Probability as Extended Logic

Modern probability theory is typically derived from the Kolmogorov axioms, using measure theory with concepts like events and sample space. In one way, it's intuitive to understand how this works as Laplace wrote:

The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible, when [the cases are] equally possible. ... Probability is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.

However, the intuition of this view of probability breaks down when we want to do more complex reasoning. After learning probability from the lens of coins, dice and urns full of red and white balls, I still didn't feel that I had have a strong grasp about how to apply it to other situations -- especially ones where it was difficult or too abstract to apply the idea of "a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible". And then I read Probability Theory: The Logic of Science by E. T. Jayne.

Jayne takes a drastically different approach to probability, not with events and sample spaces, but rather as an extension of Boolean logic. Taking this view made a great deal of sense to me since I spent a lot of time studying and reasoning in Boolean logic. The following post is my attempt to explain Jayne's view of probability theory, where he derives it from "common sense" extensions to Boolean logic. (Spoiler alert: he ends up with pretty much the same mathematical system as Kolmogorov's probability theory.) I'll stay away from any heavy derivations and stick with the intuition, which is exactly where I think this view of probability theory is most useful.


Boolean Logic

Note: Feel free to skip this section if you're already comfortable with Boolean logic.

Before we begin with probability, let's do a quick review of Boolean logic (sometimes also called propositional logic or propositional calculus). In the context of modeling real-world situations, we usually define propositions to describe things we may want to reason about, denoted by \(\{A, B, C \ldots\}\). Propositions have an unambiguous meaning, and must either true or false. For example the following two sentences could be propositions:

\begin{align*} A &:= \text{It is raining.} \\ B &:= \text{It is cloudy.} \end{align*}

We could also define a logical relation between the two propositions using an implication operator (colloquially if-then statement):

\begin{equation*} \text{if }A\text{ is true, then }B\text{ is true} := \text{if it is raining, then it is cloudy} \end{equation*}

Rules of Inference

To reason about propositions, we usually use two forms of inference, modus ponens (Rule R1), which uses a premise (the "if-then" statement), and an antecedent (the "if" part), to derive the consequent (the "then" part):

\begin{equation*} \text{if }A\text{ is true, then }B\text{ is true} \end{equation*}
\begin{equation*} \frac{A\text{ is true}}{\text{therefore, }B\text{ is true}} \tag{R1} \end{equation*}

and similarly with modus tollens (Rule R2), which is the contrapositive and a logically equivalent statement:

\begin{equation*} \text{if }A\text{ is true, then }B\text{ is true} \end{equation*}
\begin{equation*} \frac{B\text{ is false}}{\text{therefore, }A\text{ is false}} \tag{R2} \end{equation*}

Both make intuitive sense when you try to apply it to examples above:

\begin{equation*} \text{if it is raining, then it is cloudy} \end{equation*}
\begin{equation*} \frac{\text{it is raining}}{\text{therefore, it is cloudy}} \end{equation*}

and:

\begin{equation*} \text{if it is raining, then it is cloudy} \end{equation*}
\begin{equation*} \frac{\text{it is not cloudy}}{\text{therefore, it is not raining}} \end{equation*}

Basic Boolean Operators

There are several basic Boolean operators which arise very naturally when discussing propositions. The most basic one is the negation (or "not") operator, usually drawn with a bar above the proposition (or expression):

\begin{equation*} \bar{A} \end{equation*}

The next one is conjunction (or the "and" operator) meaning "both A and B are true", denoted by:

\begin{equation*} AB \end{equation*}

The final one is disjunction (or the "or" operator) meaning "at least one the propositions A, B are true", denoted with a "+" sign:

\begin{equation*} A + B \end{equation*}

With the above examples, our intuition isn't too far off from the natural English interpretation (except for "or", which is the inclusive one instead of the exclusive one usually used in English):

\begin{align*} \bar{A} &:= \text{it is }\textbf{not}\text{ raining} \\ AB &:= \text{it is raining }\textbf{and}\text{ it is cloudy} \\ A + B &:= \text{it is raining }\textbf{or}\text{ it is cloudy (or both)} \end{align*}

Limitations of Boolean Logic

Boolean logic has wide applications in many areas. It is one of the fundamental ideas used in modern computing and one of the simplest symbolic logic systems in modern use. From one point of view, it's quite a natural way to rationally reason about real-world problems. With repeated applications of Rules R1 or R2, we can logically "prove" a fact from a set of premises. In fact, this type of reasoning system has been used for centuries with Aristotelian logic. However, it's not hard to see that it has some limitations on the kinds of things that can be modeled with it.

