Monday, 27 July 2015

Women in Logic: two new initiatives

For those who haven't yet come across these, I have two new initiatives relating to women in logic to advertise:

  • Women in Logic group on Facebook: "A group for women in Logic, philosophical, mathematical or computational. or any other kind of formal logic that you care about." Membership is not restricted to women.
  • Female Professors of Logic, an editable google spreadsheet. One outcome of this will be to give a list of people who should have wikipedia pages if they don't already.

Please share widely and contribute as you can.

© 2015 Sara L. Uckelman

Tuesday, 21 July 2015

Reductio arguments from a dialogical perspective: final considerations

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

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.


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.

Monday, 20 July 2015

Conference on Belief, Rationality, and Action over Time

University of Wisconsin-Madison, September 5-7The goal is to get action theorists and epistemologists (especially formal epistemologists) together to think about topics related to diachronic rationality and belief.  All are welcome, but attendees are expected to have read the papers beforehand.  Register for free here.

Organizers:  Mike Titelbaum, Sergio Tenenbaum, Chrisoula Andreou, and Sarah Paul
Funded by the Canadian Journal of Philosophy, the University of Wisconsin, and a gift from Rodney J. Blackman. 

Friday, 17 July 2015

Dialectical refutations and reductio ad absurdum

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

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.


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:

Thursday, 16 July 2015

A precis of the dialogical account of deduction

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

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.


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)

Wednesday, 15 July 2015

Problems with reductio proofs: "jumping to conclusions"

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

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.


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?

Tuesday, 14 July 2015

Problems with reductio proofs: assuming the impossible

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

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.


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.

Monday, 13 July 2015

Problems with reductio proofs: cognitive aspects

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

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.


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.

Logical parenting, balloons, and Abelard's insights on quantifiers

My 3.5 year old daughter has apparently been learning about opposites at nursery, because all weekend she was popping out such gems as "You know what are opposites? Big and little!" (Hot and cold, up and down, in and out, etc., etc., etc.). Sunday evening while we were getting supper read, she proceeded to play underfoot with a balloon she'd been given at a birthday party earlier in the day. This was increasingly irritating until she came out with:

"Do you know what are opposites? No balloon and some balloon!"

Logical parenting: Ur doin it right.

Of course, I was curious to know if she could extrapolate, so I asked her what the opposite of "All balloons" was. Her reply was "No balloon", which I couldn't complain about, because, after all, I hadn't specified whether I was looking for the contradictory opposite or the contrary opposite. Being the proud parent I was, I relayed the story on FB, and was amused at the selection of half-joking, half-serious suggestions I got for the opposite of "all balloons": Negative balloons? Impossible balloons? The square root of minus one balloons? i balloons? But it also made me think: The usual Aristotelian quantifier opposed to 'all' is 'some____not'. But "Some balloons not" doesn't make any sense. You can have "all balloons", you can have "no balloons", you can have "some balloons", but you can't have "some balloons not" [1]; if you want to use that quantifier, there needs to be more than just a quantified subject, there has to be a predicate, too. The same is not true of the non-Aristotelian form of the negation, 'not all': While you can't have "some balloons not", you can have #notallballoons.

Reflecting on this on the way home this evening, I was reminded of how Abelard made this very distinction, between non omnis and quidam non, arguing that these two are not equivalent with each other: non omnis does not have existential import, while quidam non always does. Many people think that making a distinction between 'not all' and 'some____not' is only necessary in a context where 'all' has existential import; but perhaps Abelard's insight that non omnis and quidam non are not equivalent reflects something deeper than just logical machinery to deal with a problematic assumption about universal quantifiers.

[1] This is, essentially, just the well-known observation that there is no single natural language English term 'nall'.

© 2015, Sara L. Uckelman