# Monthly Archives: September 2010

## The Spectrum From Logic to Probability

Let $\Omega$ be the set of propositions considered by some rational logician (call her Sue).  Further, suppose that $\Omega$ is closed under the propositional connectives $\vee$, $\wedge$, $\neg$.  Here are two related but different preorders on $\Omega$:

1. $p\leq q$ if $p$ logically entails $q$.
2. $p \preceq q$ if Sue considers $q$ at least as likely to be true as $p$ is.

Let $\sim$ be the equivalence relation defined by $p \sim q$ iff $p \leq q \wedge q \leq p$ and let $\approx$ similarly be defined by $p \approx q$ iff $p\preceq q\wedge q\preceq p$.

Then we know what type of structure $\Omega/{\sim}$ is: since we’re assuming classical logic in this article, it’s a Boolean algebra.  What type of structure is $\Omega/{\approx}$?

We can at least come up with a couple of examples.  Since Sue is a perfect logician, it must be that if $p\leq q$, then $p\preceq q$.  If Sue is extremely conservative, she may decline to offer opinions about whether one proposition is more likely to be true than another except when she’s forced to by logic.  In this case, $\Omega/{\approx}$ is equal to $\Omega/{\sim}$ and therefore again a Boolean algebra.

In the other extreme, Sue may have opinions about every pair of propositions, making $\preceq$ a total ordering.  A principal example of this is where $\Omega/{\approx}$ is isomorphic to a subset of $[0,1]$ and Sue’s opinions about the propositions were generated by her assigning a probability $P(p)\in [0,1]$ to every proposition $p$.

What’s in between on the spectrum from logic to probability?  Are there totally ordered structures not isomorphic to $[0,1]$ or a subset? More ambitiously: every Boolean algebra has operations $\vee$, $\wedge$, $\neg$, while $[0,1]$ has operations ${+}$, $\times$, $(x\mapsto 1-x)$ which play similar roles in the computation of probabilities (note that $+$ is partial on $[0,1]$).  How are these related and does every structure on the spectrum from logic to probability have analogous operations?

These structures (i.e., structures of the form $\Omega/{\approx}$ for some acceptable $\preceq$ in a sense to be defined below) were called scales and defined and explored in a very nice paper by Michael Hardy.