For example, given our above proposition "it is raining", using Boolean logic, we would have to assign this either a true or false value. If we think a bit, we can probably come up with a situation where it's not so clear whether the statement should be clearly true or false. Perhaps I'm in my bedroom and my curtains are closed but I can see that it looks kind of grey outside. Am I 100% certain that it is raining, or is there more like a 50/50 chance that it is raining? Clearly, Boolean logic isn't quite ready to handle these situations. However, if we relaxed the criteria that each proposition had to be 100% true or false and instead had a range values corresponding to how "true" we think it is, we could come up with a reasoning system that could be used to model a wider variety of real-world situations. In the next section, we'll introduce some ideas to get us closer to this type of system.


Plausible Reasoning

By relaxing the constraint of Boolean logic's strict true or false values, we end up with a reasoning system that is more widely applicable. For a proposition such as "it is raining", no longer will we assign it just true or false values, we instead want to assign it a value that represents to what degree we believe it to be true. We will call this degree of belief the plausibility of a proposition. Along with these extended truth values, we'd also like to develop rules so we can reason about them while, ideally, still maintaining the same type of deductive reasoning we have with Boolean logic. Let's see how it works out.

Weaker Rules of Inference

We already saw two forms of inference from Boolean logic, Rule R1 and R2:

\begin{equation*} \text{if }A\text{ is true, then }B\text{ is true} \end{equation*}
\begin{equation*} \frac{A\text{ is true}}{\text{therefore, }B\text{ is true}} \tag{R1} \end{equation*}
\begin{equation*} \frac{B\text{ is false}}{\text{therefore, }A\text{ is false}} \tag{R2} \end{equation*}

These rules extend quite naturally to our degrees of plausibility. For R1, if we think that A is plausible (to some degree), then it intuitively makes sense that B becomes more plausible. Similarly for R2, if we think B is implausible (to some degree), then A should also become more implausible. Using this line of reasoning, we can come up with some more rules of inference that, while in Boolean logic would be non-sensical, do make sense in our new system of reasoning with plausibilities. Consider these new rules R3 and R4:

\begin{equation*} \text{if }A\text{ is true, then }B\text{ is true} \end{equation*}
\begin{equation*} \frac{B\text{ is true}}{\text{therefore, }A\text{ is more plausible}} \tag{R3} \end{equation*}
\begin{equation*} \frac{A\text{ is false}}{\text{therefore, }B\text{ is less plausible}} \tag{R4} \end{equation*}

If we try to apply it to our example above, it passes our simplest smoke test of a rational line of reasoning:

\begin{equation*} \text{if it is raining, then it is cloudy} \end{equation*}
\begin{equation*} \frac{\text{it is cloudy}}{\text{therefore, it is more plausible that it is raining}} \end{equation*}
\begin{equation*} \frac{\text{it is not raining}}{\text{therefore, it is less plausible that it is cloudy}} \end{equation*}

Here, if it's cloudy, we're not positive that it's raining but somehow it has increased our belief that it will rain ("Is it going to rain? It might, it looks cloudy."). Alternatively, if it's not raining there is definitely some degree of plausibility that it is not cloudy. With Boolean logic and it's strict true/false dichotomy, we cannot really make any conclusions from the premise but with plausible reasoning we can change our degree of belief about the propositions.

Of course, there is not much precision (read: mathematics) in what we've said, we're just trying to gain some intuition on how we would ideally reason about propositions with varying degrees of plausibility. In whatever system we end up designing, we'd like to keep the spirit of R1-R4 in tact because it follows what we would expect a smart rational person to conclude.

Introducing the Robot

In all of the above discussion about plausible reasoning, we've been trying to build "a mathematical model of human common sense" as Jayne puts it. However, we need to be careful because human judgment has many properties (that while useful) may not be ideal for us to include in our system of reasoning such as emotion and misunderstandings. Here is where Jayne introduces a really neat concept, the robot, in order to make it clear what we're trying to achieve:

In order to direct attention to constructive things and away from controversial irrelevancies, we shall invent an imaginary being. Its brain is to be designed by us, so that it reasons according to certain definite rules. These rules will be deduced from simple desiderata which, it appears to us, would be desirable in human brains; i.e. we think that a rational person, on discovering that they were violating one of these desiderata, would wish to revise their thinking. ... To each proposition about which it reasons, our robot must assign some degree of plausibility, based on the evidence we have given it; and whenever it receives new evidence it must revise these assignments to take that new evidence into account.

Sounds like a pretty cool robot! So our goal now is to build a reasoning system for this hypothetical robot that that will be consistent with how an ideal rational person would reason. Here are the three requirements (desiderata) that Jayne states for our robot:

  1. Degrees of plausibility are represented by real numbers.
  2. Qualitative correspondence with common sense.
  3. Consistency:
    1. If a conclusion can be reasoned out in more than one way, then every possible way must lead to the same result.
    2. The robot always takes into account all of the evidence it has relevant to the question. It does not arbitrarily, ignore some of the information, basing its conclusions only on what remains. In other words, the robot is nonideological.
    3. The robot always represents equivalent states of knowledge by equivalent plausibility assignments. That is, if in two problems the robot's state of knowledge is the same (except perhaps for the labeling of the propositions), then it must assign the same plausibilities in both.

