More on syllogisms

I can’t say I expected my last post on logic to make into File770’s daily Puppy roundup but it did 🙂 which gives me the tiny excuse I needed to write more about syllogisms and propositional logic.

Syllogism are certainly an attractive logical structure. They provide a neat logical structure in natural language and forming a good syllogism is lying writing a short poetic form. They also provide a neat way of understanding how some apparently plausible logical arguments are fallacious. For example this is a fallacy:

  • All Greeks are mortal
  • Socrates is mortal
  • Therefore Socrates is Greek

Each part is true but the logical connection between them has not been made properly. The form looks superficially correct but the reasoning is off. The study of syllogisms allowed Artistotle and then later scholars to study not only specific kinds of incorrect arguments but also to establish as key abstract idea that an argument could be logically flawed even when the individual parts looked correct.

Modern (i.e. 19th century) propositional logic looks less amenable to informal arguments. Propositional logic was revived precisely because it allowed for logic to be formed like algebraic equations. However to understand the power of propositional logic it is worth considering it in terms of informal arguments.

I’ll start with a syllogism:

  • All SJWs lie
  • Camestros is an SJW
  • Camestros lies

As I mentioned in my last post, this syllogism has some serious advantages over propositional logic. I’ll recast it in an pseudo-algebraic form of a kind of informal propositional logic.

  • statement P: all SJWs lie
  • statement Q: Camestros is an SJW
  • statement R: Camestros lies
  • (P and Q) imply R

In propositional logic we’ve lost the connection between the individual terms. We would have to unpack each statement to see why P and Q together would imply R. We’ve gained comparatively less it would seem: (P and Q) imply R shows a general structure – all syllogisms follow this structure (but not everything that follows this structure is a syllogism). The difference is we can now see two important logical operators “and” and “implies”. How is that an advantage. Try a different syllogism(ish)

  • All SJWs are liars or stupid (or both)
  • Camestros is an SJW
  • Camestros is either a liar or stupid (or both)

Now imagine we discover that Camestros is not stupid. Unfortunately this additional information doesn’t help us. We can throw the information away and just revise our syllogism to avoid the bit about being stupid but this is a rather ad-hoc approach to logic. Propositional logic handles this new information more easily.

  • statement P: SJWs are stupid or liars
  • statement Q: Camestros is an SJW
  • statement R: Camestros is a liar
  • statement S: Camestros is stupid
  • (P and Q) implies (R or S)

In other words if SJWs are stupid or liars then Camestros is either stupid or a liar. Now let us add in the additional information.

((P and Q) implies (R or S)) and not S) implies R {where not S means “it is not the case that S is true”}

We gain a some important things. Firstly even this simple toy argument is more complex than a syllogism but we also have a form of argument that allows us to consider particular new evidence.

Syllogisms work by arguing from the general to the particular. This can work well if you have some fundamental truths and some reliable connections between them. However, for those of us without access to the fundamental truths of the universe, we often have access to particular truths more readily than general principles.

Consider this syllogism:

  • All SJWs are stupid
  • Camestros is an SJW
  • Camestros is stupid

Now we check and find that Camestros is not stupid. Now if we reason with syllogisms we can still cope with that because we know that therefore at least one of the initial two clauses must be faulty. However we can’t express this in a syllogism. Propositional logic allows us to make this more explicit.

((P and Q) imply R) and not R) imply not (P and Q)
In words: If all SJWs are stupid and Camestros is an SJW is a liar then Camestros would be stupid, but Camestros is not stupid so either it isn’t the case that Camestros is an SJW or that SJWs are all stupid (or both).

Meanwhile on the Internet you may well encounter people arguing syllogistically become somewhat intractable. i.e. they take the initial clauses of their syllogism to be axiomatic truths and hence the conclusion HAS to be true. Contrary evidence about their conclusion is therefore hard to process.

Mind you that is very much a mistake that Aristotle would NOT have made easily.