# Does anyone know a no-nonsense intro to “logic for mathematics” that I can give to a Year 11 student?

I'm looking for material on propositional and first-order logic to give to a Year 11 student that explains how they're used "in practice." For example, I want to be able to write the null-factor law as $$\forall (a \in \mathbb{R},b \in \mathbb{R}) \,ab = 0 \rightarrow a = 0 \vee b = 0,$$ and have the student nod their head and say "yep, I get it" and be able to use this to deduce $$(x-1)(x-2) = 0 \rightarrow x-1 = 0 \vee x-2 = 0.$$ More generally, I want to be able to explain the laws necessary to solve Year 11 and 12 problems using precise logical notation, and have the student follow what I'm saying.

The material should:

1. Be phrased in straightforward language that any clever student in Year 11+ can follow.
2. Get straight to the point and not dawdle too much on the philosophy or analysis of language.
3. Emphasize the kind of reasoning needed in high-school problem solving, as opposed to "proofs." (Although in some sense, everything is just proofs.)
4. Not introduce too many words or phrases that aren't strictly necessary: e.g. the use of words like "proposition, statement, sentence, formula, expression, term, validity, and tautology" should be kept to a minimum.
5. Avoid truth trees. (Don't get me wrong, they're super cool. But inappropriate in this context.)

Any of the following would be fine: a short book, an online video series, or even a free online course (if it isn't too long).

• IMO there is no "magic bullet" that will solve the problem of students who don't yet know how to think clearly. One book that I use for certain classes is "How to think like a mathematician" by Kevin Houston from Leeds. That is not a logic book - it is a book intended to help math students learn to read and write rigorous math, and I think the tone is engaging and accessible. It may be worth looking at. I doubt you will find a book "on logic" that meets the 5 bullets you mentioned. – Carl Mummert Jul 23 '16 at 15:11
• (2) is likely to be a sticking point. Very many attempts to explain logic to beginners fall into the trap of assuming that English words map directly to logical connectives, even going as far as asking students to "translate" English statements that make very little sense in natural language until one realises they've been translated backwards from logic to create the exercise. On the other hand, without tackling that problem head-on, it will be difficult to explain why we don't write things like $ab=0 \iff (a\lor b)=0$. – Henning Makholm Jul 23 '16 at 15:16
• @HenningMakholm, I'd probably try to explain that by pointing out that in $(a \vee b),$ we're applying $\vee$ to real numbers. But since $$\vee : \mathbb{B},\mathbb{B} \rightarrow \mathbb{B},$$ this isn't allowed. – goblin Jul 23 '16 at 15:20
• Some introductory texts on Discrete mathematics (Grimaldi, Biggs, I guess Jonsonbaugh, et cetera) tend to have a short and mainly computational chapter on (mostly) propositional and (more uncommonly) first-order logic. In fact, that's how I learned using practical (formal) logic in high-school (the book was in Swedish though…). – A.Sh Jul 23 '16 at 15:21
• I don't know if it exists, but I'd look for a text that teaches boolean arithmetic the same way that natural number arithmetic is taught. $\text{true} \land \text{true} = \text{true}$ just like $1 + 1 = 2$. Don't bother with quantifiers, not even a little, until propositional concepts are 100% solid. Afterwords, some simple set notation, like $\{\dots\}$ for finite sets, and some set arithmetic problems like $\{1, 2\} \cup \{2,3\} = \{1, 2, 3\}$. Even if such a text doesn't exist, anyone who knows how to use a computer can make worksheets for arithmetic practice problems. – DanielV Jul 23 '16 at 21:09