The fundamental theorem of logic briefly stated

The fundamental theorem of logic briefly stated

John Bjarne Grover

The theorem can be summed up thus: Since the equation (see the derivation)

- ∑_i∑_j
p(a_ib_j)
p(A)p(B|a_i) log
p(a_ib_j)
p(A)p(B|a_i) = - ∑_i∑_j
p(a_ib_j)
p(A)p(B|A) log
p(a_ib_j)
p(A)p(B|A)

is satisfied only by complete equiprobability, and since this is impossible for any code conveying anything at all, the human mind shifts this principle of equiprobability from occurrences to a phenomenon called 'semantics' - in the sense that the probability that A means B is the same every time it occurs. A poor substitute perhaps - but typical of human beings. It can be said that 'meaning' is a sort of 'humbug' that drives humans onto the slippery slope.

A 'non-euclidean logic' could be formed from the assumption that the equation can be rescued on distributional occurrence basis (without switching into 'semantics') by assuming that the sum of the probabilities need not be 1.0, which it per definition is in standard probability theory, but that it can be more or less than unity. Although this will create a temporary mythological reality which is not likely to last very long, since it quickly returns to virtual unity, it could nevertheless be the origin of the classic parts of speech - such as plus/minus context and plus/minus nesting for the N,V,A,P. It can be guessed that certain probability distributions will be more favourable for these than others.

My yellow-metre german poetry book 'SNEEFT/COEIL' presents 8 chapters which are likely to be a theory on the synchronic parts of speech.

I have discussed the historic development of nesting/context in my PhD dissertation book 2 (in volume 3) - a discussion which also allowed for the conclusion that a grammatical category is the same as a logical paradox (volume 3 page 328). It is easy to see how this could be systematically related to 'the problem of semantics'.

The theorem eliminates the last logical operator 'NEGATION' and defines TRUTH in terms of what slips in through the opening when the sum of probabilities differs from unity, that is, when the code opens to other realities - an opening which the feeble human mind normally cannot keep up very long before it returns to unity. Since it thereby also eliminates or rather avoids a quasi 'humbug' semantics, this tells why it can be conceived as a fundamental theorem of logic.

Non-euclidean geometry (Bolyai, Lobachevski) solves the problem with the fifth postulate (among the five postulates) in Euclidean geometry. The present fundamental theorem of logic can be compared with a similar solution to Euclid's fifth 'common notion' (sometimes also called a fifth 'axiom' - see this overview wherein the fifth common notion here is recognized also as Euclid's 9th axiom) in the sense that the whole need not be bigger than the part if the sum of probabilities can be different from unity.

The equation above seems to measure the information flux on the border between history and eternity - that is, 'sub specie eternitatis'. I discovered the principle in Budapest in the spring 1991 when I was trying to find a distributional basis for grammar.

Added 12 may 2024: As far as the briefly opening mythological reality is concerned, I refer also to this brief observation - see the 'helicopters' in my thresholded photo #1 from the Danube island. One could think that the fundamental theorem of logic could allow for a fourth or fifth dimension concerning isomorphies from cosmic to subatomic levels - through a tellurian origo - since it expectedly would be a faculty of faith that would regulate the conception and recognition of these isomorphies (not in the sense of 'justification for terror', though - by 'isomorphies' of human ideas with patterns on Mars).

(When I recorded these knockings sounds in january 2020, I believed that they probably were 'ex nihilo' sounds from another reality - here possibly telling: From 00:23 there are 8 'equiprobable' knockings and then follows 2 'non-equiprobable').