Propositional Calculus for Computer Programmers: Background and First Part

This post is the inaugural element in what I plan to be an ongoing project into attempting to develop a framework for translating philosophical propositional calculus (PC) into terms that computer programmers can easily understand.

Personally, I find the confluence of factors that motivate this project incredibly interesting; but of course that’s because they’re largely about my life. The next section will talk about some of my motivations, as well as some of the sources I plan to draw on for this project. If you want to skip this section, you can click here.

Background

  1. I told you before that my philosophical background is in the so-called continental tradition. What does this mean? Well, one thing that it means is that in my previous philosophical career, I never had to understand the sentential calculus. It’s just not a form of arugmentation that people in that tradition are prone to use.
  2. At the same time, I told you that I was applying for PhD studies in American philosophy programs. Most of these programs are couched in the analytic tradition, which is a tradition that is deeply enamored of propositional logic as an argumentation tool. American analytic philosophy in the Russelian tradition - at least in some (admittedly much rarer now) cases - actually situates logic as first philosophy. This is quite a change for a person who comes very distinctly from a tradition that has historically situated everything but logic (i.e. metaphysics, epistemology, ethics) in the big chair.
  3. Luckily, in addition to my philosophical interest, I have also been a computer programmer for ten years. This means that I have spent a lot of time negotiating symbolic logic, and as such I imagine or — at least hope — that there’s got to be a strategy to map these things together.

Pedagogy

When I was eighteen, I took a class called Formal Foundations of Computer Science, which was a symbolic logic class. I did rather poorly at this, which should confuse you, given the aforementioned. It confused the heck out of me for just long enough to be disastrous. Only recently, spurred by my partner Nick’s research project in programming pedagogy, did I understand this: the pedagogical model for teaching formal logic to computer programmers should be way more practical than it is (or at least was in my case).

Hence, the following attempt to make logic work. Throughout the following, I will be using the class notes and assignments most kindly provided by the Massachusetts Institute of Technology’s Open Courseware project. The syllabi will be provided courtesy of Prof. Vann McGee’s Logic I and Logic II classes in the MIT Linguistics and Philosophy department. The pedagogical materials will be provided courtesy of Delicious.com and Nicholas Senske of the University of Michigan.

Finally, expect this series to take some time to complete. Let’s get started.

Propositional Calculus for Computer Programmers: Part 1

Today’s course is easy. All we’re going to do is look at the basic syntax of the propositional calculus, and then the truth assignments for its basic operations. Really, nothing to it. I’ll be doing this using C-style syntax.

Syntax

PC symbol English C Operator or Fragment
and &&
or (inclusive) ||
¬ not !
if…then (only if) onlyIf(a,b)
if and only iff ifOnlyIf(a,b)

You’ll notice that the last two rows above are logical operations that don’t have C operators. Programming languages usually handle these operations by selection statements. For brevity’s sake, I’ll be using the ternary operation.

function onlyIf(a,b)   { a ? (b ? return true : return false) : return true; }
function ifOnlyIf(a,b) { a ? (b ? return true : return false) : (b ? return false : return true); }

Now, you may remember from way back in your beginning CS career drawing truth tables. Well, that’s what we’re going to be working on now. A final note on my conventions is that you should just assume, from now on that 1 == true and 0 == false.

Truth Assignments for simple PC operations

a b (a ∧ b) (a ¬ b) (a → b) (a ↔ b) ¬a
1 1 1 1 1 1 0
1 0 0 1 0 0 0
0 1 0 1 1 0 1
0 0 0 0 1 1 1

One last thing to note that’s really convenient for programming types is that when you draw truth tables, an easy algorithm for mapping out your variable columns (in the above case, the first two columns: a, b), you just start from the bottom and count in binary (above: 00,01,10,11). This holds true for n variables.

So the plan is that in the next one of these, we’ll look at the axioms for the PC. I’m hoping that we can figure out ways to make this stuff feel more intuitive for the pragmatic programmer.

No, /you’re/ dumb.

