Impossibility reasoning


Summary: It's typical to teach and use sequential reasoning, but all sequential arguments can be reforumalted as impossibility results. Thinking and presenting arguments in terms of impossibility results rather than sequential arguments can be more fruitful than sequential reasoning.

This is niche topic but I want to write about it because it’s something that I’ve found useful. ‘Syllogistic reasoning’ is sequential reasoning from premises to a conclusion, and it’s the type of reasoning most people use. For example:

  1. If the tax plan will increase jobs then it’s worth passing
  2. The tax plan will increase jobs
  3. Therefore it’s worth passing the tax plan

I prefer to use a slight variant of syllogistic reasoning, which I’ll call ‘impossibility reasoning’. To use impossibility reasoning, you just convert every argument like the one above into a set of things that you can’t have: you can’t have all of the premises and also have the negation of the conclusion. In other words, the argument above is saying that the following is an impossibility set (we’re not reasoning sequentially so the order doesn’t matter):

  • It’s not worth passing the tax plan
  • If the tax plan will increase jobs then it’s worth passing
  • The tax plan will increase jobs

If you can look at this set and see some way to make all of the things in it true then you immediately know that the original argument is invalid. And if you notice that the member of the set that you find least plausible is one of the original premises then you can pick that out and make a counterargument. You’ve also identified your point of disagreement with the original author. (’Actually, I don’t agree that if the tax plan will increase jobs then it’s worth passing - that’s only true if the cost per job created isn’t too high.’)

This might seem like a very minor adjustment, especially in an example this simple, but I find it much easier to work with impossibility sets than with sequential arguments. I also like presenting arguments as impossibility sets because it lets people explicitly see the trade-offs they have to make. Underneath it all, arguments are just statements of the form “you can’t have all of these things”. I think it’s better to present them as such and make your case, but let your reader decide what they want to give up.

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