Counterfactual conditionals

Counterfactual reasoning is a thorny problem that interests many logicians (cf. Lewis [38], for example). Roughly speaking, counterfactual reasoning is drawing a conclusion based on antecedents which are not (or not yet) the case. For example, ``if I paint the moon red, it is not green.'' The state of affairs of counterfactual reasoning is therefore not (or not yet) actual in this world. For a long time, it has been taken as an epiphenomenon of sound logical reasoning and should be at best tolerated and at worst totally removed. After all, as many believe, a sound argument should be based on facts but not fiction, and this is what logic is all about.

However, this view misses a very important point of counterfactual reasoning, which plays a crucial part in our life. In fact, every decision made, when carefully thought over, is based on some sort of counterfactual reasoning. Consider the following example. A university senior had to decide whether she should attend graduate school or get a job. At the moment of this decision, her attending graduate school was definitely not actuality, but neither was her taking a job. In order to make a better decision, she had to imagine a thread of the future in which she attended graduate school (the `school'-thread) and another one in which she did not (the `job'-thread). In the first thread she would have to branch further to threads in which, for example, she took either computer science or physics as her major. Likewise if she followed the `job'-thread. Her typical reasoning about this would probably be,

... If I had a graduate degree, I would get a better job.
... But if I attend graduate school, I would have to take a loan because I don't have enough money at this time.
... I don't really like the life in the academic world. But if I attended graduate school, I'll have to live with it, at least for a while.
... But if I get a good job now, I'll actually have a better chance in my career than if I go on the job market in two years (e.g. the Internet boom in the late 1990s.)

To accommodate this kind of reasoning, one needs a competent model of, as well as and information about, the actual world, such as how much one gets paid with an undergraduate degree and how much graduate school costs, etc. In this regard, it has become an expert system problem. Classical logic and optimization are sufficient to solve this reasoning problem.

But this is not all that counterfactual is about. Consider the following example (cf. [35]). Jack and Jim are old friends. Under normal circumstances they help each other. But Jim is very proud, so he will never ask for help from someone with whom he has recently quarreled. Jack, on the other hand, is very unforgiving. So he will never help someone with whom he has just quarreled. Jack and Jim have a quarrel. Now an interesting question is this:

Example 10   If Jim asks Jack for help, then Jack will help him. .

First of all, this sentence seems to mean something. There is a state of affairs being addressed. But is this statement true? Since Jack and Jim just had a quarrel, Jim wouldn't ask Jack for help and of course Jack wouldn't help him. To answer the question, however, we have to envisage proper counterfactual conditions. We have to either ignore the fact that they just had a quarrel, or ignore the fact that Jim is very proud. Now if they haven't had a quarrel, Jim would ask Jack for help and Jack would help him. So the sentence is true. But if Jim is not proud (but they did have a quarrel), Jack wouldn't help him since Jack is unforgiving. So the sentence is false. This sentence seems to be true and false at the same time.

It turns out that counterfactual reasoning is not only common in everyday life, as well as carefully conceived logic games, it is also one of the most important activities involved in constructing scientific theories. A boy imagines that he can run at the speed of light (surely he does not and cannot), and he might ask himself what will happen. Similar thought experiments can lead to the Theory of Relativity. But if one tries to argue along the same line, à la Einstein, but assumes that one can accelerate oneself to a speed faster than light, and then asks what will happen, the question becomes physically meaningless and cannot be answered. (As long as the Theory of Relativity is correct in this regard.)

So, is there any way to accommodate all these varieties of counterfactual reasoning in classical frameworks? Let us try. To begin with, counterfactual is a problem of representation. According to a physicalist account, in any reasoning process, everything must indeed take place in the brain and the brain must have access to a similar or identical physical environment which is represented by the counterfactual state of affairs. In short, counterfactual is a ``simulated'' situation. It can be argued that, in this simulated environment the information (or probability) is taken into account in order to tell which is a sound or plausible argument and which is not. The parameters are acquired through processing all the sensorial data the brain has ever encountered. Let us call this a bottom-up approach to counterfactual reasoning.

For a cognitive scientist who uses computation as a model of the mind, representation plays a similar role. Moreover, the cognitive scientist may contend that a ``theory'' is built into the head like a program in a computer. And there is a set of rules with which he can manipulate the components of the theory. The soundness of a conclusion is evaluated, for example, according to how it violates the original theory. Let us call this a top-down approach.