Every now and then, I really enjoy picking up something new by an analytic philosopher, and this week I have opened the new book Freedom Evolves by Daniel C. Dennett. Dennett is often a thoughtful and interesting person, and although I don’t always understand why he chooses his certain domain of knowledge to pursue, I do usually enjoy the way he does it. I am not very far along in his book now, but I was very disappointed by the following move:

As is the wont (I have noticed) of many American philosophers, he mentions and dismisses the work of continental philosophy - under the umbrella of “post-modernism” - in one paragraph located in the first five pages of his book.

Look around at those who are participating in this quest for further scientific knowledge and eagerly digesting the new discoveries; they are manifestly not short on optimism, moral conviction, engagment in life, commitment to society. In fact, if you want to find anxiety, despair, and anomie among intellectuals today, look to the recently fashionable tribe of post-modernists, who like to claim that modern science is just another in a long line of myths, its institutions and expensive apparatus just the rituals and accoutrements of yet another religion. That intelligent people can take this seriously is a testimony to the power that fearful thinking still has, in spite of our advances in self-knowledge. (pp. 5)

And that’s what he has to say about that. No big surprise here, we have all seen the so-called analytic-continental divide play out like this over and over again, and this kind of dismissive behavior, and wild generalization (or complete misreading, depending on how tight your focus is) is nothing new. I found myself thinking about it again, though, not fifteen pages later, as Dennett addresses his own interlocuters:

They are prepared to take whatever steps to discourage, squelch, or discredit those [among whom, him, Dennett] they see as breaking the spell [of belief in an indeterminism which provides the condition of possibility for their conception of free will] before some serious harm is done. They have been at it for many years, and while their campaigns have grown threadbare, and their simple fallacies have been exposed over and over by their scientific colleagues, the debris from their campaigns continues to pollute the atmosphere of the discussions, distorting the understanding of the general public on these topics. (pp. 19)

You see where I’m going here? For some reason I found the way this whole discussion, which is about a sort of responsibility to thought and writing, to be profoundly sad after Dennett had so quickly dismissed a couple hundred years of a certain kind of knowledge-building/writing. Naturally, he would probably say that that certain thread which followed the speculative elements of Hegel in Europe really has no claim on producing or accessing any kind of knowledge, but that is just more of the same. Later on the same page, he notes:

I agree with the critics on both left and right that there have been some unfortunate overstatements and oversimplifications by some of those they target, and I also agree that such lapses from responsibility can have truly pernicious effects. Moreover, I don’t challenge their motives or even their tactics; if I encountered people conveying a message I thought was so dangerous that I could not risk giving it a fair hearing, I would be at least strongly tempted to misrepresent it, to caricature it for the public good. (Ibid.)

Self-aware, I suppose, if not entirely honest. Either way, a giant sigh escapes here in the first twenty pages of the book.

Zero-sum piracy.

1. Ask me how I became a pirate…

I recently ran into something that happened about a year ago (this not being my area of interest, I am a little late with it): Namely, a usenet post:

Information from microsoft.public.windowsmedia.drm
Posted by Satoru Koshiba (JP):
“As far as I survey at this time some popular DRM Protected Video
Providers’s contents were cracked…
Many many unlocked video files were distributiong with no payment…
And …at 17:00(GMT+9:00) anonymous cracker named ‘lark’ upload source code of ‘DRM crack software’.
Software name is “DrmDbg.exe” and “DRM2WMV”.

DrmDbg.exe will pick up “KID” and “Seed” from PC’s memory when user playback DRM Protected contents , and make “xxxxx.key” file. DRM2WMV will unlock ProtectedFile using “xxxxx.key”.”

ask me how i became a pirate I located and downloaded application, tried it on a sample DRM’ed WMV and it worked gangbusters. Which, of course, was not surprising - it makes a sort of intuitive sense how this works when you see it in action, and we are all used at this point to every ludicrous content-security attempt by corporations (heretofore referred to as "Microsoft" as a sort of cute nickname for "corporations forced into a specific model of software/content creation and distribution perpetuated by operating system monopolization on behalf of the Microsoft corporation") being cracked in half. So, fine. I went to sleep. I don’t really download any DRM’ed content anyway: I don’t like it. Zzz.*

* Please note that I slept emphatically.

