# User:IssaRice/Logical induction notation

Term Notation Type Definition Notes $\mathcal F$-combination $A$ $\mathcal S \cup \{1\} \to \mathcal F_n$ Function application of an $\mathcal F$-combination uses square brackets instead of parentheses. Why? As far as I can tell, this is because each coefficient is in $\mathcal F$ so is itself a function. This means we have two senses of "application": we can pick out the specific coefficient we want (square brackets), or we can apply each coefficient to return something (parentheses).
Holdings from $T$ against $\overline{\mathbb P}$ (a $\mathbb Q$-combination) $T(\overline{\mathbb P})$ $\mathcal S \cup \{1\} \to \mathbb Q$
Trading strategy $T$ $\mathcal S \cup \{1\} \to \mathcal{E\!F}$
Feature $\alpha$ $[0,1]^{\mathcal S \times \mathbb N^{+}} \to \mathbb R$ or equivalently $(\mathcal S \times \mathbb N^{+} \to [0,1]) \to \mathbb R$ or equivalently $\mathcal F$

Example of a 5-strategy given on p. 18 of the paper: $\underbrace{\left[(\neg\neg\phi)^{*5} -\phi^{*5}\right]}_{\xi_1} \cdot (\phi - \phi^{*5}) + \underbrace{\left[\phi^{*5} - (\neg \neg \phi)^{*5}\right]}_{\xi_2} \cdot \left(\neg\neg\phi - (\neg\neg\phi)^{*5}\right)$

Since the coefficients ( $\xi_1$ and $\xi_2$) are in $\mathcal{E\!F}_5$, this is an $\mathcal{E\!F}_5$-combination. Let's call this 5-strategy $T_5$. We can pick out the coefficient for the $\phi$ term like $T_5[\phi] = (\neg\neg\phi)^{*5} -\phi^{*5}$. But since each coefficient is a feature (which is a function), we can also apply each coefficient to some valuation sequence $\overline{\mathbb V}$, like this: $T_5(\overline{\mathbb V}) = \left[(\neg\neg\phi)^{*5}(\overline{\mathbb V}) -\phi^{*5}(\overline{\mathbb V})\right] \cdot (\phi - \phi^{*5}(\overline{\mathbb V})) + \left[\phi^{*5}(\overline{\mathbb V}) - (\neg \neg \phi)^{*5}(\overline{\mathbb V})\right] \cdot \left(\neg\neg\phi - (\neg\neg\phi)^{*5}(\overline{\mathbb V})\right)$

Now each coefficient is a real number, so $T_5(\overline{\mathbb V})$ is an $\mathbb R$-combination. Note that since $T_5\colon \mathcal S \cup \{1\} \to \mathcal{E\!F}_5$ is a function that takes a sentence or the number $1$ and $\overline{\mathbb V}$ is a valuation sequence (not a sentence or number), there appears to be a type error in writing $T_5(\overline{\mathbb V})$. What is going on is that we aren't evaluating $T_5$ at $\overline{\mathbb V}$; rather, we are evaluating each coefficient of $T_5$, to convert the range of $T_5$ from $\mathcal{E\!F}_5$ to $\mathbb R$.

To summarize the types:

• $T_5 \colon \mathcal S \cup \{1\} \to \mathcal{E\!F}_5$
• $T_5[\phi] \in \mathcal{E\!F}_5$ in other words $T_5[\phi] \colon [0,1]^{\mathcal S\times \mathbb N^{+}} \to \mathbb R$
• $T_5(\overline{\mathbb V}) \colon \mathcal S \cup \{1\} \to \mathbb R$

If $T = c + \xi_1\phi_1 + \cdots + \xi_k\phi_k \colon \mathcal S \cup \{1\} \to \mathcal{E\!F}_n$, then $\mathbb V(T) = c + \xi_1\mathbb V(\phi_1) + \cdots + \xi_k\mathbb V(\phi_k) \in \mathcal{E\!F}_n$

and $T(\overline{\mathbb V}) = c(\overline{\mathbb V})+ \xi_1(\overline{\mathbb V})\phi_1 + \cdots + \xi_k(\overline{\mathbb V})\phi_k \colon \mathcal S \cup \{1\} \to \mathbb R$

and $\mathbb W(T(\overline{\mathbb V})) = c(\overline{\mathbb V})+ \xi_1(\overline{\mathbb V})\mathbb W(\phi_1) + \cdots + \xi_k(\overline{\mathbb V})\mathbb W(\phi_k) \in \mathbb R$

I think $\mathbb W(T(\overline{\mathbb V})) = (\mathbb W(T))(\overline{\mathbb V})$ but the former notation seems to be preferred in the paper.