Part 1: A demonstration of deductive logic
If Violet is a mathematician, then she is smart.
If Violet is not smart, then she is not a mathematician.
If Violet is not a mathematician, then she is not smart.
If Violet is smart, then she is a mathematician.
This little poem illustrates a basic set of logical forms. In order, they are the statement, its contrapositive, the inverse, and the converse.
There are three key insights here.
- Any statement implies the contrapositive, and the contrapositive implies the statement. They go together - if one holds, then the other is guaranteed to hold. They are saying the exact same thing.
- A statement does not imply the inverse or the converse, as you can see in the poem.
- The converse is the contrapositive of the inverse. That basically means that we can call If Violet is not a mathematician, then she is not smart "the statement," which will make If Violet is smart, then she is a mathematician the contrapositive. Of course, just because we've changed the "roles" of these statements doesn't mean they've become true! We've simply made an illogical statement. But if it were true - if we lived in a heaven/hell where all smart people were mathematicians - then these statements would "go together" and mean the exact same thing. And our original statement and contrapositive would become the inverse and converse, respectively. (Yes, this means that the inverse implies the converse!).
Once you know this, you can use this "logic machine" as a tool to restate informal arguments in multiple ways in order to understand it better, find the weak points, check for consistency, and see if it accords with your understanding of how the world works.
Only fair to make a victim of my own writing. In Let the AI teach you how to flirt, I wrote the following statement:
If you can get your partner to engage in their own natural flirting style, and get good at detecting it, then you can guess their intentions with much more confidence than the average person is capable of.
If somebody else wrote this post and I wanted to make sure I understood it, I might put it through the logic machine, being a bit generous about the word choice as I modify the statement into the contrapositive, inverse, and converse.
Statement: If you can get your partner to engage in their own natural flirting style, and get good at detecting it, then you can guess their intentions with much more confidence than the average person is capable of.
Contrapositive: If you can't guess your partner's intentions with much more confidence than the average person is capable of, then either you're not engaging them in their own natural flirting style, or you're not good at detecting it.
Inverse: If either you're not engaging your partner in their own natural flirting style, or you're not good at detecting it, then you cannot guess their intentions with much more confidence than the average person is capable of.
Converse: If you can guess your partner's intentions with much more confidence than the average person is capable of, then you can get them to engage in their own natural flirting style, and are good at detecting it.
What I notice on a phenomenological level is that after rewriting this statement in the contrapositive, inverse, and converse, I lose track of the point. When I wrote the original article, I had a visual, intuitive model foremost in my mind. It was based on my own memories, as well as the article the blog post was inspired by.
After converting the argument into a logical format, I have a very hard time getting back on the horse. It's hard to see how the statement I picked out leads to the next one, or follows from the previous one.
This is because my article is based on inductive reasoning, rather than deductive logic. You can't symbolically compute the next statement from the previous one.
That seems like a problem. Surprisingly, it's almost like deductive logic has temporarily killed my ability to figure out how to flirt...
Well, maybe that's not soooo surprising.
It seems like it would be useful to find a way to distinguish between whether engaging your fast, inductive reasoning mode, or your slow, logical mode, would be more useful.
Part 2: Deductive logic saves us from misfires of intuition
Recently, I had a fit of anxiety in applying to graduate schools. My anxious thought went something like this:
I don't know exactly what I want to research in grad school. But I know that what I can research effectively will depend a lot on the faculty's expertise. What if I get forced into a topic that isn't actually the most effective thing to research? What if the foundation in that ineffective research topic that I build in my MS program then forces me to continue pursuing it at the PHD level? What if I'm jumping into a program that will determine my future career path? What if I need to figure out the most effective thing to research in before applying, but I can't figure that out until after getting into grad school? Augh!
No worries if you didn't quite follow - it was an anxious headspace.
I found the application of deductive logic here to be very helpful. Shoehorning these thoughts into an "if... then" format and shifting them around to the contrapositive, inverse, and converse forms forced me to slow down and consider carefully the statement I was making, wording definitions with care. It went something like this:
The outcome of my MS depends on two things: the program, and my work. The outcome of my MS determines what I can attempt to accomplish in my next role. Therefore, if the program determines my work, then it determines my next role in school or work. If the programs I enter determine my work at every stage of my working life, then choosing my MS program determines what I can attempt to accomplish for the rest of my life.
Statement: If the programs I enter determine my work at every stage of my working life, then choosing my MS program determines what I can attempt to accomplish for the rest of my life.
Contrapositive: If choosing my MS program does not determine what I can attempt to accomplish for the rest of my life, then the programs I enter do not determine my work at every stage of my working life.
Inverse: If the programs I enter do not determine my work at every stage of my working life, then choosing my MS program does not determine what I can attempt to accomplish for the rest of my life.
Converse: If choosing my MS program determines what I can attempt to accomplish for the rest of my life, then the programs I enter determine my work at every stage of my working life.
Having my original anxious thought forced into this logical format allowed me to stand apart from it, and give me a gut check on whether the statement, or the inverse, felt more true. The latter felt much more true than the original statement. Of course, my choice of program is influential, but there must be a meaningful degree of freedom within that. From there, I was able to proceed with my planning in a much more calm and constructive manner.
