The Nature of Consciousness

Piero Scaruffi

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

Intuitionism

The limits and inadequacies of classical Logic have been known for decades and numerous alternatives or improvements have been proposed. There are two main approaches: one criticizes the very concept of "truth", while the other simply extends Logic by considering more than two truth values. 

As an example of the first kind, “Intuitionism” (a school of thought started in 1925 by the Dutch mathematician Luitzen Brouwer) prescribes that all proofs of theorems must be constructive.  Unlike classical Logic, in which the proof of a theorem is only based on rules of inference, in Intuitionistic Logic only “constructable” objects are legitimate.  Classical Logic exhibits properties that are at least bizarre. For example, the logical OR operation yields “true” if at least one of the two terms is true; but this means that the proposition “my name is Piero Scaruffi or 1=2” is to be considered true, even if intuitively there is something false in it. Because of this rule, the logical implication between two terms can yield even more bizarre outcomes. A logical implication can be reduced to an OR operation between the negation of the first terms and the second term. The sentence "if x is a bird than x flies" is logically equivalent to "NOT (x is a bird) OR (x flies)". The two sentences yield the same truth values (they are both true or false at the same time). The problem is that the sentence “if the week has eight days then today is Tuesday” is to be considered true because the first term (“the week has eight days”) is false, therefore its negation is true, therefore its OR with the second term is true. By the same token, the sentence “Every unicorn is an eagle” is to be considered true (because unicorns do not exist, a fact that makes that formula true).

On the contrary, intuitionists accept formulas only as assertions that can be built mentally. For example, the negation of a true fact is not admissible. Since classical Logic often proves theorems by proving that the opposite of the theorem is false (an operation which is highly illegal in Intuitionistic Logic), some theorems of classical Logic are not theorems anymore.

Intuitionists argue that the meaning of a statement resides not in its truth conditions but in the means of proof or verification.

The “Theory of Types” introduced by the Swedish mathematician Per Martin-Lof in 1970 is an indirect consequence of this approach to demonstration. A “type” is the set of all propositions which are demonstrations of a theorem. Any element of a type can be interpreted as a computer program that can solve the problem represented (or “specified”) by the type. This formalizes the obvious connection between Intuitionistic Logic and computer programs, whose task is precisely to "build" proofs.

Alan Gupta's "revisionist theory of truth" also highlights how difficult it is to pin down what “true” really means. Truth is actually impossible to define: in order to determine all the sentences of a language that are true when that language includes a truth predicate (a predicate that refers to truth), one needs to determine whether that predicate is true, which in turn requires one to know what the extension of true is, while such extension is precisely the goal.  The solution is to assume an initial extension of "true" and then gradually refine it. Gupta suggests that truth can only be refined step by step.  An indirect, but not negligible, advantage of Gupta’s approach is that truth becomes a circular concept: therefore all paradoxes that arise from circular reasoning in classical Logic fall into normality.

Frederick and Barbara Hayes-Roth’s form of opportunistic reasoning (the “blackboard model” of 1985) stems from the same principles, albeit in a computational scenario. Reasoning  is viewed as a cooperative process carried out by a community of agents, each specialized in processing a type of knowledge. Each agent communicates the outcome of its inferential process to the other agents and all agents can use that information to continue their inferential process. Each agent contributes a little bit of truth, that other agents can build on. Truth is built in an incremental and opportunistic manner. Searching for truth is reduced to matching actions: the set of actions the community wants to perform (necessary actions) and the set of actions the community can perform (possible actions). An agent adds a necessary action whenever it runs out of knowledge and has to stop. An agent adds a possible action whenever new knowledge enables it. When an action is made possible that is also in the list of the necessary actions, all the agents that were waiting for it  resume their processing. The search for a solution is efficient and more natural, because the only actions undertaken are those that are both possible and necessary. Furthermore, opportunistic reasoning can deal with an evolving situation, unlike classical Logic that considers the world as static.

Classical Logic only admits two “values”: true or false. Either a proposition or its negation are true (the “law of the excluded middle”). In 1920 the Polish mathematician Jan Łukasiewicz worked out a logic based on more than just two values. First he added “possible” to “true” and “false”. Then he extended the idea to any number of truth values. A logic with more than “true” and “false” is not as “exact” as classical Logic, but it has a higher expressive power. It can be used to better mirror the human experience.

 


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