Routledge has now offered a contract, so I can make official the anticipated publication of the book for which this blog is named:
Kenneth Arrow is a well known economist, logician, statistician, and political theorist. While his scholarly contributions are numerous, his best known was his first, published as a part of his dissertation. This is the above titled “paradox of voting,” which is also referred to has his “impossibility theorem.” This latter is evidently the technically correct title. However, I learned about it as the paradox of voting, and that’s the title I’ll stick with here. For one thing, calling it his “paradox of voting” makes it more clear at the outset what the theorem is about, and suggests what is really at stake. Details of the impossibility theorem are readily found for no more effort than looking, so my intention here is to provide a non-technical gloss of the topic. Still, enough of what I say here is about basic logic (and not merely political screed) that I am satisfied that this topic falls within my basic parameters for this blog.
The stakes here could scarcely be any higher, as they effect the very foundation of our nominally democratic system. Because of how our voting and electoral system is set up, we have a “winner take all” format that can (and often enough, does) allow a person to be elected even thought that person did not receive a majority of the votes. Once you have more than two candidates (or more than two parties) involved in any particular election, it is no longer possible to representatively distribute preferences in the election. This is the somewhat fancy way of saying things. The simpler way of saying it is that the more widely detested candidate can win. Continue reading
An oddity about philosophers, and especially logicians, is that when they talk about “quantity” they are not talking about numbers, or numerical counts. Rather, they are talking about the ways things can be gathered together (or singled out) using words like “all” or “some.” These ideas are called “quantifiers.” I want to do three things (briefly, as always) here: say a little about the “basic” quantifiers (“all” and “some”), say a little about how they get dropped from common discourse and argument – whether from laziness or deliberate obfuscation – leading to much gratuitous confusion. Finally, I want to say something about quantifiers that typically do not make it onto philosophers’ or logicians’ lists, yet are at least as common in ordinary discourse and argument as the “principal” two are. My purpose here (as always) is not to lead you onto the path of righteous proof making, but simply alert the reader to the importance of these operators so that they might not slip by quite so stealthily in the future.
The second greatest sin in logic is to allow things to pass implicitly; the greatest sin is to block the road of inquiry, which is one of the things that happens when concepts are allowed to pass implicitly. Allowing things to remain implicit means that vague statements are permitted, by innuendo, to become concrete, thus leading us astray (blocking inquiry) from the directly stated vagueness. Sometimes things really are ambiguous, and they must be allowed to stay that way until real data, rather than jumping at conclusions, enables us to clear up the ambiguity. That, or recognize that the ambiguity is not – or, at least, not yet – cleared. Continue reading
A number of years ago I got into a discussion with an acquaintance about what kind of symbol system tells us “the truth” about the world. This is not how my interlocutor expressed the problem; she simply insisted that mathematics gives us the truth. I tried many different approaches to get her to understand that what she was saying made absolutely no sense, because the first thing that must happen (once any collection of symbols is at hand) in order to talk about truth was that those symbols have to be interpreted, and such interpretation is not given in advance. Thus, I have a modest background in some advanced forms of mathematics (mainly formal logic, abstract algebra, and a touch of differential geometry), and I understand that simply having a bunch of squiggles in front of you is not enough to adjudicate whether those squiggles say anything at all, much less anything that is true. Meanings must be assigned to those squiggles such that they hang together to form some kind of model, and that model then must be associated with the world in some form such that the model can be interpreted as making claims about the world which then can be interpreted as to its truth content. And here, “world” can mean either the world of concrete experience or a purely abstract “world” which is itself something of a mathematical construct. Also, my choice of the term “truth content” rather than “truth value” is not an innocent one: I wish to leave open the possibility that truth evaluations can be more complex and multi-dimensional than the mere assignment of values.
