fair amount of popularity in the last decades. Some of those who have defended views of this kind are Cooper, Maddy, and more recently Joshua Schechter. The basic idea is that an explanation for why we have developed the ability to reason deductively (if indeed we have developed this ability!) is that it conferred a survival advantage to those individuals who possessed it among our ancestors, who in turn were reproductively more successful than those individuals in the ancestral population who did not possess this ability. In other words, deductive reasoning would have arisen as an adaptation in humans (and possibly in non-human animals too, but I will leave this question aside). Attractive though it may seem at first sight (and I confess having had a fair amount of sympathy for it for a while), this approach faces a number of difficulties, and in my opinion is ultimately untenable. (Some readers will not be surprised to hear this, if they recall a previous post where I argue that deductive reasoning is best seen as a cultural product, not as a biological, genetically encoded endowment in humans.)

In this post, I will spell out what I take to be the main flaw of such accounts, namely the fact that they seem incompatible with the empirical evidence on deductive reasoning in human reasoners as produced by experimental psychology. In this sense, these accounts fall prey to the same mistake that plagues many evolutionary accounts of female orgasm, in particular those according to which female orgasm has arisen as an adaptation in the human species. To draw the parallel between the case for deductive reasoning and the case for the female orgasm, I will rely on Elisabeth Lloyd’s fantastic book The Case of the Female Orgasm (which, as it so happens, I had the pleasure of re-reading during my vacation last week).

A dialogical analysis of structural rules - Part I

(Cross-posted at M-Phi)

As some of you may have seen, we will be hosting the workshop ‘Proof theory and philosophy’ in Groningen at the beginning of December. The idea is to focus on the philosophical significance and import of proof theory, rather than exclusively on technical aspects. An impressive team of philosophically inclined proof theorists will be joining us, so it promises to be a very exciting event (titles of talks will be made available shortly).

For my own talk, I’m planning to discuss the main structural rules as defined in sequent calculus – weakening, contraction, exchange, cut – from the point of view of the dialogical conception of deduction that I’ve been developing, inspired in particular (but not exclusively) by Aristotle’s logical texts. In this post, I'll do a bit of preparatory brainstorming, and I look forward to any comments readers may have!

Further thoughts on reductio proofs

(Cross-posted at M-Phi)

Some months ago I wrote two posts on the concept of indirect proofs: one presenting a dialogical conception of these proofs, and the other analyzing the concept of ‘proofs through the impossible’ in the Prior Analytics. Since then I gave a few talks on this material, receiving useful feedback from audiences in Groningen and Paris. Moreover, this week we hosted the conference ‘Dialectic and Aristotle’s Logic’ in Groningen, and after various talks and discussions I have come to formulate some new ideas on the topic of reductio proofs and their dialectical/dialogical underpinnings. So for those of you who enjoyed the previous posts, here are some further thoughts and tentative answers to lingering questions. 

Recall that the dialogical conception I presented in previous posts was meant to address the awkwardness of the first speech act in a reductio proof, namely that of supposing precisely that which you intend to refute by showing that it entails an absurdity. From studies in the literature on math education, it is known that this first step can be very confusing to students learning the technique of reductio proofs. On the dialogical conception, however, no such awkwardness arises, as there is a division of roles between the agent who supposes the initial thesis to be refuted, and the agent who in fact derives an absurdity from the thesis.

Request for literature suggestions on ancient Greek mathematics

Those of you who have been following some of my blog posts will recall my current research project ‘Roots of Deduction’, which aims at unearthing (hopefully without damaging!) the conceptual and historical origins of the very concept of a deductive argument as one where the truth of the premises necessitates the truth of the conclusion. In particular, this past year we’ve been reading the Prior Analytics in a reading group, which has been a fantastic experience (highly recommended!). For next year, the plan is to switch from logic to mathematics, and look more closely into the development of deductive arguments in Greek mathematics.