The following day, today in fact, I happened to have some time to browse the internet, and I had developed a nagging interest in how Microsoft took the news of Satoru’s discovery. To my surprise, they virtually ignored it. I will learn, very soon, not to be surprised at things like this. At this point, however, I need to backtrack.

DRM (digital rights management) is a system that applies to digital media, which, as we all know, is for all intents and purposes infinitely reproducible at zero degradation. This means that I can email you a copy of a document, still have the document, and both of our copies will be identical within a reasonable scope. This as opposed to a photocopy of a book, or a copy of a vhs tape, which lose quality with each successive copy. This fact has lead to a widespread phenomenon called piracy, which is quite similar to the undisambiguated phenomenon.

Traditionally, pirates are those who rob or plunder at sea, or sometimes the shore, without a commission from a recognised sovereign nation. They are now also those who commit acts of copyright infringement against sovereign coporate bodies, many of which own audio recordings or other media produced by artists and entertainers (human beings). In this way, and in sympathy with Marx, sovereignty in both cases boils down to owning human labor. We will return to this.

Now: Digital/copyright piracy has become such hit in the so-called bit-economy, that it has caused a major brouhaha in what might be called the dominant world order. This is a powerful phenomenon and very rich white men are losing sleep over it. Zzz.

To protect the sovereign bodies of those corporations now wildly threatened by this new breed of pirates, many zany schemes have been hatched. It has often reminded me of Rube Goldberg, who was famously incapable of doing very simple things like opening a door without the aid of preposterously complicated machinery.

To be fair to our corporate overlords, however, the task set before them is substantially more difficult than you or I would find opening a door. In fact, given that the program of stamping out human creativity and ingenuity has not been going nearly according to schedule, many have been so audacious as to suggest that their task is impossible. As a kind of funny joke, and keeping in the spirit of corporate creativity, a series of independent like-minded corporations developed systems of mandatory access control to protect things such as television commercials and pornography.

None of these systems, to date, address in principle (that is to say, some may address some elements on accident) what is called in the archaic fashion fair use. This queer expression is part of United States copyright law which is meant to suggest that:

Notwithstanding the provisions of sections 106 and 106A, the fair use of a copyrighted work, including such use by reproduction in copies or phonorecords or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use the factors to be considered shall include:

  1. the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
  2. the nature of the copyrighted work;
  3. the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
  4. the effect of the use upon the potential market for or value of the copyrighted work.

The fact that a work is unpublished shall not itself bar a finding of fair use if such finding is made upon consideration of all the above factors.

Copyright Act of 1976, 17 U.S.C. ¶ 107

2. Ouroboros

Which is all to say that if you or I were to purchase a song from the iTunes music emporium over the internet, we would be able to play it on exactly one specific platform, namely, the one we authorized. So, if we were to perchance have a non-Apple network music player attached to [our] stereo(s), we would be out of luck. This is a nonsensically limiting phenomenon, which has irritated a great many legal purchasers of music and or video content. Almost across the board, these "fair use"-related restrictions have lead to the technological breakthroughs which open the door for a flood of piracy on the so-called protected files.

So whatever, this point has been talked to death, and it seems clear at this point that one side isn’t going to change the other’s minds. The technology cycle is also obviously ouroboral (is this a word?). The question I want to insist on at this point is whether or not this dialogue has ever been allowed to play out on something like fair terms. Let’s take as an example of what I’m talking about a famous dialogue between the American philosopher John R. Searle and Jacques Derrida. Ostensibly what happens here is this: These two guys have very different thoughts about the world, and about language, which they are talking about specifically in this case through the lens of the analytical staple (this is a mixed metaphor) J.L. Austin. The content of this bitter exchange is not what interests me so much as something that I percieve to be an underlying incompatibility or better, a disinterest on the part of Searle that leads to something like a dogmatic barrier. Here’s how it goes:

