Replacing truth tables and Boolean equivalences | MathFoundations274 | N J Wildberger

While Propositional Logic is a branch of philosophy, concerned with systematizing reasoning using connectives such as AND, OR, NOT, IMPLIES and EQUIVALENT, the Algebra of Boole provides a mathematical framework for modelling some of this. With this approach we ignore the issue of the meanings of the various atomic propositions, and replace philosophical discussion of the meaning of "true (T)" and "false (F)" with a cut and dried algebraic approach using just 0's and 1's.

With the Algebra of Boole framework, we move away from using Truth Tables and Boolean equivalences to verify statements: it is simpler and preferable to just compute Boole polynumbers for ingredient clauses. We illustrate this technology with specific examples, including the non-obvious Distributive Law, and also Transitivity of Implication, otherwise known as Hypothetical Syllogism.

Correction: On slide 3, the last column (not P) should be 1,1,0,0.

Video Content:
00:00 Introduction
4:49 Imbedding PL into the Algebra of Boole
10:52 Example: The non - obvious distributive law
18:40 Example: Transitivity of lmplication
21:50 Hypothetical syllogism
27:13 Evaluating using the Algebra of Boole

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That last example has made me a believer. What a wonderful system. This really does deserve more prominence in the teaching of logic.

The simplification that this brings to logic is analogous to the arithemetization of geometry. After Descartes, being able to prove many geometric theorems no longer takes flashes of ingenuity--just the ability to translate the problem into a new language where routine methods can resolve it. Brilliant.

I teach a high school logic and computation class--there's already a lot packed into the curriculum, but I'd love to show this to the students after we learn about truth tables.

[cropotkin]

I did Propositional Logic in my Philosophy course 20 years ago. It was clearly an entrenched incantational technology that was crying out to be rationalised and simplified. If only this algebra of Boole insight could mean its days were over.

[KineHjeldnes]

Your videos are such a resource and inspiration! ❤

[Putrycz]

For months I was messing around with truth tables trying to find shortcuts for calculating argument validity. I remember trying to make a "row function" that worked like the way I was doing proof by contradictions, but gave up on it and haven't worked on it since. I can immediately see how your method would be amenable to the kind of computation I was trying to achieve before! I liked your demonstration and am going to have to try out "The Algebra of Boole" for myself. Thank you so much for sharing.

[PeterHarremoes]

I had the same experience when I had my first logic course except that it was at a mathematical department. First we have the normal language. Then we introduce symbols for some of the connectives like "and" and "imply". Then we prove theorems about these formal propersitions involving symbols for "and" and "imply" using imference rules where formal propersitions are written below each other. If nothing is written between the lines it means "and" and if a horisontal line is written inbetween it means implies. Sometimes some dots in triangular configuration also appear. What is really missing is a clear explanation of what we achieve by introcing these different metalevels of description. I think we should focus on the study of mathematical models. One such model is the algebra of Boole, which has important applications that has nothing to do with philosophy. We may also develop and study other mathematical models like lattice theory, which is also useful even if there were no applications in philosophy. If these models can be used to say something interesting about philosophy or the language then it is fine, but one should expect that there are other aspect of philosophy that are not captured by these models and that is also fine. As long as we are studying models that have obvious applications outside philosophy the problem the indtroduction of various meta levels of reasoning disappear. The problem is philosophy where they are only interested reasoning about their own reasoning.

[KipIngram]

Maybe it was just my particular professor, but I don't really recall great emphasis being placed on AND, OR, and NOT as being "special" or privileged in any way. It was just one set of functions from which you could create any function you wished - one "cover" of the logical space. It was always clear there were others, and it was interesting that NAND can do the job all by itself, as can NOR. These are like coordinate systems in physics - no one of them is special. Maybe you have one you're most comfortable using, but that's all it is - a "comfort zone."