SYNTHESIS AND SYSTEMS

I. Introduction

1. Monotony in the language of first-order predicates

Any attempt to construct a new formal language that works, including formulas of the form x=1, that is, describing the fact that a variable has some of its permissible value, raises the question whether it will be useful to use the existing language of predicates of the first order? Doubts about this cause attempts to analyze whether this language, which lies in the Foundation of modern mathematical logic, to a proper extent has the property of monotony?

Quite convincing is the generally accepted interpretation of monotony, as the preservation of the truth estimates of the already obtained formulas in the logical conclusion, that is, when adding new formulas to the previously recognized as "correct". However, the following three formulas in example 1 encourage a broader understanding of It:

And there is a conjunction of two atoms

(x=1) & (y=1)

In there is a disjunction of two atoms

(y=1) v (y=2)

C is the conjunction of the formulas A and b

A & B

Then, doable (!) formula C with free variables x and y can not correspond to any state in which y=2.

So, the knowledge that y=2, " was "in B, but "disappeared" in C.

One can therefore admit that in some sense the language of first-order predicates is NOT monotonous.

Since such" nonmonotonicity " of knowledge growth inherent in the language of first-order predicates often manifests itself as a lack of language, then further:

the root cause of "loss of knowledge" , "facts" will be identified (see Chapter 2)

the ways are outlined (see Chapter 3), and in the end a new formal system is built, called the language AND-or systems, which does not allow in the correct context the joint existence of formulas like A, B and C, and which provides a new interpretation of the "monotony of the growth of knowledge".

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