Derrida writes an article about J.L. Austin, like I said, and it’s called Signature, Event, Context. Searle, who feels himself an heir of Austin, comes in to defend him against the "lunatic fringe" (he literally refers to Derrida and the estimable Richard Rorty in this way here). Ok: The point, and skipping right to Searle’s first response to SEC: (SEC itself is a complicated text and I don’t really think we can do it justice in the scope of this document - suffice it to say it makes considerable headway in the "reframing" of Austin). We are going to quote Marcel Dascal here, who says this very well:

In his ‘Reiterating the Differences: A Reply to Derrida’, Searle makes it quite clear that he doesn’t want to play the game in the ground and by the rules introduced by Derrida’s reframing of Austin. He presents his task in terms of ‘correcting’ Derrida’s mistakes - particularly those that led him to present a picture of Austin that is ‘unrercognizable’ (p. 204). He undertakes to do so by pointing out Derrida’s equivocations, by returning to well-established and familiar distinctions, and by denouncing invalid arguments. In so far as he, along with other analytic philosophers, assumes that these requirements form a universal and neutral baseline for any serious philosophical debate, seeing to it that they are fulfilled is an operation that might be properly called ‘deframing’. For those who doubt the assumption of a frameless neutrality and universality, the operation might be seen simply as an attempt to return to an arena considered (by Searle and others) to be the appropriate frame to discuss Austin, an arena where Searle’s proven skills can be put to full use.

"How Rational…" International Journal of Philosophical Studies Vol.9(3), 323-24.

Now, it seems clear to me that anything like "frameless neturality and universality" is at the bottom line dogmatic, but this actually ends up not making a difference as Searle’s first reply is fraught with allusion to "rules" - it begins and ends with such allusions. Dascal again: "It is because the proper rules for understanding and stating Austin’s position are not followed that Derrida’s confrontation with Austin ‘never quite takes place’" (326). In other words, according to Searle, Derrida’s engagement with Austin "never quite takes place" precisely because it does not follow some preset guidelines for the way a thinker might interact with Austin. Radically novel thought is excluded as a possibility from the start: Dogma.

3. The Zero sum

I have heard some variation of this idea dozens of times in relationship to computer piracy: "Intellectual property theft is (or is not) a zero-sum game." In general, when corporate types say: Computer piracy is a zero sum game, what they are really saying, as I understand it, is that the loss of the owner of the stolen copyrighted media is equal to the gain of the pirate. And what is this supposed to tell us? Let’s take an example: If you were to purchase a copy of a compact disc, which I recommend that you do at some point, you would ostensibly be making this trade: $15, say, one half or one or two or three hours of your labor for a piece of musical media. The reason that trade works in general is because this is categorically not a zero-sum gain - which is to say, that both you and the producer of the purchased disc feel like you are getting a positive deal. (Elementary arithmetic: a positive number plus a positive number never equals zero.) Again, the suggestion of major retailers of media is that computer piracy, much like piracy on the high seas, amounts to a zero-sum game in favor of the pirate - the company loses as much as the consumer gains. And of course, there have been many very fine rebuttals to this argument and either side may or may not seem whiny or silly to you personally.

Now: Is any of this "zero sum" stuff even reasonable in the scope of the discussion? Why or why not?