The first requirement is mostly for practicality. If we're building a machine, we'd like some standard way to tell it about plausibility (and vice versa), real numbers seem appropriate. The second requirement tells us that the robot should at least qualitatively reason like humans do. For example, the robot should be able to reason somewhat like our rules R1-R4 above, which is precisely the whole point of our exercise. The last requirement is obvious since if we're trying to build a robot to reason, it has to be consistent (or what use is it?).

What is surprising is that from these three desiderata, Jayne goes on to derive probability theory (extending it from Boolean logic)! If you're interested, I encourage you to check out his book Probability Theory: The Logic of Science (see link below), where in Chapter 2 he goes over all the gory details. It's quite an interesting read and pretty accessible if you know a bit of calculus and are comfortable with some algebraic manipulation. I'll spare you the details here on how the derivation plays out (as I'm probably not the right person to explain it) but instead I want to focus on how probability theory can be viewed as an extension of Boolean logic.


Probability as Extended Logic

The rules of probability have direct analogues with our Boolean operators above (as it can be viewed as an extension of them). Now our propositions don't have 0 or 1 truth values, they can take on any value in the range 0 (false) to 1 (true) representing their plausibility. The symbol \(P(A|B)\) is used to denote the degree of plausibility we assign proposition A, given our background or prior knowledge B (remember the robot will take all relevant known information into account).

The really interesting insight is that all the concepts from Boolean logic are just limiting cases of our extension (i.e. probability theory) where our robot becomes more and more certain of itself. Let's take a look.

Extended Boolean Operators

Consider negation ("not" operator). The analogue in probability theory is the basic sum rule:

\begin{equation*} P(A|B) + P(\bar{A}|B) = 1 \end{equation*}

If we are entirely confident in proposition A (i.e. \(P(A|B)=1\) or A is true), then from the above rule, we can conclude \(P(\bar{A}|B) = 1 - P(A|B) = 0\), or \(\bar{A}\) is false.

This works equally well with our two basic Boolean operators. Consider the "and" operator, it's analogue is the basic form of the product rule:

\begin{equation*} P(AB|C) = P(A|BC)P(B|C) = P(B|AC)P(A|C) \end{equation*}

Let's try a few cases out. If A is true and B is true, we should see that AB is true. Translating that to probabilities, we get \(P(A|C)=1\) and \(P(B|C)=1\). Now this doesn't fit as nicely into our product rule but we just need to go back to the concept of our robot taking all known information into account.

Consider the second form of the product rule: \(P(AB|C) = P(B|AC)P(A|C)\). We know that \(P(B|C)=1\), this means that given background information \(C\), we know enough to conclude that \(B\) is plausible with absolute certainty. When we add the additional information that A is plausible with absolute certainty (i.e. \(B|AC\)), it doesn't have any affect on \(B\) (because C is already telling us that \(B\) is true) [1]. From this, we can conclude that \(P(B|AC)=1\) because the fact \(A\) is irrelevant to our robot when computing \(P(B|AC)\).

Plugging that along with \(P(A|C)=1\) into the formula we get the desired result of \(P(AB|C)=1\). And since the "and" operator is commutative, we could have easily used the first form and reached the same conclusion. Alternatively, if we try \(P(A|C)=1\) and \(P(B|C)=0\), we can see through a similar line of reasoning that the result should be \(P(AB|C)=0\).

The last basic Boolean operator "or" also has a direct analogue in the extended sum rule:

\begin{equation*} P(A + B|C) = P(A|C) + P(B|C) - P(AB|C) \end{equation*}

Taking a similar line of reasoning, if we have \(P(A|C)=0\) and \(P(B|C)=1\), we have \(P(AB|C)=0\) from the above line of reasoning. With these three quantities, we can easily compute \(P(A + B|C)=1\), as we would expect (If A is false and B is true, then "A or B" is true). The other combinations of truth values for \(A\) and \(B\) yield a similar result.

Extended Reasoning

As we saw before, we would ideally like our original rules (R1 and R2) as well as our extended rules (R1-R4) to be included in our new system. As expected, these common sense interpretations are preserved in probability theory with a modified form of the product rule.

