Notes on Trivalent Logic

I've been reading a lot about trivalent logics to help with understanding Peter Vranas' work on a three-valued imperative logic. One thing in particular that I'm puzzled by is how we should understand negation in a trivalent logic. In this post I'm going to write some thoughts on this problem.

Consider that there are only six out of the 27 possible unary truth functions that are non-degenerate. By non-degenerate I mean that they do not reduce the number of possible outputs, given an arbitrary input. These are as follows:

INPUT IDENTITY 1.1 1.2 1.3 2.1 2.2
- - - 0 1 0 1
0 0 1 - 0 1 -
1 1 0 1 - - 0

It's a little simpler to tell what's going on here when you graph the relationship between the truth values. Graphically, the idea is that if each truth function is a relation between 3 truth values, then these six are the only relations where there is only one incoming edge and one outgoing edge for each truth value. In the case of identity and 1.1-3, some of these are the same edge.

Kleene, Priest, and others interpret 1.1 as negation (depending on how we interpret the truth values FALSE and UNKNOWN) but I'm not sure this is right. For one thing, this is what lets us preserve double negation elimination as a theorem/rule and with it DeMorgan's laws, etc. I think for a genuinely trivalent logic we should interpret either 2.1 or 2.2 as negation.

We can start to see the effects of this interpretation by looking at the binary truth functions. Since there are almost 20,000 of them in a trivalent logic, we're going to have to rely on symmetries to help make sense of things.

1 2 3 4 5 6
000000 000000 ------ 111111 111111 ------
0----0 011110 -1111- 1----1 100001 -0000-
0-11-0 01--10 -1001- 1-00-1 -1001- 01--10
0-11-0 01--10 -1001- 1-00-1 -1001- 01--10
0----0 011110 -1111- 1----1 100001 -0000-
000000 000000 ------ 111111 111111 ------

This is a collection of the characteristic truth tables for 24 binary truth functions in trivalent logic, divided into six groups of four, based on symmetries between them. Starting with the leftmost set of four (labeled '1'), the top left 3x3 square of truth values is the characteristic truth table for conjunction. Moving counterclockwise in that set of four, we see the characteristic truth tables for conjunction, nonimplication, sheffer stroke (nand), and converse nonimplication. In the fourth collection we have peirce's arrow (nor), converse implication, disjunction, and implication.

Likewise, the relationship between the unary function 1.1 and the binary functions in the first and fourth collections is as we would expect. Negating one or the other inputs rotates around the collection, and a negation in front of the function moves us to the other collection (and, importantly, back again).

What about the other collections of binary truth functions? These are truth functions that we can produce by applying the unary functions in 2.1 and 2.2. These, in some sense, rotate us through the pair of characteristic truth tables in (5 and 6) and (2 and 3), respectively.

OK, so what does all this show us? Honestly, I'm not sure. But I think that these truth tables show that there is a kind of symmetry that would give us two unary truth functions that capture many, and perhaps all, of the properties of negation that we care about. But instead of negation, in a trivalent logic we have left-handed negation and right-handed negation, or something like that.

These "handed" negation functions would have some interesting properties. First of all, they would cancel each other out:

For any proposition P, LRP = RLP = P.

Also, they would obey a triple negation elimination rule:

For any proposition P, LLLP = RRRP = P.

This would imply a modification to the DeMorgan's Laws for operator duality. Duals would no longer be defined in the typical way, and instead each operator would have two intermediate stages (not represented in the truth tables above) that it would have to pass through on the way to its classical dual.

I'm going to have to spend some more time on figuring out what these intermediate steps are, but it seems plausible to me that this is a more thoroughgoing trivalent logic than the traditional interpretation.

