The law of non-contradiction in The law of non-contradiction in the combined calculus of the combined calculus of sentences, situations and sentences, situations and

contexts (TLL)contexts (TLL)

Victor GorbatovVictor Gorbatov

HSE, MoscowHSE, Moscowvic-gorbatov@yandex.ruvic-gorbatov@yandex.ru

ContentsContents

Vasiliev’s “imaginary logic” and Vasiliev’s “imaginary logic” and Smirnov’s combined calculiSmirnov’s combined calculi

Pragmatics of language as an Pragmatics of language as an instrument of empirical instrument of empirical determination of our thoughtdetermination of our thought

Three-levelled logic (TLL)Three-levelled logic (TLL) Where did contradictions come from?Where did contradictions come from? Three senses of the law of non-Three senses of the law of non-

contradictioncontradiction

Vasiliev’s “imaginary logic”Vasiliev’s “imaginary logic”

««Imaginary logic allows us to Imaginary logic allows us to penetrate into the nature of penetrate into the nature of our logic in a more profound our logic in a more profound way and to distinguish within way and to distinguish within it empirical (eliminable) it empirical (eliminable) elements and non-empirical elements and non-empirical ones, which cannot be ones, which cannot be eliminated. All the non-eliminated. All the non-empirical elements and empirical elements and relations are to be regarded relations are to be regarded as metalogicas metalogic.».»

NN.. А. А. VasilievVasiliev

(1880-1940)(1880-1940)

Two senses of the law of non-Two senses of the law of non-contradictioncontradiction

Law of non-contradiction Law of non-contradiction (empirical)(empirical)

Law of self-consistancy Law of self-consistancy (metalogical)(metalogical)

Комбинированное Комбинированное исчисление высказываний и исчисление высказываний и

событий (1989)событий (1989) «Let us distinguish to acts:

the act of predication (a synthesis of object with it’s property) – and the act of assertion (correlation of our thought content and reality)”

P(a) – act of predication (constitute the logic of situations)

ΘΘP(a) – act of assertion (constitute the logic of truth)

VV.. А. А. SmirnovSmirnov

(1931-1996)(1931-1996)

Two senses of the law of non-Two senses of the law of non-contradictioncontradiction

Law of non-contradiction Law of non-contradiction (empirical)(empirical)

(Θ(Θxx & Θ & Θ~x~x) )

Law of self-consistancy Law of self-consistancy (metalogical)(metalogical)

(Θ(Θxx & & ΘΘxx) )

Extension and anti-extension Extension and anti-extension of situational termsof situational terms

М = <М = <WW, П, ПП, П, > >– WW – – set of possible worlds set of possible worlds – П П ⊆⊆ 2 2ww – set of its subsets– set of its subsets – – functionfunction that ascribes to each that ascribes to each

situational term situational term a pair of a pair of elements of elements of П: П: (() = <) = <11((), ), 22(()>, )>, which are to be regarded as which are to be regarded as extension and anti-extension of extension and anti-extension of

Extension and anti-extensionExtension and anti-extension are are defined independently, so defined independently, so it is not it is not postulated thatpostulated that

(i) (i) 11(() ) 22(() = ) = WW

(ii) (ii) 11(() ) 22(() = ) =

Truth-definition for atomic formulasTruth-definition for atomic formulas: : ww ⊨ ⊨ Θ Θ w w 11(())

Extension and anti-extension Extension and anti-extension of situational termsof situational terms

Haw to make your logic non-Haw to make your logic non-classical (V. A. Smirnov)classical (V. A. Smirnov)

Vary the abstract logic (the logic of truth) without varying the empirical one (the logic of situations)

Vary the logic of situations (e.g. accepting or abolishing (i) and (ii)), the abstract logic remaining the same

Vary both (no metalogic in that case)

Traditional interpretation of Traditional interpretation of situational termssituational terms

aa ~a~a

WW

Interpretation of situational Interpretation of situational terms on pairs of situationsterms on pairs of situations

WW

aa ~a~a

Some questionsSome questions

Do extensions and anti-extensions depend Do extensions and anti-extensions depend on context? If so, what is the nature of such on context? If so, what is the nature of such correlation?correlation?

