Non-monotonicity

One of the ``bugs'' (or ``features'') of common sense reasoning is non-monotonicity. Let us begin with an example. Suppose I have an appointment at 8 a.m. on a certain Sunday and I take a look at my watch. The scenario is:

Example 5   My watch shows 7:30 ($p$), so I still have enough time ($q$).

However, on my way to the meeting I quickly learn that it is the Sunday when daylight-saving time takes effect, so I am certainly too late for the appointment. The scenario changes to:

Example 6   My watch shows 7:30 ($p$) and today is the first day of DST ($r$), so I am too late ($\neg q$).

Since both reasonings seem to be sound, in symbolic logic, the two sentences above can be written as follows:

$$p \to q$$

$$p\wedge r \to \neg q$$

where $p$ is the proposition ``My watch shows 7:30;'' $q$, ``my having enough time for being punctual;'' $r$, ``it is the first day of DST.'' Asserting both formulas as sound is to assert that the following formula is true:

$$(p \to q) \wedge (p\wedge r \to \neg q)$$ (5.1)

However, it is a theorem in classical propositional logic:

$$(p \to q) \to (p \wedge r \to q)$$

which is clearly contradictory to Equation 5.1. Here we can see that classical logic is monotonic but common sense is not.

More generally speaking, non-monotonicity is a situation in which the facts which can be derived from a collection of premises $\mathfrak C$ is less than those that can be derived from an extension of $\mathfrak C$, $\mathfrak C'$. In plain language, non-monotonicity implies: the more one believes, the less one knows.

Apparently, if classical logic is the most reliable form of reasoning (many implicitly believe it is the case), something must have gone wrong in the above argument. Of course, unless we abandon classical logic, we have to find a way to accommodate non-monotonicity. Indeed, it is a common practice to treat the situation in Example 5 as ``normal'' and that in Example 6 as ``pathological.'' More specifically, the apparent non-monotonicity, it may be argued, is a situation in which one does not have complete information. For instance, one can devise a ``situation logic'' to correct the ``fallacy'' of common sense above. This is done by treating ``My watch says 7:30'' as a situation (among every possible situation) in which my watch indeed agrees with an objective measurement of time. The measurement, then, is a situation in which the sunlight makes a particular angle in relation to the meridian. The premise ``my watch says 7:30'' is then true and Example 5 is applicable. But if some social institution offers another convention for synchronizing clocks, as the additional fact in Example 6 shows, the premise is no longer true and should be considered a different situation. Example 5 should not automatically be applied. With this approach, it is hoped, non-monotonicity can be eliminated.

In fact, any endeavor to accommodate non-monotonicity mathematically is an endeavor to eliminate non-monotonicity (this is because mathematics itself is monotonic). This is possible only if we have a physically sound information theory. Unfortunately, in the same vein as the arguments in Section 5.2, this is impossible.

We can look at the last statement in another, somewhat amusing, way. Notice that many instances of non-monotonicity come from false belief. For instance, I believed that my watch told the correct time in Example 5, but it turns out to be a false belief. What, then, is ``genuine'' belief? Shouldn't it grounded in reality? I think most physicists do believe that there is always something new to discover in physics. A newly discovered scientific fact may falsify an existing theory and therefore modify existing valid statements (by predicting the facts more precisely, for example), which makes the whole scientific endeavor non-monotonic. This indicates why the reduction of information down to physical properties is pointless.

Another interesting example is to be found in physics itself. For one thing, non-monotonicity is very common in theories of physics. Consider the following statement of the law of motion (Newton's second law):

Example 7   The acceleration of a rigid body is proportional to its mass and the force acted upon it.

It is a very effective tool. We know that ballistic rockets and communication satellites can be designed according to this law, but we also know it is not applicable if the body is moving at a very high speed or if the force of gravitation is very strong. In such situations, the concept of a rigid body is not even remotely correct and the whole statement in Example 7 becomes a false belief.

So Example 7 is not true. Its ``truthfulness'' depends on the situation. For example, in the situation described above, the ``truthfulness'' of Newton's second law can be very ``low,'' in contrast to the ``normal'' situation, where it is very ``high.'' In this case our crucial question becomes how to find a situation (or a class of situations) in which a particular physical law is applicable. Interestingly, as far as the applicability of physics is concerned, the last question is exactly the question that we want to solve at the outset, in the hope that the knowledge of ``situation'' may eliminate non-monotonicity.

Since a rigorous (pure classical logical plus physical) way of eliminating non-monotonicity is impossible, one might want to suggest a statistical approach (actually it is an informational approach without delving ``too'' deeply into how one gets the information -- let us for the moment say that the probability is given a priori.). For example, the ``truthfulness'' of Example 5 can be transformed into a formula of conditional probability:

$$P(q \vert p)={P(q, p) \over P(p)}$$ (5.2)

and that of Example 6

$$P(q \vert p,r) = {P(q,p,r) \over P(p,r)}$$ (5.3)

where $P(x)$ is the probability of proposition $x$ being true; $P(x \vert y)$ is the conditional probability of $x$ being true given the fact that $y$ is true. So if the probability of a situation can be known a priori, the seeming contradiction to classical logic can be technically explained away.