But here’s the catch: the members of the project are all much more versed in the history of logic than in the history of mathematics, so we can’t count on as much previous expertise for mathematics as we could in the case of (Aristotelian) logic. Moreover, the history of ancient Greek mathematics is a rather intimidating topic, with an enormous amount of secondary literature and a notorious scarcity of primary sources (at least for the earlier pre-Euclid period, which is what we would be interested in). So it seems prudent to focus on a few specific aspects of the topic, and for now I have in mind specifically the connections between mathematics and logic (and philosophy) in ancient Greece. More generally, our main interest is not on the ‘contentual’ part of mathematical theories, but rather on the ‘structural’ part, in particular the general structure of arguments and the emergence of necessarily truth-preserving arguments.

One reason (among many) to like Homotopy Type Theory

(Cross-posted at M-Phi)

Last week I was in Munich for the excellent ‘Carnap on logic’ conference, which brought together pretty much everyone who’s someone in the world of Carnap scholarship. (And that excludes me -- still don’t know exactly why I was invited in the first place…) My talk was a comparison between Carnap’s notion of explication and my own conception of formalization, as developed in my book Formal Languages in Logic. In particular, I proposed a cognitive, empirically informed account of Carnap’s notion of the fruitfulness of an explication.

Anyway, I learned an awful lot about Carnap, and got to meet some great people I hadn’t yet met. But perhaps the talk I enjoyed most was Steve Awodey’s ‘On the invariance of logical truth’ (for those of you who have seen Steve lecturing before, this will come as no surprise…). The main point of Steve’s talk was to defend the claim that the notion of (logical) invariance that is now more readily associated with Tarski, in particular his lecture ‘What are logical notions?’ (1966, published posthumously in 1986), is already to be found in the work of Carnap of the 1930s. This in itself was already fascinating, but then Steve ended his talk by drawing some connections between the invariance debate in philosophy of logic and his current work on homotopy type theory. Now, some of you will remember that I am truly excited about this new research program, and since I’ve also spent quite some time thinking about invariance criteria for logicality (more on which below), it was a real treat to hear Steve relating the two debates. In particular, he gave me (yet another) reason to be excited about the homotopy type theory program, which is the topic of this blog post.

The weirdest passage from the Prior Analytics

As some readers will recall, we’ve been holding a reading group of the Prior Analytics in Groningen over the last academic year, which then prompted me to write (too?) many posts inspired by this venerable work (here and here, for example). We are nearly finished, only three more chapters to go (so just one more session). But interestingly, towards the end things are getting increasingly strange. Up to chapter 18 of book B (which traditionally receives much less attention than its more famous sibling, book A), things were still following the usual Aristotelian pattern of extreme systematicity and strenuous examination of cases. But as we got to chapter B19, there was a sudden change of gears: B19 and B20 are explicit applications of the theory of syllogistic to dialectical situations (needless to say, these made me very happy), and B21 is really about epistemology and quite out of place in the context of the Prior Analytics (though also very interesting). (Some scholars think that these are older layers of the text, which then somehow ended up being placed at the very end.)

At B22 it looked like we were back on track with the usual analysis of cases in the figures, but there was still a surprise in store. Towards the end of the chapter, Aristotle presents a puzzling discussion of ‘opposites’, one of which is preferable over the other. He writes (Smith translation):

When A and B are two opposites, of which A is preferable to B, and D is preferable in the same way to its opposite C, then if <the combination of> A and C is preferable to <the combination of> B and D, then A is preferable to D. (68a25-28)

Deductive proofs: from premises to conclusion, or from conclusion to premises?

(Cross-posted at M-Phi)

It is fair to say that the ‘received view’ about deductive inference, and about inference in general, is that it proceeds from premises to conclusion so as to produce new information (the conclusion) from previously available information (the premises). It is this conception of deductive inference that gives rise to the so-called ‘scandal of deduction’, which concerns the apparent lack of usefulness of a deductive inference, given that in a valid deductive inference the conclusion is already ‘contained’, in some sense or another, in the premises. This is also the conception of inference underpinning e.g. Frege’s logicist project, and much (if not all) of the discussions in the philosophy of logic of the last many decades. (In fact, it is also the conception of deduction of the most famous ‘deducer’ of all times, Sherlock Holmes.)