Part 3: Synthesizing babble and prune
For those familiar with the framework, deductive logic seems to be a form of pruning, while my anxious stream of thought was babble. Sometimes, as in the case of my flirtation article, it is valuable to allow ourselves to babble more and prune less. In the case of my graduate school planning, I needed to prune. If we do one when we ought to do the other, it'll be hard to make intellectual progress. We might draw a blank and fail to even start understanding the issue at hand, a failure of babble. Or we might make decisions based around anxious babble: an idea that feels wrong, but seems like it might be a frightening truth of how the real world works.
Clearly, babble has to come first, or else there would be nothing to prune. Intuition has to precede logic in the sequence of thought.
So the question is, "when is it time to prune, and how do we go about it?"
My guess is there are many forms of pruning. As I wrote this post, I would frequently reword sentences, adjust examples, and sometimes throw out whole sub-ideas that seemed to be leading in the wrong direction. I didn't know where I'd end up when I started writing. So there's constant, lightweight pruning going on as I babble, and it's not all logical, deductive reasoning. Sometimes, it's just choosing a different word, or a gut feeling that leads me to say "nah, I can do better than that," or a new idea that's better and wants to replace an old idea.
In fact, I almost want to introduce a distinction between "pruning," as a form of criticism, and "pivoting," as a change of direction. Pivoting is what I do as I go along with my babble. It's a part of babble that looks a bit like pruning, but isn't. It is part and parcel of developing an idea from the hazy intuitions in my head into a legible form. Even offering counterexamples can be a pivot, a form of babble. It leads the conversation forward.
Pruning, on the other hand, is when I've gotten a thought down, or even a whole series of thoughts. Now, I stand back, and turn them around and around. What am I actually saying here? Are my definitions clear? What if I put a statement through the logic-machine, or shoehorn it into a series of math-like "if... then" statements, just to see what it spits out? Does the cited article actually support the claim that uses it as a reference? This takes place on the level of symbolic manipulation, and kills the progress of thought. It is not generative. It won't allow you to build a map of any more territory than you've already charted. It only allows you to check the symbols you've already put down on paper for accuracy and consistency.
Observe, babble, and prune
I also want to add one more category on top of babble/pivot and prune: observe.
If I sit down and say "I want to babble about X," I might have a very hard time. I can't just demand that my brain come up with all kinds of thoughts about X. And of course, if I can't babble, then I can't prune. What if I wanted to babble about X, though? How do I get this party started?
The first thing is to observe X. This might seem obvious, but it really isn't. It has powerful implications for building good, productive mental habits. Building a strong practice of observation is a foundational skill for a mind, just like being able to babble a model, or prune it.
I want to draw a big circle around observation. It encompasses many things: direct sensory perception, scientific experiment, memory. Even scholarship counts, since it is on some level an attempt to transmit a direct observation or mental experience. Just absorbing observations that seem related to X has to come first before we can start to babble.
My guess is that building a systematic habit of sequencing observe-babble-prune in the proper order would be the fundamental skill for any human being. Probably most people do it on an intuitive level, but having an explicit 1-2-3 map seems like a very useful thing to have.
When I'm trying to think about X, I usually start by trying to babble. In the future, I'm going to try to take it a step back and start to observe.
When I read other people's babble, I know that it will usually include some observations. I'm going to try and bear in mind that their babble didn't start as babble. In fact, if it's good babble, it certainly started with many observations, the vast majority of which are probably not included in the babble. There's a pre-screening process of picking and choosing, synthesizing, and putting forth, that happened in the mind of the writer or speaker before they put words on a page. I haven't gone through that same period of observation, and I can't reconstruct it from the babble.
Conclusion: What to do with babble?
If babble can't let us step into the mind of the author, then what's it good for beyond entertainment and parroting the author's opinions?
- Babble can function as an observation-inspiring machine. It suggests a new way of observing the world: new books to read, new patterns to recognize, new experiences to pursue, or old memories to revisit. It can also transmit select observations directly to you, whether it's a story, a mathematical proof, a graph, a recipe, a recommendation, or a picture in a picture-book.
- Babble can function as a babble-inspiring machine. If it has done the work of inspiring your own observations, you might expand on it, offer counter-examples, criticize it fairly or unfairly, or crack a joke.
- Babble can function as a pruning-machine. It goes into your mind, finds an idea, and addresses it on the level of sheer deductive logic. It points out direct contradictions on a symbolic level, or restates an idea in words that are equally accurate but render it less (or more) sympathetic.
I think it's exciting to imagine a shift from rationality as primarily addressing misfires of the mind - statistical mishaps, cognitive biases, and so on - to rationality as a constructive practice. I would feel comfortable offering "observe, then babble/pivot, then prune" as a comprehensive prescription for literally any question or topic that anybody wanted to explore.
All the other techniques of a rational practice fit within it. My guess is that great mappers of the territory understand clearly how to move between all three phases and have a big toolkit of practices for each one. They don't shirk observation before babbling, and they have a lot of background observation beyond what they choose to convey in their written work.
This framework is neatly recursive and can be applied somewhat mechanically: "I want to get better at observing. How do I do that? Well, first, I need to observe how I observe. And to do that, I need to observe what I observe. Oh, OK - what specifically do I want to get better at observing?" and so on.
I can't promise that it's a magic bullet, but I plan to use it as both a lens when I read other people's writing, and as a guide when I'm following my own curiosity.