Recall the rules R1 and R2:

\begin{equation*} \text{if }A\text{ is true, then }B\text{ is true} \end{equation*}
\begin{equation*} \frac{A\text{ is true}}{\text{therefore, }B\text{ is true}} \tag{R1} \end{equation*}
\begin{equation*} \frac{B\text{ is false}}{\text{therefore, }A\text{ is false}} \tag{R2} \end{equation*}

The premise can be encoded in our background information \(C\):

\begin{equation*} C \equiv A \implies B \end{equation*}

Given this background information, we can use these forms of the product rule to encode R1, R2 as rules PR1 and PR2, respectively:

\begin{align*} P(B|AC) = \frac{P(AB|C)}{P(A|C)} \tag{PR1} \\ P(A|\bar{B}C) = \frac{P(A\bar{B}|C)}{P(\bar{B}|C)} \tag{PR2} \end{align*}

This is not all that obvious because we lose some of the nice one-to-one correspondence like the operators above. However, treating A, B, C as propositions aids us in understanding these equations. Given our major premise \(C \equiv A \implies B\), let's look at the truth table for the relevant propositions.

A B \(C \equiv A \implies B\) \(AB | C\) \(A\bar{B}|C\)
False False True False False
False True True False False
True False False Impossible Impossible
True True True True False

Notice that this truth table is a bit special in that I am mixing our extended logic with Boolean logic (e.g. \(|\) symbol). Although it's not really proper to do so, this is more an exercise in intuition than anything else so I'll stick with the sloppiness for sake of explanation. Next, we see that I have filled in a special notation for the third row using the term "impossible". This is to indicate, given the premise \(C\), this situation cannot possibly occur (or else our premise would be false).

Now given this truth table, we can see that \(AB | C\) simplifies to the expression \(A|C\) by ignoring the impossible row from our premise (the first, second and fourth rows match). Similarly, \(A\bar{B}|C\) simplifies to "False" (by ignoring the third row). Plugging these back into PR1 and PR2:

\begin{align*} P(B|AC) = \frac{P(AB|C)}{P(A|C)} = \frac{P(A|C)}{P(A|C)} = 1 \\ P(A|\bar{B}C) = \frac{P(A\bar{B}|C)}{P(\bar{B}|C)} = \frac{0}{P(\bar{B}|C)} = 0 \end{align*}

we get the desired result. In particular, \(P(B|AC)\) tells us the same thing that \(A \implies B\) combined with \(A\text{ is True}\) tells us: \(B\) is true. Similarly, \(P(A|\bar{B}C)\) resolves to the same thing that \(A \implies B\) combined with \(\bar{B}\) resolves to: \(A\) is false. Pretty neat, huh?

The rules R3 and R4 also extend quite naturally from our product rule. Recall rules R3 and R4:

\begin{equation*} \text{if }A\text{ is true, then }B\text{ is true} \end{equation*}
\begin{equation*} \frac{B\text{ is true}}{\text{therefore, }A\text{ is more plausible}} \tag{R3} \end{equation*}
\begin{equation*} \frac{A\text{ is false}}{\text{therefore, }B\text{ is less plausible}} \tag{R4} \end{equation*}

R3 can be encoded as this form of the product rule:

\begin{equation*} P(A|BC) = P(A|C)\frac{P(B|AC)}{P(B|C)} \end{equation*}

But from the discussion above, we know \(P(B|AC)=1\) and \(P(B|C) \leq 1\) (from the definition of a probability), so it must be the case that:

\begin{equation*} P(A|BC) \geq P(A|C) \tag{E1} \end{equation*}

In other words, given new information \(B\), we now think \(A\) is more plausible. We can build upon this reasoning to understand R4 using this form of the product rule:

\begin{equation*} P(B|\bar{A}C) = P(B|C)\frac{P(\bar{A}|BC)}{P(\bar{A}|C)} \end{equation*}

From E1, we know that \(P(\bar{A}|BC) \leq P(\bar{A}|C)\) (remember the "not" rule), so we can conclude that:

\begin{equation*} P(B|\bar{A}C) \leq P(B|C) \end{equation*}

which says that given \(\bar{A}\), proposition \(B\) becomes less plausible.


Conclusion

Probability as an extension of logic is quite a different approach compared to the traditional treatment of the subject. I've tried to shed some light on this view of probability and hopefully have provided some intuition on how it all works. For me, probability as an extension of logic is much more natural way of looking at the subject while also much more philosophically satisfying. It also directly leads to a Bayesian interpretation of data (because you're just updating our robot's prior knowledge), which also makes a lot of sense to me. It's a shame that probability isn't taught (or even mentioned) in the context of extended logic because I think it would help people internalize the concepts a lot better and, dare I say, even start to like the subject!


Further Reading


[1] You might wonder what happens when \(A\) and \(C\) are mutually exclusive propositions (i.e. impossible to happen at the same time). In this case, \(P(B|AC)\) is not defined but also our original question is ill formed because we couldn't have the case \(P(A|C)=1\) (we would instead have \(P(A|C)=1\)).

I'm Brian Keng, a former academic, current data scientist and engineer. This is the place where I write about all things technical.

Twitter: @bjlkeng


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