# Help me go from English to Logic

The positive-definiteness axiom used for just about all the definitions of inner-product spaces that I've seen goes like this:

$$\langle \mathbf{x},\mathbf{x}\rangle \ge 0 \text{ with equality only for } \mathbf{x}=\mathbf{0}$$

For example here. I'm having trouble rewriting that statement using logical connectives. Is this it?

$$\langle \mathbf{x},\mathbf{x}\rangle = 0 \Leftrightarrow \mathbf{x}=\mathbf{0}$$ and

$$\langle \mathbf{x},\mathbf{x}\rangle > 0 \Leftrightarrow \mathbf{x}\neq\mathbf{0}$$

I feel that my second statement can be scrapped, since it is basically the contrapositive of the first. (I was looking for a "minimal axiom" in the sense of assuming the least.)

In general, I feel that "property $P$ holds only for $x$" means "$x$ iff $P$". Am I right?

• The two statements that you wrote are equivalent (given that $(x,x)\geq 0$) and so one is indeed superfluous. This is the correct statement essentially. You also need to point out that $(x,x)\geq 0$. A more complete answer is the following: $$(x,x)\geq 0\land ((x,x)=0\iff x=0).$$ Yes when we say that this holds "only when" or "exactly when" then this is translated with "if and only if". Commented Sep 13, 2013 at 15:57
• Thanks. What bothers me about "only when" is that since "when" = "if", I translate it as "only if" which also means "then". Thoughts? (btw, why do you power users leave great answers in the comments where I cannot accept them!? :D Commented Sep 13, 2013 at 16:01
• "Only if" is also used as "if and only if". The first "if" in "if and only if" is there to stress that the implication actually holds. Commented Sep 13, 2013 at 16:13
• @Apostolos ?? "p only if q" I thought meant q is necessary for p, i.e., "if p then q." I don't understand what you write. Commented Sep 13, 2013 at 16:21
• Nevermind. English is not my native language. I've seen this being used differently but in other languages. Just ignore my last comment. Commented Sep 13, 2013 at 17:35

Let $I_{x y}$ denote the inner product operation $\langle x, y \rangle$. Then we have that $$\forall x (I_{x x} \ge 0 \wedge (I_{xx} = 0 \Leftrightarrow x=0))$$
• Why would you introduce the notation $I_{xy}$? Commented Sep 13, 2013 at 16:14
• @sasha Any time that you want to do formal logic, you must have a formal language, $\mathscr{L}$, which is basically just a set of symbols, some of which are constants, some variables, some operators, etc etc., and you restrict yourself to using only those symbols when you translate from natural language to $\mathscr{L}$. I can't say what symbols are usual in mathematics, but in philosophy things like $\langle\$, $\rangle\$, and $\ge$ are not usually taken as part of $\mathscr{L}$ (depending obviously on your application). If those symbols aren't in your language, you can't use them. Commented Sep 13, 2013 at 18:49