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The Size of the World of Logic. Jan Woleński Jagiellonian University , Krakow , Poland. Talk Outline. W hat is the world of logic ; Different a ccount s; Other logics ; T, (BI) and propositional calculus ; The general f orm of the wor l d of logic ; Argument for bivalence ;

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The size of the world of logic

The Size of the World of Logic

Jan Woleński

JagiellonianUniversity, Krakow, Poland


Talk outline
Talk Outline

  • What is the world of logic;

  • Differentaccounts;

  • Other logics;

  • T, (BI) and propositional calculus;

  • The general formof the world of logic;

  • Argument for bivalence;

  • Other.


W hat is the w orld of l ogic
What is the Worldof Logic

The problem: what is the world of logic

Russell: Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.

But what are more abstract and general features of the world?

Logic as consisting of tautologies.

Frege: Logic is concerned with the predicate “true”

Frege’ssemantics of sentences: the True and the False as references

(senses) of sentences.


W hat is the w orld of l ogic1
What is the Worldof Logic

Example:

  • p p q.

  • 1 (10),

  • 0 (01), 0 (00) and 0 (01).

  • p r q.

  • A A B as the way out.

    “Dual” logic

  • w(A B) = 1 wtww(A) = w(B) = 0; otherwisew(A B) = 0.

  • A B  A.

  • 1 as the distinguished values.


Different a ccount s
DifferentAccounts

The world of logic consists of logical value.

(BI) every sentence is either true or false.

9. (BI)  (CN)  (EM).

(BI) and the theorems of PC.


Different a ccount s1
DifferentAccounts

Béziau on conditions for (BI):

  • Counter-domain of w is two-elements;

  • Domain of w - the set of sentences;

  • w is a total function.


Different a ccount s2
DifferentAccounts

Other account:

(a) card(V) = 0; no A is a theorem;

(b) card(V) = 1; every A is a theorem;

(c) card(V)  2; some A are theorems, some A are not theorems.

D – distinguished values, D’ – non-distinguished values, L is consistent if

card(V)  card(D) V = D  D’, D D’ = .


Different a ccount s3
DifferentAccounts

Truth and falsehood as modalities:

 

 


Different a ccount s4
DifferentAccounts

 – 1A, – 1(A),  – 1(A),  – 1(A).

(10)(a) 

(b) 

(c) () ;

(d) 

(e) ;

(f)) .

What about 0? Either  or .

(11)(a) 0A 1A;

(b) 0A 1A.

(12) 1A 1A 0A.


Different a ccount s5
DifferentAccounts

(11), (12) (BI) and more than 2 values.

 

 


Different a ccount s6
DifferentAccounts

 – 0,  – 1A0A,  – 1A 0A

(11) and the triangle.

(13) A(1A0A),

Conclusion: (BI) is not a tautology.


Other l ogics
Other Logics

Assumption: the only designated value.

Is possible to save (BI)?

(14) A(D(A) D(A), D’, DA

(15) A(D(A)  DA(A)),

DA(A) D(A) and its legitimization.

T-scheme : TAA,

DA and A


Other logics
OtherLogics

.

 

 

 


Other l ogics1
Other Logics

 , – A i A.

(16) D(A) A,

holds for every value, but reverse dependence not.


T bi and p ropositional c alculus
T, (BI) and PropositionalCalculus.

(17) w(T(A)) = 1iffw(A) = 1; otherwise w(A) = 0.

The formula (17) is not generalized to predicate calculus.


The g eneral form of the w or l d of l ogic
The GeneralForm of the World of Logic

(WL) {w1, w2 ,…, wn, …}.


Argument for b ivalence
Argument for Bivalence

Argument for bivalence: metalogic (the role of classical logic, simplicity.


Other
Other

  • Truth – facts, falsehood – the lack of facts’;

  • Various oppositions, spatial, temporal;

  • Biological oppositions;

  • Passive- active;

  • Possession and its lack; Inner – outer;

  • Modal contrasts;

  • Biological rhythms are binary;

  • Perceptual contrasts;

  • Binary structure of the helix and genetic codes;

  • 0-1 nature of information;

  • Truth protects information, falsehood results in its dispersion;

  • Ordinary quantifiers are dual.


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