But first (and really, this is my answer to that question), why are we allowing the scope of this dicussion to be so insanely limited? In other words, isn’t the possibility of the zero-sum gain extant entirely in a de jure economic plane of immanence, or, isn’t it that an actual zero-sum situation occurs only in an arbitrarily Cartesian (or whatever the economic equivalent of that is, Smithian?) economic setting, a setting, I should note, that is defaulted on even in the most likely candidate at least ontologically-speaking, the math machine itself - the computer. The evidence of this bankruptcy seems clearly provided mutatis mutandis the possibility of p2p technology itself! That is, in an economic climate of competition between digital piracy and drm technology, an economy decidedly unlike the one that produced a certain economic "rule set" that favors and protects copyright holders, this rule-set is still the only model we use to think this problem. It fails to question, from the get-go, the legitimacy of an entire model of trade and ownership (a model that things like p2p and open source software undermine on technological merit alone) - a rule-set which has, I might add, become the watermark for the entire world-currency system to begin with in a "virtual" economy extending far beyond the reach of networked technologies. The zero sum is, in other words, basically an extremely complex and internally differentiated concept in an economy in which value is assessed arbitrarily based on dollar hegemony.

To wit: The value of a widget is assessed at x dollars. The value of a dollar is assessed based on keeping the strength of a certain nation state’s economy afloat. This value translates into other values, y euro, z yuan, etc., but the dollar is the rosetta stone for this translation - there is no actual good or service-based value involved. Value here is pure politics. So, we are used to purchasing one widget for x dollars, where for us, as the consumers, x dollars has the value of b hours of work. Our piece of the bargain is that we work b hours in trade for, in the end, one widget. The owner of the copyright’s assessed value of the one widget, again, x dollars, is based in a sort of nebulous space whereby they determine what how big x can be without causing us to stop thinking that it’s a fair trade. “X” is apparently to be set at a higher value in the instance when a company’s profits suffer from the supposed loss based on piracy - that is, people who steal media with no intention of ever purchasing it. So who suffers? The legitimate consumer, and the company: namely, if, to prevent piracy, a company revalues its widgets at a higher cost to the consumer, what is the incentive for an extant pirate to start purchasing widgets? Hence, DRM: In which companies try to protect their intellectual property by creating software that basically walls off dozens of legitimate uses in order to prevent the few illegitimate ones. Which, if you remember what happens back up at the top of the page or about a year ago, doesn’t work in the first place. Anything like whether or not this all amounts to a "zero-sum game" would have to be worked out on an impossibly large scale.

So it’s not that it’s "wrong" per-se to argue one way or the other about whether digital piracy is (not) a zero-sum game, it’s just that it’s the wrong argument to have. The argument is really, in the first place, about the world economy and the status of the arbitrarily assessed value of a widget versus the actual labor cost to the legitimate consumer. It is unrealistic and unreasonable to think that people are going to put up with this forever, which is what digital piracy stands for - we will note here that almost as soon as the technology became widespread, it reached critical mass. Which is not to say that I advocate piracy - I don’t - but rather, that I’d avocate a discussion that steps back once or twice to take a look at the real issue here, the same absolutely political issue that we’ve always had: a hugely disproportionate assessment of the value of thing a, an seemingly insatiable desire to eradicate fair use and property-ownership by the individual, and of course, a set of not only archaic and corporate-friendly but completely unfair copyright laws.

Closing this (now getting long) document up, let’s say that framing computer piracy in terms of a fair economic model renders it ‘unrecognizable’, as was Derrida’s Austin to Searle. Of course piracy is wrong, but it is only wrong so decidedly in such a ‘recognizable’ economic framework - that framework where a corporation’s skills can be put to full use.

A justice love will not annul

Last year, as Cardinal Joseph Ratzinger, Pope Benedict XVI wrote this document to commemorate the occasion of the 60th anniversary of the Allied forces landing in Normandy in 1944. It is, as one might expect, an almost stoically well-written text. Toward the front:

They [the first-generation post-war politicians] did not want to found a state upon religious faith, but rather a state informed by moral reason, yet it was their faith that helped them to raise up again a reason once distorted by, and held in thrall to ideological tyranny.

Across Europe ran a frontier, and not just across our continent, but dividing the entire world. A great part of Central Europe and Eastern Europe came under the domination of an ideology that subjected state to party, in the end, effacing the difference. Here, again, the result was the rule of lies.

