1 / 18

# The Size of the World of Logic - PowerPoint PPT Presentation

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 ;

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' The Size of the World of Logic' - midori

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### The Size of the World of Logic

Jan Woleński

JagiellonianUniversity, Krakow, Poland

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.

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.

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.

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.

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.

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’ = .

DifferentAccounts

Truth and falsehood as modalities:

 

 

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.

DifferentAccounts

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

 

 

DifferentAccounts

 – 0,  – 1A0A,  – 1A 0A

(11) and the triangle.

(13) A(1A0A),

Conclusion: (BI) is not a tautology.

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

OtherLogics

.

 

 

 

Other Logics

 , – A i A.

(16) D(A) A,

holds for every value, but reverse dependence not.

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 GeneralForm of the World of Logic

(WL) {w1, w2 ,…, wn, …}.

Argument for Bivalence

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

• 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.