# Fundamental equations in Philosophy and Logic

For the natural sciences it is easy to indicate the most important fundamental equations. If I want to explain Philosophy and Logic to a physicist or a chemist, what are considered the most important equations that define the subject that I should introduce and try to explain?

• There is the Fundamental Theorem of Philosophy of course since there is always a "Fundamental Theorem of...", but it only says that there is a non-countable infinity of "-isms" in philosophy ;-) Mar 10, 2023 at 18:50
• there are no equations underlying philosophy Mar 10, 2023 at 21:56
• "For the natural sciences it is easy to indicate the most important equations that underlie the discipline." I don't believe this is the case. Are the most important equations in physics from QM, relativity, or thermodynamics? What kind of equations are fundamental in the biological or social sciences? Mar 11, 2023 at 0:56
• Another fundamental principle of philosophy is that to every difficult question there is a simple answer... and it's wrong. ;-) Mar 11, 2023 at 5:35
• Equations depend on phenomenon (the way we perceive). For example, you can write an equation adding apples because you take a range of weights and sizes by granted in the definition of an apple (there are no apples the size of a car). But philosophy is beyond such assumptions, so, equations don't apply in philosophy. Same happens with logic: you can describe phenomenon using logic, but you can't describe logic using logic, that would be circular. Mar 11, 2023 at 10:06

Philosophy can be amorphous enough to be "about" questions more than axioms/theorems. There are seemingly plausible symbolisms that can end up yielding undesired (by some people) results, like the converse Barcan formula: this variously says that {there exists a possible X, so possibly there exists an X} or, for necessity, you can get {there exists a necessary X, so necessarily there exists an X}.

But there are some results that are strongly accepted by, I think, most of the people who are aware of the relevant mathematical questions, and who are interested in answering them. Maybe more disagreement would manifest eventually, but "for now," a possible counterexample to "not even weak consensus" in philosophy: a "general axiom" of normal modal logic. It's perhaps an obscure-seeming matter, but anyway, another arguable example comes from second-order logic: there are logical structures that are in some sense reducible to structures in a different-order logic. And this doesn't seem controversial, as far as I know.

Most of what is currently called philosophy has very little to do with learning how to do much formal logic, and everything to do with learning what historical people who also didn't know how to do much formal logic had to say about things. Most of those people died before most of this stuff was discovered, so it's hardly their fault - but it is what it is. Learning how to work with those historical ideas has value, but learning formal logic is something else - mathematics.

The core laws of formal logic are identity: A equals A; non-contradiction: A does not equal not-A; and the excluded middle: either B equals A or B does not equal A.

From there you'll want to build up through binary logic, with which almost every physicist and many chemists will be intimately familiar from writing computer code and/or circuit design. Wikipedia has a comprehensive guide to the binary logical operators, but learning to use them properly will take a lot of repetitions on practice problems.

The language of logic is mathematics, and the chances are that the physicist and chemist are pretty good at it. All scientists and philosophers should share a grounding in statistics, which you can't do properly without at least basic calculus.

Philosophy makes outsized use use of Bayes' Theorem, which would probably be intimately familiar to a biologist, but may be new to or forgotten by a physicist or chemist. Bayes' theorem doesn't require any math beyond basic algebra in and of itself.

To go farther in formal logic (even for application to real-world problems) one will want a basic foundation in abstract mathematics, by which point you'd probably be ready to start a masters degree program in mathematics, but may never have read a word of philosophy so labeled.