Backwards Design

Backwards design is a method of approaching course design by starting "at the end" and working your way through the course backwards, so to speak. The basic idea is that you should begin the process by asking the following four questions (from Ken Bain's What the Best College Teachers Do):

1) What should my students be able to do intellectually, physically, or emotionally as a result of their learning? 2) How can I best help and encourage them to develop those abilities and the habits of mind to use them? 3) How can my students and I best understand the nature, quality, and progress of their learning? 4) How can I evaluate my efforts to foster that learning? (Bain 49)

Fair enough. It seems good to me to begin designing a course by figuring out how people should be different after having interacted with the course. This comports with design advice in other areas. How should players feel at the end of a game? How should listeners feel at the end of a play, podcast, or song? How should readers update their beliefs at the end of an article or book? How should people interact with your park bench, tea kettle, headphones, traffic intersection, or whatever? How should the world be different after you've made your thing?

My concern about backwards design is that it presumes that we can predict how students will change during the semester. Maybe the thing that will change for them after interacting with your course is that they will have been evicted, or their kid will be sick, or they'll lose a parent. Maybe the most important thing about the semester won't be your course. My point isn't that life events are outside of your control or that they are unpredictable. My point is that you can't predict how your course will interact with the rest of a student's life.

Obviously as instructors we should consider who we are teaching, what their lives are like, and the material and social circumstances of their lives as they participate in the class. But what I don't think we can predict with any accuracy is how the course will effect them, either in the short term or the long term. This is why I think that the best approach to course design accommodates a largely individualized approach to education.

Accelerated Andragogy

As part of my ongoing teaching training I'm taking an accelerated course in andragogy -- or adult education -- and I'm going to use this space to talk about some of what I learn.

First, a historical note. "Andragogy" was coined by German education theorist Alexander Kapp in 1833 to describe Plato's theory of education.

Malcolm Knowles picked up the term and presented five assumptions made by andragogues(?):

  1. Self-Concept: As a person matures their self concept moves from one of being a dependent personality toward one of being a self-directed human being.
  2. Adult Learner Experience: As a person matures their accumulates a growing reservoir of experience that becomes an increasing resource for learning.
  3. Readiness to Learn: As a person matures their readiness to learn becomes oriented increasingly to the developmental tasks of his/her social roles.
  4. Orientation to Learning: As a person matures their time perspective changes from one of postponed application of knowledge to immediacy of application. As a result their orientation toward learning shifts from one of subject-centeredness to one of problem-centeredness.
  5. Motivation to Learn: As a person matures the motivation to learn is internal. (Knowles 1984:12)

These assumptions also suggest four principles that instructors should adhere to in designing their courses.

  1. Adults need to be involved in the planning and evaluation of their instruction.
  2. Experience (including mistakes) provides the basis for the learning activities.
  3. Adults are most interested in learning subjects that have immediate relevance and impact to their job or personal life.
  4. Adult learning is problem-centered rather than content-oriented. (Kearsley, 2010)

It seems to me that the andragogical approach is less paternalistic than its pedagogical counterpart. It also seems to be less driven by a need to instruct values, since it's presumed that the adult learners already have a relatively fixed set of values.

I'm not sure to what degree any of Knowles assumptions are borne out, at least insofar as they apply to my own experience as a student. I guess I see how my learning has become more independent and informed by more experience. But just that fact, on its own, doesn't suggest to me that my own education ought to be any different. In fact, it might be better to reframe at least some aspects of my education in the opposite direction. Try to learn things for their own sake or to ignore my past experience.

I'm also curious what, if any, explanation Knowles gives for these assumptions. Are they just supposed to be axiomatic? Is there some reason why adult learners tend to be this way?

Recent Writing

Now that the sound of labor chants and bucket drums has died down, I wanted to link to some of the writing I did for the strike. My wonderful partner Ashli Anda and I coauthored a piece on the strike and on the graduate worker/student dichotomy and how it is misleading. I also wrote a short piece while I was sitting on the tile floor outside of the University president's office for the second day in a row.

Now that the strike is over I plan to get back to writing over here more often.