User:IssaRice/Logical induction notation
|-combination||Function application of an -combination uses square brackets instead of parentheses. Why? As far as I can tell, this is because each coefficient is in 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 against (a -combination)|
|Feature||or equivalently or equivalently|
Example of a 5-strategy given on p. 18 of the paper:
Since the coefficients ( and ) are in , this is an -combination. Let's call this 5-strategy . We can pick out the coefficient for the term like . But since each coefficient is a feature (which is a function), we can also apply each coefficient to some valuation sequence , like this:
Now each coefficient is a real number, so is an -combination. Note that since is a function that takes a sentence or the number and is a valuation sequence (not a sentence or number), there appears to be a type error in writing . What is going on is that we aren't evaluating at ; rather, we are evaluating each coefficient of , to convert the range of from to .
To summarize the types:
- in other words
If , then
I think but the former notation seems to be preferred in the paper.