That an inference, and a deductive inference in particular, proceeds from premises to conclusion may appear to be such an obvious truism that no one in their sane mind would want to question it. But is this really how it works when an agent is formulating a deductive argument, say a mathematical demonstration?

Ted Sider's "Knee-Jerk Realism" and the Continuum Hypothesis.

[The following is the consequence of discussion with F.A. Muller, Lieven Decock, and Victor Gijsbers. They should be blamed for my mistakes.--ES]

A few weeks ago I complained that Ted Sider's approach to "knee-jerk realism" is dismissive toward views that do not share his (ahum) fundamental outlook (and I mused a bit about the sociology of knowledge that facilitates such dismissiveness). One worrisome consequence is that Sider fails to see objections to his view when they ought to be staring him in the face. Consider the following two passages from Ted Sider's Writing the Book of the World:

I hold that the fundamental is also determinate. "The fundamental is determinate" is not particularly clear, and improving the situation is difficult because there are so many different ways to understand what "determinacy" amounts to, but perhaps we can put it thus. First, no special-purpose vocabulary that is distinctive of indeterminacy...carves at the joints. Second, fundamental languages obey classical logic. The combination of these two claims is perhaps the best way to cash out the elusive dogma that vagueness and other forms of indeterminacy are not "in the world." (137)

The continuum hypothesis is sometimes said to be indeterminate. But suppose that mundane set-theoretic truths, such as the axiom of extensionality, are fundamental. Then by the combinatorial principle, the continuum hypothesis must be determinate, since it can be stated using only expressions that occur in mundane set-theoretic truths (namely, logical expressions and the predicate ∈). Thus we have a surprising result: the fundamentality of the mundane truths of set-theory requires the non-mundane continuum hypothesis to be determinate. (151)

Now, first,  the "sometimes said to be," is an odd locution. After all, it was proven that if ZFC is consistent then the continuum hypothesis can neither be proven nor disproven in it (see here for a good intro). Second, in the context of Sider's program ("mundane set-theoretic truths"), abandoning ZFC is not on the table.  Third, I know that One's modus ponens is another's modus tollens, but Sider has no "result" here--he ought to be facing up to the fact that there is a straightforward objection against his claim that the "fundamental languages" obey classical logic and mundane set theory: there is no reason to think the continuum hypothesis is determinate. To think otherwise is an act of faith (recall my observation about the odd religiosity of his so-called "knee-jerk realism"). So, I stand by my earlier claim that there is something troubling about an agenda-setting book that wishes away obvious problems with the program.

What's wrong with Mochizuki's 'proof' of the ABC conjecture?

(Cross-posted at M-Phi)

A few days ago Eric had a post about an insightful text that has been making the rounds on the internet, which narrates the story of a mathematical ‘proof’ that is for now sitting somewhere in a limbo between the world of proofs and the world of non-proofs. The ‘proof’ in question purports to establish the famous ABC conjecture, one of the (thus far) main open questions in number theory. (Luckily, a while back Dennis posted an extremely helpful and precise exposition of the ABC conjecture, so I need not rehearse the details here.) It has been proposed by the Japanese mathematician Shinichi Mochizuki, who is widely regarded as an extremely talented mathematician. This is important, as crackpot ‘proofs’ are proposed on a daily basis, but in many cases nobody bothers to check them; a modicum of credibility is required to get your peers to spend time checking your purported proof. (Whether this is fair or not is beside the point; it is a sociological fact about the practice of mathematics.) Now, Mochizuki most certainly does not lack credibility, but his ‘proof’ has been made public quite a few months ago, and yet so far there is no verdict as to whether it is indeed a proof of the ABC conjecture or not. How could this be?

As it turns out, Mochizuki has been working pretty much on his own for the last 10 years, developing new concepts and techniques by mixing-and-matching elements from different areas of mathematics. The result is that he created his own private mathematical world, so to speak, which no one else seems able (or willing) to venture into for now. So effectively, as it stands his ‘proof’ is not communicable, and thus cannot be surveyed by his peers.

What is logic?