What intentional act produces the very What intentional act produces the very distinction between extension and anti-distinction between extension and anti-extension? extension?

Is it only the properties of reality that Is it only the properties of reality that explain the paraconsistency of internal logic, explain the paraconsistency of internal logic, or some properties of our language as wellor some properties of our language as well? ?

How is it possible to have no metalogic?How is it possible to have no metalogic?

Pragmatics of language as an Pragmatics of language as an instrument of empirical determination instrument of empirical determination

of our thoughtof our thought Ordinary language is highly context-Ordinary language is highly context-

dependent and it is not just the dependent and it is not just the matter of its alleged “irregularity”matter of its alleged “irregularity”

The usual way to “free” the language The usual way to “free” the language of this dependence is to pretend that of this dependence is to pretend that our context is the universal oneour context is the universal one

But in fact all that we can But in fact all that we can meaningfully assert we must first meaningfully assert we must first expose in some contextexpose in some context

Pragmatics of language as an instrument of Pragmatics of language as an instrument of empirical determination of our thoughtempirical determination of our thought

The pair <extensionThe pair <extension, , anti-extension>anti-extension> constitute the real meaning of sentence – constitute the real meaning of sentence – proposition proposition

The proposition is a result of special The proposition is a result of special intentional act – act of putting a situation intentional act – act of putting a situation into the context of other oneinto the context of other one

Situations themselves mean nothing, they Situations themselves mean nothing, they obtain meaning only through each otherobtain meaning only through each other

That is why we cannot assert situations, That is why we cannot assert situations, but only propositions but only propositions

Pragmatics of language as an Pragmatics of language as an instrument of empirical determination instrument of empirical determination

of our thoughtof our thought The act of putting a situation into a The act of putting a situation into a

context can be regarded as operation that context can be regarded as operation that prepares our thought to the assertionprepares our thought to the assertion

This act induces a kind of polarization This act induces a kind of polarization within our thought, distinguishing the within our thought, distinguishing the “possibilities of truth” (Wittgenstein, TLP “possibilities of truth” (Wittgenstein, TLP 4.41)4.41)

In general there could be n such In general there could be n such possibilities, not just two (Vasiliev’s idea of possibilities, not just two (Vasiliev’s idea of n-dimensional logics)n-dimensional logics)

Three-levelled logic (TLL)Three-levelled logic (TLL)

SituationsSituations

PropositionsPropositions

SentencesSentences

Ontological Ontological levellevel

Pragmatic Pragmatic levellevel

Gnoseological Gnoseological levellevel

TLL-languageTLL-language а, а, bb, , cc … – … – situational variablessituational variables 1, 0 – 1, 0 – situational constantssituational constants ~, ~, , , – – situational operatorssituational operators – – ““positioning in context” operator positioning in context” operator ((it it

converts situational terms into converts situational terms into propositional onespropositional ones))

, , , , – – propositional operators (not to propositional operators (not to be confused with connectives of PL)be confused with connectives of PL)

Θ – Θ – assertion operator assertion operator ((it converts it converts propositional terms into formulaspropositional terms into formulas))

, , &&, , , , – – sententional connectivessententional connectives

Axiom schemes of TLLAxiom schemes of TLL

AA0. 0. Axiom schemes of PLAxiom schemes of PL

А1. Θ(1А1. Θ(11)1)

А2. А2. Θ(0Θ(01)1)

АА3. 3. ΘΘ(x(xx) x) ΘΘ(x(x1)1)

АА4. 4. ΘΘ(x(x(y(yz)) z)) ΘΘ((x((xz)z)y)y)

АА5. 5. ΘΘ((x((xx)x)(y(yy))y)) ΘΘ((y((yy)y)(x(xx))x))

