The Nature of Consciousness

Piero Scaruffi

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These are excerpts and elaborations from my book "The Nature of Consciousness"

The Impossibility of Reasoning

A very powerful argument in favor of common sense is that logical reasoning alone would be utterly impossible.

Classical Logic deduces all that is possible from all that is available, but in the real world the amount of information that is available is infinite: the domain must be somehow artificially “closed” to be able to do any reasoning at all. And this can be achieved in a number of ways: the “closed-world assumption” (all relations relevant to the problem are mentioned in the problem statement), “circumscription” (which extends the closed-world assumption to “non-ground” formulas as well, i.e. assumes that as few objects as possible have a given property), “default” theory (all members of a class have all the properties characteristic of the class if it is not otherwise specified).  For example, a form of default theory allows us to make use of notions such as “birds fly” in our daily lives. It is obviously not true that all birds fly (think of penguins), but that statement is still very useful for practical purposes. And, in a sense, it is true, even if, in an absolute sense, it is not true. It is “plausible” to claim that birds fly (unless they are penguins).

At the same time, common sense reasoning introduces new problems in the realm of Logic. For example, John McCarthy's "frame problem”: it is not possible to represent what does “not” change in the universe as a result of an action, because there is always an infinite set of things that do not change. What is really important to know about the new state of the universe, after an action has been performed?  Most likely, the position of the stars has not changed, my name has not changed, the color of my socks has not changed, Italy’s borders have not changed, etc. Nevertheless, any reasoning system, including our mind, must know what has changed before it can calculate the next move. A reasoning system must continuously update its model of the world, but McCarthy suggests that this is an impossible task: how does our mind manage?

Complementary paradoxes are the "ramification problem" (infinite things change, because one can go into greater and greater detail of description) and the "qualification problem" (the number of preconditions to the execution of any action is also infinite,  as the number of things  that can go wrong is infinite). Somehow we are only interested in things that change and that can affect future actions (not just all things that change) and in things that are likely to go wrong (not just all things that can go wrong).

“Circumscription” (McCarthy’s solution to the frame problem) deals with default inference by minimizing abnormality: an axiom that states what is abnormal is added to the theory of what is.  This reads as: the objects that can be shown to have a certain property, from what is known of the world, are all the objects that satisfy that property. Or: the only individuals for which that property holds are those individuals for which it must hold.  (This definition involves a second-order quantifier.  Technically, this is analogous to Frege's method of forming the second-order definition of a set of axioms: such a definition allows both the derivation of the original recursive axioms and an induction scheme stating that nothing else satisfies those axioms).

For similar reasons Raymond Reiter introduced the "closed-world axiom" (what is not true is false), or "negation as failure to derive": if a formula cannot be proven using the premises, then assume the formula's negation. In other words, everything that cannot be proven to be true must be assumed to be false. His “Default Logic” employs the following inference rule: "if A is true and it is consistent that B is true, then assume that B is also true" (or "if a premise is true, then the consequence is also true unless a condition contradicts what is known").

Ultimately, these are all tricks to account for how the mind can do any reasoning at all in the face of the gigantic complexity that surrounds it.

 


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