A few weeks ago Jeff Bell had a post with the suggestive title ‘What is philosophy?’, referring to the eponymous work by Deleuze and Guattari. Now, as it turns out, a similar question, ‘What is logic?’, has preoccupied David Marans, a lecturer in logic at St. Thomas University in Miami, for quite some time. He thus created the Logic Gallery, a compilation of statements by philosophers and logicians across the centuries (starting with Aristotle and ending with Tim Williamson) on the nature of logic (or else comments from which one can infer the author’s conception of logic). It is a useful glimpse at the different views on logic held across time, and as such well worth spending some time on. In particular, it illustrates the early influence of the conception of logic as pertaining to debating and discussing, and the later predominance of the conception of logic as pertaining to thought and mental processes.

Marans welcomes suggestions for improvement and additional submissions. (I should perhaps add the proviso that there are no exact references to the original location of the quotes, and also that there may be some issues of translation.)

Desiderata for formal theories of truth

I have a fairly technical post over at M-Phi on desiderata for formal/axiomatic theories of truth, picking up on some ideas I discussed in a NewAPPS blog post of two years ago. My feeling is that it is a bit too technical/specific for cross-posting here at NewAPPS, but do check it out if formal theories of truth is what you like to read about on a Sunday afternoon!

Indirect proofs in the Prior Analytics

(Cross-posted at M-Phi)

A few days ago I wrote a post on a dialogical conceptualization of indirect proofs. Not coincidentally, much of my thinking on this topic at the moment is prompted by the Prior Analytics, as we are currently holding a reading group of the text in Groningen. We are still making our way through the text, but here are some potentially interesting preliminary findings.

I am deeply convinced that the emergence of the technique of indirect proofs marks the very birth of the deductive method, as it is a significant departure from more ‘mundane’ forms of argumentation (as I argued before). So it is perhaps not surprising that the first fully-fledged logical text in history, the Prior Analytics, offers a sophisticated account of indirect proofs. 

A dialogical conception of indirect proofs

(Cross-posted at M-Phi)

In his commentary on Euclid, the 5^th century Greek philosopher Proclus defines indirect proofs, or ‘reductions to impossibility’, in the following way (I owe this passage to W. Hodges, from this paper): 

Every reduction to impossibility takes the contradictory of what it intends to prove and from this as a hypothesis proceeds until it encounters something admitted to be absurd and, by thus destroying its hypothesis, confirms the proposition it set out to establish.

Schematically, a proof by reduction is often represented as follows:

[~A]

.

.

.

⊥

------

A

 It is well know that indirect proofs pose interesting philosophical questions. What does it mean to assert something with the precise goal of then showing it to be false, i.e. because it leads to absurd conclusions? Why assert it in the first place? What kind of speech act is that? It has been pointed out that the initial statement is not an assertion, but rather an assumption, a supposition. But while we may, and in fact do, suppose things that we know are not true in everyday life (say, in the kind of counterfactual reasoning involved in planning), to suppose something precisely with the goal of demonstrating its falsity is a somewhat awkward move, both cognitively and pragmatically.

The complexity of poker

(A second in a series, drawn from joint work with K. Joseph Mourad.)
How do we measure the complexity of decision procedures in poker?  This is a question that is both complex and subtle, and seems to me interesting in thinking about the interplay between formal modeling of epistemological situations and more concrete strategic epistemic thinking.

impredicativity in finite (empirical) contexts

(This will be the first in a series of posts designed to suggest that the mathematics of impredicativity - especially methods of definition that make use of revision-theoretic procedures - are relevant to empirical contexts.  Everything I say in these grows out of joint work with my math colleague Joe Mourad.)

Two basic points about the notion of impredicativity: first, it is much broader than what non-expert philosophers tend to think of under the rubric of paradoxes, vicious circularity, and the like.  Second, it is a property of definitions - or, more generally, procedures - not of concepts or sets, in the first instance.  Given an appreciation of these points, it is not hard to see that the general phenomenon can pose important epistemological issues in contexts in which there are no infinite totalities in play, indeed, in the context of various empirical discussions.  