It became very clear that while I understood my acquaintance’s position, she in no way understood mine. This was because while I was repeatedly able to paraphrase – that is, interpret – her argument, when asked to do the same for mine she was unable to do anything other than repeat her own position, which addressed none of the points I had made. In later years, she was known to crow a bit about how she “won” the argument. To be fair, in retrospect I realize that there were a number of ways I could have made my own position clearer, as it was burdened by a much greater degree of philosophical nuance than the position she was presenting. And I confess that I do not think quickly on my feet; indeed, I’ve only ever suggested that, given time, I can think thoroughly. (One of the reasons I went into philosophy is because a line like, “Herstein! If we don’t get this metaphysical principle out the door by end of business today, our competition is going to crucify us!” is not something one is ever likely to hear from one’s department head.) Continue reading
So, what is it that makes something true? (Trust me, this ties in with this post’s title.) If I say that “X is the case,” and it, indeed, turns out that X IS the case, then my saying so was true. Or, rather, the thing I said was true, and my saying it was said truly. (Actually, my saying it was said truly, because I truly said it, regardless of whether what I said was actually true.) But what establishes the connection(s) between my saying it is the case, and its actually being the case? Well, presumably it is reality that makes that establishment; but how is that reality, how is that establishment, established in experience such that the truth-saying and the truth-being converge in a truth making?
Because even as (and insofar) as “the truth is out there,” our having, getting, finding, or whatever, that truth involves a substantial amount of making. If you take the idea of truth seriously, then you must take seriously the fact that we have to go out and make that truth apparent through significant and substantive inquiry. Where this is going (and it will go fast) is that the maker that connects the truth as said with the truth as found, looks a lot like a successful “strategy” in a “game.” This is a formal, logical concept, which brings scientific inquiry into a dirty-dance with that part of formal logic known as model-theory. (Somewhere, somebody has sheet music on this stuff … ) Continue reading
I was reminded again this other day that the varieties of ways that things can be “together” easily exceed the kinds of ways that even smart people will often notice or imagine are possible. The issue I have in mind here is not a matter of relationship advice but rather of logic (although more than a few relationships would profit from even a smattering of basic reasoning.) In this instance, some things can be analyzed into genuine parts that can be separated in fact, while other things can only be analyzed into abstract “parts,” which are not ever separable in reality; there is yet a third type that can only be taken as a whole, even in analysis, without doing violence to the nature and meaning of the thing in question. Failure to recognize what type of thing or idea one is dealing with can lead one into fundamental errors which, while often terribly clever are, for all of that, still just flat wrong. My interest here will be with the first two of the above three.
Various common phrases are easily recognizable in this context, most especially the old saw about, “the whole is greater than the sum of its parts.” This is especially true of organic unities. For while we’ve achieved a level of surgical finesse that can, under extremely delicate and rigorously right sets of circumstances, permit us to, say, remove an organ from a living being and replace it with another, this generally cannot occur without considerable trauma, frequent enough failures, and extraordinary skill to reassemble the whole that has been torn apart by the procedure. Such holistic entities are what the Greeks referred to as a-tomos, a word that roughly translates as “uncut.” It is from this Greek root that we get our term “atom,” which originally meant an undivided unity. Continue reading
Regardless of what Mick said, it is not on your side.
I’ve been in the position to observe a number of significant transitions of late – from which there will be no coming back – and the thought of time is once again on my mind. Saint Augustine – a fairly bright fellow, for a psychotic authoritarian – mused in his Confessions something to the effect (I quote from memory, so this is only analogously correct) that, “As long as no one asks me, I know what time is; as soon as anyone asks, I have no idea.”
Time is something like THE fundamental mystery. “Intention” is right up there with it, except that intention is a logical/semantical category, whereas time is more about ontology – what genuinely IS (ontology), rather than what must be taken into account for the rational possibility of inquiry and discourse (logic/semantics). Moreover, it is not clear that intentionality (which includes things like “meaning,” “believing,” “interpreting,” “intending,” “wanting,” and so on) has any logical – much less ontological – possibility, that is not already thoroughly infused with time and temporality. Certainly this seems true in the human world; perhaps gods, devils, and their associated helpmates suffer no such limitations. I should add here that persons involved with phenomenological philosophy would require 200 pages of densely packed and, often enough, uninterpretable obfuscation and hand-wringing to ask the above question; but I am not a phenomenologist, and as such I labor under no such constraints. Continue reading
“A lie can travel halfway around the world before the truth can get its boots on.” This well known saying is variously and unreliably attributed to a number of persons, from Mark Twain to Winston Churchill. But as long as one is not trying to steal the words for one’s self, it is less important who said a true thing, than that the thing said be true. Credit should be given, of course, when credit is due, and identifiable. But just because, say, Abraham Lincoln said a thing, that thing is not automatically true, any more than if Richard Nixon said something, it is automatically false. Now, it is not an ad hominem to call a liar a liar, nor is it a fallacy to question the credibility of a person whose credibility has been shredded by repeated abuses of the truth. Still, one must be very careful when it comes to either accepting or dismissing a statement merely on account of its source. If you dismiss an alcoholic’s statement that drinking is bad for you, on account of the fact that the person making the statement is an alcoholic (who is still drinking), you’ve committed the tu quoque version of the argumentum ad hominem. If anything, the alcoholic is better situated to speak with genuine expertise on the damage of alcoholism than, say, a more sober member of society.
