Axiomatic systems Revisiting the rules of inference Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see section 2.1, pp 51-62. Example: A theorem and its proof in an abstract axiomatic system: Suppose the following have been either established or assumed: Theorem 1: q (r s) Theorem: If p, then s. Given: p Prove: s (1) p Given (2) q Axiom 1 (3) r s Theorem 1 (4) s Simplification The above is an example in the style of Kay s text. There is a good bit going on there, some of which is implicit and goes unexplained in the text, so the point to this lecture is to work through, clarify, and hopefully answer some questions about what is happening in these proofs. Furthermore, Kay states [p 55] [...] but we do not need to get too fancy here in our terminology. Furthermore, in proofs we just go ahead and use this logical procedure without making reference to it. What Kay is referring to is a rule of inference, some of which we ve already seen. While I d agree that we don t need to do an in depth tour of the branch of logic known as the propositional calculus, it s a bad idea to overlook the rules entirely - do so, and you ll either end up looking at a proof and thinking Now where did THAT come from... or working on a proof and wondering What s a legal step for me to take? The notes provide an outline to some questions and answers. Although there aren t a lot of examples for you to work, please do work through the associated lecture - it ll be very talkative, and I ll use it to elaborate on what s going on.
What s the point? We ve been looking at axiomatic systems in the context of Euclid s geometry; the goal of this section is to examine them at a more abstract level. Recall that an axiomatic system has four components: undefined terms axioms defined terms theorems plus the rules of inference that allow us to progress from the building blocks (the undefined terms, axioms, and defined terms) to the results which are the theorems. Although it gets glossed over in the Kay, what you really are focusing on here are the rules of inference. You won t see much in the way of undefined or defined terms, since we re working at a symbolic level. What you will see are axioms and theorems getting connected to form more theorems in a logical structure. What s the distinction between an axiom and a theorem? An axiom is a statement which is assumed to be a true in the system, and is accepted without proof. A theorem is proven using the terms and axioms of a system as steps in a chain of deductive logic. However, in the examples and homework, this is arbitrary, because the individual problems are set up in a vacuum, with an element of let s pretend - suppose these are the axioms, and suppose that somebody somewhere has already proven the theorems. From your perspective, they could all be axioms. Or theorems. Don t get hung up on it. The theorems are phrased as conditionals. How do I prove a conditional? Assume the hypothesis as a given, and work through until the conclusion appears. Kay immediately breaks each theorem into two pieces, Given: and Prove:. The conclusion is s. Did we just prove that s is true? NO! We just proved that if p is true, then s will follow. We don t actually know if p is true or not (it doesn t appear as one of the axioms or theorems of the system); we only assumed that it was to see what would happen. I ll do an example of this distinction at the end, after I finish picking through this one.
What rule of inference keeps getting used (implicitly) here? Modus ponens. Used a lot, since this is the basic way that an implication works. What you have going on here is p p q q What you see in the proof is the immediate progression from p to q, with Axiom 1 (p q) cited as the justification. Personally, I d prefer a little parenthetical remark ( modus ponens ) when you do this, because you re allowed to use other rules of inference as well. What are the other rules of inference? You ve met some of them (modus tollens, disjunctive syllogism, transitivity) already. I ve posted a complete listing of all the ones you should know. Of special interest are the things you can do with and s and or s. Take a moment to print that off and look at it. Look at it right now. I m going to talk about it. Notice I m using simplification - aka conjunction subtraction - as a justification here.
When am I really proving the conclusion is a fact? Compare these two short examples: Example 1: Theorem 1: q r Theorem: If p, then r. Given: p Prove: r (1) p Given (2) q Axiom 1 (modus ponens using (1)) (3) r Theorem 1 (modus ponens using (2)) Example 2: Axiom 2: p Theorem 1: q r Theorem: r. Given: Nothing at all, other than the axioms and theorems of the system. Prove: r (1) p Axiom 2 (2) q Axiom 1 (modus ponens using (1)) (3) r Theorem 1 (modus ponens using (2)) In the first example, you ve proven the conditional p r. In the second, you ve proven that in this system, r itself is a theorem of the system.
Any other miscellaneous stuff to watch out for? Yep, terminology. And a useful trick. iff is not a typo - it s shorthand for the biconditional if and only if. p iff q (symbolized as p q) means that p and q are logically equivalent. Each implies the other. In the context of proofs, you can prove either one, and the other immediately follows. Kay uses the symbol p for negation (instead of p). This isn t unusual (pick a text at random and it s about 50/50 one way or the other); it is annoying in a geometry text, since is also used for similar. Generally, the contexts (logic vs. triangles) are so far apart that there isn t any confusion. The symbol is used to indicate a contradiction has been reached. (The imagery here is two statements smashing into each other.) And a useful tip - it is sometimes easier to prove the contrapositive ( q p) of a statement (p q) than it is to prove the original. Since a statement and its contrapositive are logically equivalent, this is perfectly legal. I ll be using this in the live example. Finally, proving things in the abstract is the one place I d rather see a two column proof than a narrative. It s pointless to write these in English, since the statements have no meanings, and the focus is on how to formally get from one step to the next.