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

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. Back to the beginning of the chapter "Common Sense" | Back to the index of all chapters |