The Modus Ponens-Modus Tollens asymmetry and non-monotonic logics

(Cross-posted at M-Phi)

A well-known phenomenon in the empirical study of human reasoning is the so-called Modus Ponens-Modus Tollens asymmetry. In reasoning experiments, participants almost invariably ‘do well’ with MP (or at least something that looks like MP – see below), but the rate for MT success drops considerably (from almost 100% for MP to around 70% for MT – Schroyens and Schaeken 2003). As a result, any theory purporting to describe human reasoning accurately must account for this asymmetry. Now, given that for classical logic (and other non-classical systems) MP and MT are equally valid, plain vanilla classical logic fails rather miserably in this respect.

As noted by Oaksford and Chater (‘Probability logic and the Modus Ponens-Modus Tollens asymmetry in conditional inference’, in this 2008 book), some theories of human reasoning (mental rules, mental models) explain the asymmetry at what is known as the algorithmic level (a terminology proposed by Marr (1982)) – that is, in terms of the mental process that (purportedly) implement deductive reasoning in a human mind. So according to these theories, performing MT is harder than performing MP (for a variety of reasons), which is why reasoners, while still trying to reason deductively, have difficulties with MT. Other theorists defend that participants are not in fact trying to reason deductively at all, so the asymmetry is not related to some presumed competence-performance gap. (Marr’s term to refer to the general goal of the processes, rather than the processes themselves, is ‘computational level’ – the terminology is somewhat unnatural, but it has now become standard.) Oaksford and Chater are among those favoring an analysis at the computational level, in their case proposing a Bayesian, probabilistic account of human reasoning as a normative theory not only explaining but also justifying the asymmetry.

The phylogeny and ontogeny of deductive reasoning

(Cross-posted at M-Phi)

In a recent paper, the eminent psychologist of reasoning P. Johnson-Laird says the following:

[T]he claim that naïve individuals can make deductions is controversial, because some logicians and some psychologists argue to the contrary (e.g., Oaksford & Chater, 2007). These arguments, however, make it much harder to understand how human beings were able to devise logic and mathematics if they were incapable of deductive reasoning beforehand.

This last claim strikes me as very odd, or at the very least as poorly formulated. (To be clear, I side with those, such as Oaksford and Chater, who think that deductive reasoning must be learned to be mastered and competently practiced by reasoners.) It looks like a doubtful inference to the best explanation: humans have in fact devised logic and mathematics, which are crucially based on the deductive method, so they must have been capable of deductive reasoning before that. Something like: birds had to have fully formed wings before they could fly – hum, I don’t think so… Instead, the wing analogy suggests that there must be some precursors to deductive reasoning skills in untrained reasoners, but the phylogeny of the deductive method (and to be clear, I’m speaking of cultural evolution here) would have been a gradual, self-feeding process.

Logical Properties of 'It Seems'

I am currently finishing a paper on the semantic and logical properties of 'seem'. As 'seem' is a subject-raising verb, we can treat 'it seems' as a sentential operator. This raises the question of how this operator behaves logically. Is it hyperintensional? Does it distribute over conjunction? Over disjunction? Over conditionals? Does it commute with negation?

I think it's fairly obvious that 'it seems' is hyperintensional. It seems to Lois Lane that Superman is not Clark Kent but it doesn't seem to her that Superman is not Superman. The other questions are harder.

Suppes on neuroscience as foundations of mathematics

(OK, so it looks like I’m over-posting a bit today… Just one more!)

Between today and tomorrow, the workshop ‘Groundedness in Semantics and Beyond’ is taking place at MCMP in Munich, co-organized with the the ERC project Plurals, Predicates, and Paradox led by Øystein Linnebo. The workshop’s program seems excellent across the board, but the opening talk is what really caught my attention: Patrick Suppes on ‘A neuroscience perspective on the foundations of mathematics’. The abstract:

I mainly ask and partially answer three questions. First, what is a number? Second, how does the brain process numbers? Third, what are the brain processes by which mathematicians discover new theorems about numbers? Of course, these three questions generalize immediately to mathematical objects and processes of a more general nature. Typical examples are abstract groups, high dimensional spaces or probability structures. But my emphasis is not on these mathematical structures as such, but how we think about them. For the grounding of mathematics, I argue that understanding how we think about mathematics and discover new results is as important as foundations of mathematics in the traditional sense.