He writes then that if post-war Europe experienced a period of relative peace, South-East Asia, the Middle East, most of Africa, and parts of the Europe on the eastern side of “the frontier” experienced nothing less than a sustained and bloody arc of armed conflict. He suggests that two (somewhat) new symptoms seemingly common among these disperate conflicts were:

  1. The collapse of the “cohesiveness of law” - that is, the usurping of conscience-based order by “the cynicism of ideology”, which he corrleates at least in part with the interests of big business. The “good,” he suggests, “is shoved aside by the expedient, and might setup in the place of right.”
  2. The phenomenon of terror, which, as he notes, “so often has its source in standing injustice, not addressed by effective measures.” The forces which have access to “right and law” of course must also have access to “carefully calibrated force,” to combat the phenomenon, but, he firmly intones, “is it important to vouchsafe forgiveness in advance, in order that the circle of violence may be broken.”

He continues, “In all these cases it is important that no one particular power act as the champion of justice. All too easily can interest interfere with action, and contaminate one’s view of what is just. Most urgent is a genuine jus genitum, free from hegemonic predominance and action which follows from it: only thus can it remain clear that what is at stake is the defense of collective law and right, and those also of them who stand, so to speak, on the other side.”

A secular viewpoint of enlightened reason, he says, is opposed to a fanatical fundamentalist religious viewpoint. There are pathologies of reason (say, Pol Pot) and of religion. Both pathologies “are life threatening for peace - indeed, in an age of global power structures, for humanity as a whole,” a formation that I think Derrida himself would have liked. In the following jarring short paragraph, he continues:

To a reason fallen ill, all recognition of definitively valid values, all that stands on the truth capacity of reason, appears finally as fundamentalism. All that remains is reason’s dissolution, its deconstruction, as, for example, Jacques Derrida has set it out for us. He has “deconstructed” hospitality, democracy, the state and finally, the concept of terrorism, only to stand in horror in the face of the events of September 11th. A form of reason that can acknowledge only itself and the empirical conscience paralyzes and dismembers itself.

The fact that he seems to accuse Derrida of relativism must be put aside. It is an old and long-standing tradition to imagine that Derrida or deconstruction in itself clearly indicates a sort of ethical relativism. Doubtless, there is more to be said, but not here. Back to the Cardinal’s ontology: “God himself is Logos,” he says, “the rational first cause of all reality, the creative reason out of which the world came to be, and which is reflected in the world. God is Logos - Meaning, Reason, Word, and so it is through the way of reason that man encounters God, through the espousal of a reason that is not blind to the moral dimension of Being.” This theology will be the condition of possibility absolute and sovereign reason and absolute justice, which is commented on shortly in the amazing formulation, “There is a justice love will not annul.”

Again, ontological arguments between the Pope and Derrida are unproductive, and at any rate I am unqualified to arbitrate them. But, is there a justice love will not annul? What a question. I really don’t know how to even begin to think it. But it is now being thought. I am intrigued.

Meanwhile, I think that short of what might be called their respective theologies, Benedict might find himself much closer to what might be called the ethical Derrida than he seems to think. This fact is littered through the above quotes. It is clear. From one small example:

I believe it is necessary, by way of a philosophical, historical analysis, to deconstruct the political theology of sovereignty…But at the same time you shouldn’t think that you must fight for the dissolution pure and simple of all sovereignty: that is neither realistic nor desirable. There are effects of sovereignty which in my view are still politically useful in the fight against certain forces or international concentrations of forces that sneer at sovereignty.

And finally,

My intent here is not anti-religious, it is not a matter of waging war on the religious messianisms properly speaking, that is to say Judaic, Christian, Islamic. But it is a matter of marking a place where these messianisms are exceeded by messianicity, that is to say by that waiting without waiting, without horizon for the event to come, the democracy to come with all its contradictions. And I believe we must seek today, very cautiously, to give force and form to this messianicity, without giving in to the old concepts of politics (sovereignism, territorialised nation-state), without giving in to the Churches or to the religious powers, theologico-political or theocratic of all orders, whether they be the theocracies of the Islamic Middle East, or whether they be, disguised, the theocracies of the West…