В1В1. . ΘΘ(~x(~xy) y) ΘΘ(x(xy)y) BB22. . ΘΘ(x(x~(y~(yz)) z)) ΘΘ(x(x~y ~y xx~z)~z) BB33. . ΘΘ(x(x~(y~(yz)) z)) ΘΘ(x(x~y + x~y + x~z)~z) BB44. . ΘΘ(x(x~~y) ~~y) ΘΘ(x(xy)y)

Axiom schemes of TLLAxiom schemes of TLL

С1. ΘС1. Θ(( ) ) ΘΘ ΘΘСС2. 2. ΘΘ(( ) ) ΘΘ ΘΘСС3. 3. ΘΘ(( )) ΘΘ ΘΘСС4. 4. ΘΘ(( )) ΘΘ ΘΘСС5. 5. ΘΘ ΘΘ

Axiom schemes of TLLAxiom schemes of TLL

Rules of inferenceRules of inference

RR1. Modus ponens1. Modus ponens RR2. 2. Principle of inverse relationPrinciple of inverse relation

Θ(Θ(xx1) 1) Θ(у Θ(у1)1)

Θ(Θ(zzyy) ) Θ( Θ(zzxx)) RR3. 3. Principle of stability of meaningPrinciple of stability of meaning

Θ(Θ(xx1) 1) Θ(у Θ(у1)1)

Θ(Θ(xxzz) ) Θ( Θ(yyzz))

Deuniversalisation of contextDeuniversalisation of context

WW

aa11 ~a~a11

Deuniversalisation of contextDeuniversalisation of context

WW

aabb ~a~abb

bb

TLL-frameTLL-frame М = <М = <WW, , SS, , PP>> WW ≠ ≠ – set of possible worlds – set of possible worlds SS = = <<SS, , , , , , , , ~, ~, , , WW,, -1 -1 > >

– S S ⊆⊆ 22WW – set of situations – set of situations– <S, <S, , , , ~> – boolean algebra, ~> – boolean algebra– <S, <S, , , , W > – dually right-ordered monoid, W > – dually right-ordered monoid– <S, <S, , W, , W, -1-1> – inverse monoid> – inverse monoid aa,,bbSS ( (aa ⊆ ⊆ b b a a-1-1 ⊆ ⊆ bb-1-1))

PP = = <<PP, , , , , , , , >> – de Morgan lattice of – de Morgan lattice of pairspairs– P P ⊆⊆ SSS – set of propositionsS – set of propositions

TLL-modelTLL-model ММ = <М, = <М, >>

(1) = (1) = WW

(0) = (0) = ((xxyy) = ) = ((xx) ) (у)(у)

((xxyy) = ) = ((xx) ) (у)(у)

(~х) = ~(~х) = ~((xx))

(х(хyy) = <) = <(х)(х)((yy))-1-1, , (~х)(~х)((yy))-1-1>>

(a + b) = (a + b) = (a) (a) (b)(b)

(a (a b) = b) = (a) (a) (b)(b)

(a(a) = ) = (a)(a)

Not two, but three senses of Not two, but three senses of the law of non-contradictionthe law of non-contradiction

⊨ ⊨ (Θ((Θ(xxyy) & ) & Θ(Θ(xxyy)) )) ⊭ ⊭ (Θ((Θ(xxyy) & Θ(~) & Θ(~xxyy))))⊭ ⊭ (Θ((Θ(xxyy) & Θ() & Θ(xx~~yy)) ))

Summary Summary

Pragmatics of language is an instrument of Pragmatics of language is an instrument of empirical determination of our thoughtempirical determination of our thought

The principle of inverse relation is the main The principle of inverse relation is the main feature of itfeature of it

We must put the pragmatical level between We must put the pragmatical level between ontological and gnoseological levellsontological and gnoseological levells

As a result we have three natural variants As a result we have three natural variants of the law of non-contradiction: the first is of the law of non-contradiction: the first is valid, the third obviuosly isn’t, the second valid, the third obviuosly isn’t, the second is invalid too but now we know is invalid too but now we know why why