The best medieval solution to the Liar ever

I posted this on M-Phi yesterday, but now it occurs to me that it might be of interest to some NewAPPS readers as well (the interest sparked by Brit's recent post suggests this much!). It is in fact a review of Stephen Read's edition and translation of Bradwardine's treatise on insolubles, which I just wrote for Speculum. 

------------------------------------------------------

Thomas Bradwardine (first half of the 14^th century) is well known for his decisive contributions to physics (he was one of the founders of the Merton School of Calculators) as well as for his theological work, in particular his defense of Augustinianism in De Causa Dei. He also led an eventful life, accompanying Edward III to the battlefield as his confessor, and dying of the Black Death in 1349 one week after a hasty return to England to take up his new appointment as the Archbishop of Canterbury.

What is thus far less well known about Bradwardine is that, prior to these adventures, in the early to mid-1320s, he worked extensively on logical topics. In this period, he composed his logical tour de force: his treatise on insolubles. Insolubles were logical puzzles to which Latin medieval authors devoted a considerable amount of attention (Spade & Read 2009). What is special about insolubles is that they often involve some kind of self-reference or self-reflection. The paradigmatic insoluble is what is now known (not a term used by the medieval authors themselves) as the liar paradox: ‘This sentence is not true’. If it is true, then it is not true; but if it is not true, then what it says about itself is correct, namely that it is not true, and thus it is true after all. Hence, we are forced to conclude that the sentence is both true and false, which violates the principle of bivalence. It is interesting to note that, in the hands of Tarski, Kripke and other towering figures, the liar and similar paradoxes re-emerged in the 20^th century as one of the main topics within philosophy of logic and philosophical logic, and remain to this day a much discussed topic.

Liars and Truth-Tellers

This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. This means you're free to copy and share the comic (but not to sell it). Thanks to Kevin Lepore for the reference.

A Few Puzzles for a Rainy Sunday

Five old puzzles to brush up on your logic skills before the GREs:

(1)
Yesterday I ran into my colleagues Eric Wiland and John Brunero, who were trying to sneak out to watch the baseball game. I asked Eric:
"Is any of you a truth-teller?"
Eric said something that sufficed for me to know the answer to my question. Are Eric and John liars or truth-tellers?

Domains of quantification - part 2: does science quantify over reals?

Short version: 
Science is often said to be committed to reals, because physics, for example, essentially makes use of sentences with real-quantifiers.  But we have perfectly good countable, well-founded, constructive models of full second order arithmetic.  So why can't physics, for example, simply explicitly embrace one of these as what they are working over and thereby radically simplify their alleged ontological commitments?

Domains of quantification - part 1

It seems to me that there is an issue with the epistemology of domains of quantification that has important implications for the epistemology and semantics of math generally, and which has received less attention than it deserves.  In quick outline, the point is this:

A quantificational sentence has a determinate meaning only if there is some determinate fact of the matter as to what its domain of quantification is.

So one knows what one is saying with such a sentence only if one knows what domain one is quantifying over.  If we are discussing anything as complex as the reals - equivalently second order arithmetic - and mean to quantify over the "intended model" - that is, do not specify some constructable model as our domain - then we do not know what we are quantifying over.

Thus, we do not know what we are saying when we make claims with second order arithmetic quantifiers.

Stebbing Vindicated: on technical expertise as a criterion in being the useful subject of attention

Mohan wrote a useful follow up to my post on the unjustly forgotten Susan Stebbing. Before I get to the core of my response to Mohan's post I want to prevent a potential confusion. Mohan suggests  "it is a pity that historians of analytic philosophy don’t give her more airtime." Well, as some commentators on my post noted, historians of analytic philosophy DO write about Stebbing (and this is how I explained how I came to know of her work). My claim is, rather, that even if we leave aside Stebbing's fascinating treatment of Spinoza, reading Stebbing can be instructive to us (by which I mean professional philosophers today): (i) by reminding us that not all early analytic philosophy was fundamentally anti-Metaphysical; following up the work of Moore and Wisdom (etc), Stebbing is developing and theorizing the tools of analysis in order to offer a distinctly analytic account of metaphysics; (ii) I call it distinctly analytic because it anticipates in rudimentary and non-trivial ways the orientation of the program of David Lewis.  (Of course, I don't want to overstate the claim--modality is not driving Stebbing's approach.) This matters even if there is no line of influence. (iii) Stebbing's understanding of the continuity between Bradley and Russell is a useful corrective to a lot of narratives within the disipline. 

