Great Again wrote:Great Again wrote:The two-valued logic is also not suitable for the treatment of propositions about future events, because it implies a false determinism and leaves no space for the freedom of man. If A is a proposition about future events, then the statement "A is true" can be more accurately described by the statement "There are (that is: present) causes that force the occurrence of A in the future", and the statement "A is false" can be more accurately described by the statement "There are causes that force the occurrence of non-A in the future".

A sentence like "Bill will be home tomorrow" will not usually be true or false in this sense, because there are usually no compelling causes that determine Bill's behavior. Thus, to deal with such cases, one must introduce a third truth value, which can be assigned the property "unknown" or the property "not yet" (cf. the

arrows in my diagram), which a proposition A about future things takes on precisely when there are no compelling causes for A or not-A to occur.

A roughly similar argument is already found in Aristotle (the famous example of tomorrow's sea battle).

Great Again wrote:Great Again wrote:Classical logic includes only propositional logic and predicate logic, in which the principles of forbidden contradiction (principium contradictionis) and excluded third party (principium exclusi tertii) and, related to them, the bivalence principle (see my earlier posts) are valid.

Non-classical logics are those in which at least one of the principles of classical logic is not valid. Particularly important are those systems in which the principle of the excluded third or the bivalence principle is invalid. Such logics were developed because they were motivated by developments in mathematics (cf. for example my earlier posts about antinomy).

Non-classical logics include e.g.:

- Multivalued logic (generic term for all other logics in which the bivalence principle is not valid).

- Modal logic (also: Alethian logic).

- Intuitionistic logic.

- Dialogic logic.

- Temporal logic.

- Deontic logic.

- Conditional propositional logic.

- Doxastic logic.

- Epistemic logic.

- Relevance logic.

- Non-monotonic logic.

- Fuzzy logic.

Fuzzy logicians say that most concepts are factually fuzzy in the sense that they can apply to different objects to different degrees. The fuzzy logicians are right. Whether or not a particular term applies to an object is often not a matter of a simple yes or no, but often a matter of degree. In fuzzy logic, one specifies the degree to which a term applies to a particular object by a number from the continuum between 0 and 1: If an object does not fall under a certain term at all, the term in degree 0 applies to it; if it falls completely under it, the term in degree1 applies to it; and if it falls only more or less under it, the term in degree g with 0<g<1 applies to it. For a term one has to specify a function which determines under which circumstances it applies to an object and in which degree. (This function determines a fuzzy set.) For example, one can specify that the predicate "x is a tall man" applies to men up to 1.60m in degree 0, to men from 1.90m in degree 1, and to men between 1.60m and 1.90m in certain (with height increasing) degrees between 0 and 1; a 1.75m tall man may be tall in degree 0.5, for example.

Since the 1980s, fuzzy logic has increasingly found its way into technical applications under the keyword "fuzzy control", especially where exact mathematical calculations of the processes to be controlled are complicated, lengthy or hardly possible due to the many and unmanageable influencing variables. In this case, precise measured variables are first translated into fuzzy terms such as "quite fast", "quite close to the target", etc. ("fuzzification" is the word for this), which then form the basis for simple rules that are easily accessible to human intuition: "If the car is quite fast and quite close to the target, then brake quite hard". The "outputs" of these rules are then transformed back into precise control instructions according to specific procedures. This procedure allows control on an "intuitive" basis without the availability of an exact mathematical model of the process to be controlled. Fuzzy control has found its way, for example, into the control of video cameras, washing machines, elevators and even subways.

Multi-valued logics have been around for a long time, and they all have one thing in common: statements that are either true or false according to the bivalence principle are not valid.

According to Gödel's results, one must presuppose an infinite number of truth values.

Ulrich Blau has given a number of reasons why the logic underlying everyday language is three-valued. I would say it is multi-valued.

If X is rational and irrational, and in addition something that is itself rational or irrational, but without precise assignment, i.e. not yet known, but with high probability assignable in the future, then the possibility can be kept open that this still undetermined will turn out to be something determined in the future. For this purpose, a truth value in terms of the future and a truth value in degrees are given by numbers from the continuum between 0 and 1, where 0 or e.g. 0-0.2 stands for "still undetermined" or "rational and irrational (because in each case determinable only in the future)".

A not entirely serious suggestion: We could declare everything irrational to be taboo.

An example from mathematics:

The idea of irrational numbers, in our notation therefore infinite decimal fractions, should remain incomprehensible to the mind, never be told in school about irrational numbers.

Euclid said - and one should have understood him better - that incommensurable distances behaved "not like numbers". In fact, in the accomplished concept of the irrational number lies the complete separation of the concept of number from the concept of magnitude, and this because such a number - pi, for example - can never be delimited or represented exactly by a distance. But it follows that in the conception of the ratio of the square side to the diagonal, for example, the number as a sensual limit, a closed quantity, suddenly touches a completely different kind of number, which remains foreign in the deepest inside and therefore uncanny, as if one were close to uncovering a dangerous secret of one's own existence. This is revealed by a strange late Greek myth, according to which the one who first brought the contemplation of the irrational out of the hidden to the public, perished by a shipwreck, "because the inexpressible and imageless should always remain hidden".

An expression like e^ix, which constantly appears in our formulas, is supposed to seem absurd to us, to be a taboo.

Only calculate with finite fractions, examine the integer ratio of two distances. Great!

Even the idea of a number zero must not even arise, because it has no sense in terms of drawing.

If one would do all that - and only all that -, such a mathematics would be already perfect, only

differently perfect.