Let us formalize non-monotonicity first. In formal terms, a non-monotonic reasoning is:

$$\begin{eqnarray*} &\mathfrak C \vdash \phi\\ &\mathfrak C, \psi \vdash \neg \phi \end{eqnarray*}$$

where $\neg \phi$ is the negation of statement $\phi$; $\mathfrak C$ is a collection of premises; $\psi$ is an additional premise. In non-monotonic reasoning, additional knowledge changes the facts that can be derived. In a classical context, this simply suggests that $\mathfrak C$ and/or the newly formed collection of premises $\{\mathfrak C,\psi\}$ is not consistent. This inconsistency cannot propagate into a statistical meta-framework. So the meta-framework remains consistent.

This approach turns out to be just as problematic. This is because the problem of non-monotonicity in fact lies much deeper. Before discussing the problem, we introduce non-monotonicity of strong kind.

Definition 5   Non-monotonicity of strong kind is a reasoning process in which $\psi$ is statistically independent of the original collection of premises $\mathfrak C$ and

$$\begin{eqnarray*} &\mathfrak C \vdash \phi\\ &\mathfrak C, \psi \vdash \neg \phi \end{eqnarray*}$$

Now suppose there is indeed non-monotonicity of strong kind in Nature, with Equation 5.2 and Equation 5.3 in lieu of Equation 5.1 we have,

$$P(q\vert p,r)={P(q,p,r) \over P(p,r)} = {P(q, p) P(r) \over P(p) P(r) } = {P(q, p) \over P(p) }=P(q\vert p)$$

That is, if the non-monotonicity described in Equation 5.1 is of strong kind, the conclusion of classical logic should hold, (in other words, $P(q\vert p,r)=P(q\vert p)$ means that both assertions have the same ``truthfulness'' -- if $p \to q$ is ``quite true'', $p \wedge r \to q$ should be ``quite true,'' too.).

Intuitively, the statistical independence of $\psi$ and $\mathfrak C$ suggests that they are compatible with each other and therefore consistent. This is a new kind of non-monotonicity which can not be accommodated in a classical framework. In short, the newly introduced knowledge seems to actively change the reasoning structure and can derive novel facts and / or falsify old facts. Of course, the crucial question becomes whether there are cases of non-monotonicity of strong kind in nature. The answer is an unequivocal yes.

For example, in the electron two-slit experiment (see Section 3.2): if both slits are open, there is an interference pattern on the plate. However, if a measuring device is placed near one of the slits and records whether an electron passes the slit or not, the interference pattern disappears. Treating information as well-defined ``things,'' the knowledge of whether an electron passes a particular slit is independent of the knowledge of other gadgets in the experiment, so it must be consistent with the original premises collection. But the experiment results prove otherwise. The consistent and independent additional knowledge changes the fact!

In fact, non-monotonicity of strong kind is not remote to our everyday reasoning. Consider the following situation: one day I opened the door of my apartment, and as I was a little bit distracted I did not notice that my neighbor was just walking by the door and I bumped into him. It is an extremely rare situation, but it happened. Now I have the following description:

Example 8   I opened the door at 3:30 p.m., thus I bumped into my neighbor.

One can imagine that given the extremely low probability of my opening the door at exactly 3:30 p.m., and equally low probability of someone's standing in front of my door at exactly that time, the probability of my bumping into him can be extremely high. However, if I had known that my neighbor was passing by the door (if I had peeked through the key-hole, for instance), I wouldn't have bumped into him. So it seems valid to say:

Example 9   I opened the door at 3:30 p.m. and as I knew my neighbor was there, I did not bump into him (but I did greet him).

But how can my knowing of his presence change the probability distribution of bumping into him? The situation would be clearer if the ``bumping event'' was replaced by another event. Suppose the pipeline in front of my apartment broke at 3:20 that afternoon, and the doorway was flooded. If I open the door, my shoes will be wet. If I know that the pipeline is broken (by peeking through the key-hole again) but still open the door, my shoes will again be wet. In this situation, my knowing of the event does not seem to change the consequence, but wherein lies the difference?

We are all able to give a folk psychological explanation: through my behavior I can change the first situation but not the second one (at least it would be very hard to change it).

A key issue is here raised implicitly: we seem to have free will so that we can do something (but we do not have to). There are situations in which free will plays a crucial role and there are situations where it does not. It is on this issue that classical mechanics (including scientific frameworks built on naive ``folk physical'' understanding) and classical logic have a hard time. In fact, many endeavor to remove this sort of folk psychology and restore the exactness of ``science.'' (that is, ``mechanistic.'') As long as the Cartesian mechanistic view is not abandoned, these endeavors are not likely to be successful.

According to quantum theory, the world at the quantum level is inherently strongly non-monotonic. Most importantly, a quantum measurement is irreversible -- the world is different before the measurement versus after the measurement. It is not to say that classical logic cannot be accommodated in a quantum computational account of cognition. In fact, monotonicity can be maintained in quantum theory in two ways: either by holding the quantum state in a pure-state (eigenstate) so that it remains invariant and reversible; or by resorting to the derived classical system with a large number of quanta. Strictly speaking, the latter case is a limiting case statistically approaching monotonicity.

A final point: while classical reduction of non-monotonicity seems quite hopeless, the probability picture (which is a classical framework nevertheless) of non-monotonic reasoning is not completely out of focus. In fact, it can be treated as a derived characteristic and can be better understood in a framework of possibility vs. actuality in a quantum mechanical context. This will become clearer after we discuss counterfactual conditionals.