The gap between a bottom-up and a top-down approach is very difficult, if not impossible, to bridge. Nevertheless, unless one wants to argue for everything based on any arbitrary imagination, the final arbitration of the validity of a counterfactual argument lies in the physical world. Whether for everyday or for serious scientific arguments, counterfactual is a physical problem and has to find its solution in physics. In a sense, a bottom-up approach has the upper hand, for it is inherently physical. A top-down approach has a hard time in ``calibrating'' a mental theory with the physical world. That is, according to a bottom-up approach, all counterfactual conditions must exist in the physical world, and the representation is only a pointer.

But the bottom-up approach has difficulty accommodating an argument such as that in Example 10. Nor can it account for a scientific argument such as the case of running at the speed of light. This is because all these situations are novel, non-existent, and, in a sense, creative. All in all, a representational account must be able to represent both possibility (for all that is counterfactual) and actuality (the factual). It also has to be efficacious in saying how likely a possibility will become actuality.

Here we come to a real strength of quantum mechanics. In quantum mechanics, a physical state can be constructed as a superposition of eigenstates. Every time a system is measured, the original physical system collapses into one of its eigenstates. In this sense, a pre-measurement quantum system can be conceived of as a superposition of possibilities in which the probability of each possibility (eigenstate) is the absolute square of the coefficient of the corresponding eigenstate. Therefore, every pre-measurement quantum system is counterfactual, although each of the eigenstates points to a physical world that has at the same time existed. A novel world can be regarded as a novel superposition of existing worlds. After measurement, however, the superposed system is actualized by collapsing into one of the eigenstates (the actuality).

In this way one can see why a quantum computational cognition model can accommodate these counterfactual arguments in a very elegant way. For one thing, a quantum brain is directly coupled to the physical world through nerves and other tissues. This makes it a physical system from the beginning (so it is bottom-up). Furthermore, quantum mechanics is a stochastic theory that can easily accommodate probabalistic reasoning but does not suffer from the weakness of most probabalistic models^5.8. Moreover, since quantum mechanics can be discrete, measured results may ``jump'' between incompatible representations -- such as truth and falsity in Example 10..

To see a quantum computational account of counterfactual reasoning, consider Example 10 again. Both of the counterfactual premises can be rendered true as a superposition of two negative eigenstates of the system (viz. ``Jim and Jack did not have a quarrel'' $E_1$ and ``Jim is not a proud person'' $E_2$). The initial quantum state is then subject to a unitary evolution (a counterfactual reasoning). The outcome of this counterfactual conditional is a superposition of eigenstates of the system, in which both the eigenstates -- ``Jack helped Jim'' and ``Jack did not help Jim'' -- have a share. The answer to the question is a measured result of the end-state, which yields either ``Jack helped Jim'' or ``Jack did not help Jim.''

To elaborate this issue, notice that we are accustomed to situations in which the answer cannot be true and false at the same time. This discernibility (the XOR function or the law of exclusive middle) is crucial to logic -- perhaps any kind of logic. In fact, a quantum mechanical framework implies that a measurement is indeed either true or false. Only if a system is not measured, can it be something in between. Moreover, if the question is asked multiple times, the outcome might jump back and forth between these two eigenstates but never stop in between.

Moreover, quantum mechanics has more to say than this sort of ``quantum leap.'' Quantum mechanics offers a numerical framework within which one can predict the frequency of true or false. For example, probability may manifest itself this way: if Jim is actually not an absolutely proud person then he may still ask Jack for help ($E_2$) if, say, the situation is very urgent or the person with whom he just had a quarrel is a very close friend. Or, if the quarrel Jack and Jim had ($E_1$) is actually just a trivial squabble and it has been quite a long time since the quarrel. Under these circumstances, we have a probabilistic model with $\left\vert E_1 \right\vert^2=p_1$ and $\left\vert E_2 \right\vert^2=p_2$, where $p_1$ and $p_2$ are the respective probabilities of $E_1$'s being true and $E_2$'s being true. After subjecting the initial state to a unitary operator, we have a result that has a different composition of being true or false.

One might argue that all these advantages may be well accommodated in a hybrid model of probabilistic approaches and symbolic approaches to counterfactual. This is only partly true. All statistical models have real parameters and are therefore accumulative. They cannot accommodate wave-like interference where the probability may vanish at certain points.