But to return to our original point, there is an intransigence to falsehoods that is not easily dislodged by anything so inconsequential as reason and truth. There are many psychological studies (I’ll not link to any – they are easy to find) that point out that, for example, climate change denialism – devoid as it is of any shred of valid or scientific justification – nevertheless becomes more stubborn when it is confronted with logic and facts that admit of no rational dispute. The lie, as it were, digs in its boots. I’ll skip over any discussion of those rhetorical techniques that do seem to work, because such methods are not my interest here and it pisses me off that I’d ever have to resort to them. Rather, I want to look at those factors that let the lie out of the starting gate before the truth even knows that there is a race today. In particular, what is it that makes the lie so easy, and the truth so hard? Continue reading
Explanations come in discrete units, logically minimum quanta. It is logically impossible for the situation to be otherwise. We can reason about continua of various different kinds (the “continuum” of the Real numbers being a prominent example, although it is to be noted that within that branch of formal logic known as “model theory,” there are examples of continua that are “more continuous” than even the Real numbers.) But we cannot reason “in” a continuum. Our ideas may have vague boundaries, but they are still unitary quanta, or at least collections of such quanta. Our concepts are even more sharply defined. We assemble these units into larger structures that become arguments (in the good, philosophical sense) and, ideally, explanations. But a continuum gives us nothing to work with. Like trying to nail mercury to the wall, every time we attempt to grasp it, it slips around and away in out grasp, so that either we (1) end up speaking about the continuum itself as a whole, at which point the continuum qua whole has become our quantum, (2) we isolate individual points on the continuum, and these become our quanta as we extrapolate connections amongst them, (3) or, alternatively, we end up spouting nothing but nonsense.
I’ve touched on this subject before. But rather than making coy suggestions in the final paragraph as a rhetorical flourish, I think it time I spoke to the subject more directly. As is often the case, I’ll barely be able to gloss the topic in this post. But, of course, the whole purpose of a blog post is to provide a small quantum of ideas that might lead interested readers off in interesting directions. Continue reading
There is a large, nested, complexly intersecting, multidimensional area of logic known as “modal logic.” Standard (“assertoric” – dealing with comparatively simple assertions) logic essentially forgoes any considerations of the modes (hence, “modal”) in which an assertion is considered to be true or false; it simply is, or it is not (true or false). Modal logics are intended to examine the ways (modes) in which a proposition or assertion might express such truth or falsity. A great deal of very good work has been done in this area of study, but it remains a long way from solving its most basic problems; indeed, most proposed “solutions” do not so much “solve” their problems as strangle them.i I am at once deeply impressed by the technical sophistication of contemporary work on modality, and profoundly dissatisfied with the narrowness of its vision. Because one of the “modes” in which an assertion or proposition might be true or false is whether it is possibly true or false.
I can certainly inundate any interested party with citations, but anyone capable of following those citations would most likely already be familiar with them. It takes years of dedicated study to bootstrap one’s self up through propositional, into quantificational, and finally on to modal logics. On the other hand, it takes nothing more than the most elementary capacity for cognition to instantly see that there is a difference between saying that “X is the case,” and “X might be the case.” Just as we can talk about Jazz without mastering the saxophone, or relativity without deriving proofs related to the Ricci tensor, we can talk about possibility without becoming research mathematicians in formal logic. One might even argue that mastering such mathematics would not ideally equip us to talk about possibility which is, after all, a metaphysical, rather than a mathematical topic. Continue reading
European Sword Arts in SW Florida / Fencing Classes & Lessons Naples, Bonita, Estero
Loving Wisdom Beyond the Academy
Your argument is invalid
Science, logic, and ethics, from a Whiteheadian Pragmatist perspective (go figure)
Science, logic, and ethics, from a Whiteheadian Pragmatist perspective (go figure)