By contrast, Mohan claims that Stebbing's "lack of technical expertise in the key tropes of analytic philosophy (Russell's theory of descriptions, no less) meant that it was not very useful for later generations to read her work." At the end of this post I say something about the criterion that Mohan applies here. Now Mohan offers a bunch of examples of Stebbing's lack of "technical expertise." Dan Kervick has usefully challenged Mohan's examples. Now while I have a healthy admiration for Mohan's technical chops (around NewAPPS I am among the techno-ignoramuses), I want to develop Kervick's points a bit. I do so in reverse order (because all my comments turn, in part, on what Stebbing means by "refer.") Mohan writes:

Susan Stebbing: The History of Metaphysics at the Dawn of Analytic Philosophy

From this point of view metaphysicians are apt to seek for an indubitable datum. Descartes, for example, saw clearly the need for such a datum. Whatever one may think of the success of his efforts, it will probably be admitted that his procedure was correct provided that the aim of metaphysics be to provide reasons for our commonsense beliefs. Curiously enough, it is Descartes, rather than Spinoza who has shown most clearly the futility of a constructed system. Spinoza, so it seems to me, used the form of a deductive system in order to exhibit his vision of the universe. I see no reason why he should be dismayed by the charge of circularity; there is no reason why he should not wrap up in his definitions and axioms all that he desired to bring forth from them. For Descartes, however, such a charge, if it could be substantiated, would be fatal.--Susan Stebbing (69).

One of the great scandals of analytic philosophy is our utter ignorance of our history. Long after my  graduate education I encountered Susan Stebbing´s name for the first time in Mike Beany´s writings on early analytic. (But don't try looking for Stebbing in Soames.) Stebbing (1885 – 1943) was a crucial organizational figure in the movement (she helped found Analysis, brought Carnap to Cambridge for the first time, etc) and important popularizer. (Stebbing is useful to those who wish to tell an alternative to my stories (and here) about Ernest Nagel as Prophet of Analytic Philosophy.) Even so, here I focus on her because she was arguably the first significant theorist of analytic philosophy and the method proper to it. In particular, it is astonishing that Stebbing's (1932) master-piece, "The Method of Analysis in Metaphysics," was (I believe) never properly anthologized in the early readers of analytic philosophy.

Now, Stebbing (and her critic, Max Black) worked hard at explicating what the technique and method of philosophical analysis amounts to. What is crucial is that long before Strawson, Benardete, and Kripke-Lewis, a pre-eminent analytic philosopher had articulated (on the basis of deep reflection on Russell and Moore) how one should think about the marriage of analytic philosophy and metaphysics, which according to Stebbing "is a systematic study concerned to show what is the structure of the facts in the world to which reference is made, with varying degrees of indirectness, whenever a true statement is made." (65) For Stebbing metaphysical analysis had a direction toward the precise understanding of absolutely simple elements (that are taken to exist). Now Stebbing insisted on a distinction between logical analysis and metaphysical analysis. Black denied the distinction and offered a deflationary model for philosophy, which is primarily interested in the "structure of sentences rather than with fact." (258) Black seems to have won the day initially (although I think he misunderstands Stebbings position as uncovering facts rather than the structure of facts), but either way from our post-Lewisian vantage point, Stebbing seems most prescient. (Black ridicules the very idea of an analysist being able to describe basic facts.) We can understand Lewis' natural properties as giving Stebbing her cake (Lewis and Stebbing share a deep respect for common sense and natural science) with little loss of...ahum...simplicity.