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20150629 ``The Meaninglessness of Algebraic Semantics''
- 1. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations The Meaninglessness of Algebraic-Semantics Rational Beings WITH-IN Logic and its Semantics Satoru HANADA shanada3007@eis.hokudai.ac.jp Graduate School of Letters(M1), Hokkaido University June 29, 2015 UNILOG2015 @Istanbul Univ. Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 2. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations 1 the Aim and the Method the Aim the Method 2 Prologue 3 A summarization and its presuppositions A Summarization its Presuppositions 4 Object- / Meta- Language and the Worlds 5 Summary and Further Investigations Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 3. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations the Aim the Method The Aim any logical system and its semantics which are object-languages do not contain any meanings. e.g. Validity doesn’t come from formal logic, but from us. e.g. John Woods(2015, p32)“(82) The value of ∼ A is just when the value of ( )A( ) is F.” Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 4. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations the Aim the Method Method Examine a possible summerization of algebraic-semantics, and Show its presupposition. Introduce a distinction between object-language and other things. Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 5. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations ‘WITH-IN’ Logic is not only inside the given system. We can think from within inside the language. (I would like to avoid demanding transcendental elements.) Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 6. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations A Summarization its Presuppositions A possible summerization A role of algebraic-semantics merely consists in substituting algebraic equations and inequalities for the axioms and rules of a given logical system. (Algebraic-semantics is no more than a paraphrase of LJ into algebra.) Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 7. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations A Summarization its Presuppositions → - Right α, β ⇒ γ β ⇒ α → γ (→ −Right) The law of residuation a ∧ b ≤ c ⇔ b ≤ (a → c) No difference! ∴Algebraic-semantics does not explain the meaning of logic? Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 8. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations A Summarization its Presuppositions McKinsey-Tarski’s Thm LJ (/LDJ) is complete for Heyting Algebra composed of the total of open sets (/Brouwer Algebra composed of the total of closed sets). Moreover, finite model property α is provable in LJ iff. α is tautology in finite arbitrary Heyting algebra. Then, we can get infinite number of models. And we do not have any criteria to select which is the meaning. Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 9. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations A Summarization its Presuppositions However, this criticism is based on naive misunderstanding. Presupposition This semantics of the given system have to show something about logical system, in addition to completeness (and vice versa). There is an order of values of systems and of their semantics. Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 10. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations (Look at a Figure) distinction Object-language is only a set of symbols and operations. Meta-language is language which we use and/or is our linguistic activity. There is outside of Meta-language, e.g. the ontic world and the ontological world. Remark LJ and HA are seen as object-language. Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 11. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations It is useful to compare the principle of bivalence with Boolean Algebra with only 2 elements B(2). Each of them is semantics of classical logic, so they are of equal value as an object-language. But that criticism might not attack the principle of bivalence. Because the principle is not only inside object-language. It connects language to the world. the criticism mentioned aboved is category mistake. Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 12. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Conclusion The meaning of logical system can be found in meta-language and the world, which is outside of object-language, by Rational Beings. So every object-language has equal value. Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 13. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Start with a possible summerization. Get some unacceptable conclusions. Introduce a distinction between object-language and other things. Cancel the presupposition and get the conclusion. Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 14. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Yet, it is not clear how to find meanings. How to get the meaning is connected to next theme. A Theme In what way do knowledge and the world connect to each of them. After acquiring the view that there is no hierarchy and meanings of logical systems, how do logics / the world / knowledge look like? Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 15. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Additional Refference Komori, et Ono (2010) an Introduction to Modern Mathematical Logic(“Gendai Suuri Ronnrigaku Josetu”written in Japanese) Aoyama, et al. (2007) Non-Classical Logics and Algebraic-Models(“Hi-Koten Ronri to Daisuu Moderu”written in Japanese) Acknowledgements Prof. Koji NAKATOGAWA(Hokkaido Univ.) and Takuma IMAMURA(Toyama Univ. B4). Satoru HANAD The Meaninglessness of Algebraic-Semantics
- 16. The Meaninglessness of Algebraic- Semantics Satoru HANAD Outline the Aim and the Method the Aim the Method Prologue A summarization and its presuppositions A Summarization its Presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Outline the Aim and the Method Prologue A summarization and its presuppositions Object- / Meta- Language and the Worlds Summary and Further Investigations Thanks for your attention! Satoru HANAD The Meaninglessness of Algebraic-Semantics