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Logical Monism: The Global Identity of Applicable Logic
 Advanced Studies in Mathematics and Logic
, 2005
"... Abstract. ‘One universe, one logic ’ takes the world as it is and leads to adjointness as the global logic of anything. The alternative approach to find a unification of known logics requires assumptions and is therefore consistent with the same conclusion for a universal logic has to be universally ..."
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Abstract. ‘One universe, one logic ’ takes the world as it is and leads to adjointness as the global logic of anything. The alternative approach to find a unification of known logics requires assumptions and is therefore consistent with the same conclusion for a universal logic has to be universally applicable. The universal characteristic of adjointness is that it has a natural construction from the concept of the arrow. The application to the test sentence, ‘John said that Mary believed he did not love her’, demonstrates adjointness as the logic of the postmodern world. 1 Unity of Applicable Logic There is one ultimate logic: it is a simple ontological but pragmatic argument of ‘one universe, one logic’. If more, how can we know unless there is a logic to compare them? If logic is a family of varying strength, what logic compares the variance? Only some ultimate logic. How do we even know this? It must still be the same logic that tells us this. And that logic must tell us about itself − − tell us that it has some recursive selfclosure. The same pragmatic cogency leads us into the world of physics and beyond into the humanities. The world must fit together according to this same ultimate logic. It is therefore an applicable logic. Universal logic means universally applicable logic. This study arises from the investigation of fundamentals in two large applied areas: one is schema design in interoperable databases, the other is in legal reasoning; both studies relate logic to realworld facts. Until we are able to identify the ultimate logic of the universe, it is not surprising that goals like unified field theory within a ”theory of everything ” are so elusive. Applicable logic is needed in new ways in biology, medicine, economics, legal science, natural computing, modern physics, etc. This means it has to be a logic which can manage the advances made in the twentieth century, many of
Volume of Abstracts Logic, Algebra and Truth Degrees 2010
"... Mathematical Fuzzy Logic (MFL) is the subdiscipline of mathematical logic devoted to the study of formal systems of fuzzy logic. It has been a fairly active research field for more than two decades, since scholars undertook the task of providing solid formal foundations for deductive systems arising ..."
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Mathematical Fuzzy Logic (MFL) is the subdiscipline of mathematical logic devoted to the study of formal systems of fuzzy logic. It has been a fairly active research field for more than two decades, since scholars undertook the task of providing solid formal foundations for deductive systems arising from Fuzzy Set Theory by realizing that these systems could be seen as a special kind of manyvalued logics. This approach turned out to be very fruitful when Petr Hájek collected the results of the first systematic study of fuzzy logics in his monograph Metamathematics of Fuzzy Logic (Kluwer, 1998), a true landmark of the field. This book, together with other influential works by prominent researchers, was the start of an ambitious scientific agenda aiming to the study of all aspects of fuzzy logics, including algebraic semantics, proof systems, gametheoretic semantics, functional representation, firstorder and higherorder logics, decidability and complexity issues, model theory, philosophical issues and applications. Moreover, it was made clear
Contemporary Mathematics A beginner’s guide to forcing
"... In 1963, Paul Cohen stunned the mathematical world with his new technique of forcing, which allowed him to solve several outstanding problems in set theory at a single stroke. Perhaps most notably, he proved the independence of the continuum hypothesis (CH) from the ZermeloFraenkelChoice (ZFC) axi ..."
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In 1963, Paul Cohen stunned the mathematical world with his new technique of forcing, which allowed him to solve several outstanding problems in set theory at a single stroke. Perhaps most notably, he proved the independence of the continuum hypothesis (CH) from the ZermeloFraenkelChoice (ZFC) axioms of set theory. The
TERM EXTRACTION AND RAMSEY’S THEOREM FOR PAIRS
"... Abstract. In this paper we study with prooftheoretic methods the function(al)s provably recursive relative to Ramsey’s theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functionals provably recursive from RCA0 + COH + Π0 1CP are primitive recursive. This ..."
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Abstract. In this paper we study with prooftheoretic methods the function(al)s provably recursive relative to Ramsey’s theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functionals provably recursive from RCA0 + COH + Π0 1CP are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that WKL0 + Π0 1CP + COH is Π0 2conservative over PRA. Recent work of the first author showed that Π0 1CP + COH is equivalent to a weak variant of the BolzanoWeierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs. For Ramsey’s theorem for pairs and two colors (RT2 2) we obtain the upper bounded that the type 2 functionals provable recursive relative to RCA0 + Σ0 2IA+RT2 2 are in T1. This is the fragment of Gödel’s system T containing only type 1 recursion — roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T1bounds from proofs that use RT2 2. Moreover, this yields a new proof of the fact that WKL0 + Σ0 2IA + RT2 2 is Π0 3conservative over RCA0 + Σ0 2IA. The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel’s functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using Π0 1comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard’s ordinal analysis of bar recursion (for RT2 2). 1.
Algebraically Closed Fields
"... Algebraic closure In the previous lecture, we have seen how to “force ” the existence of prime ideals, even in a weark framework where we don’t have choice axiom Instead of “forcing ” the existence of a point of a space (a mathematical object), we are going to “force ” the existence of a model (a ma ..."
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Algebraic closure In the previous lecture, we have seen how to “force ” the existence of prime ideals, even in a weark framework where we don’t have choice axiom Instead of “forcing ” the existence of a point of a space (a mathematical object), we are going to “force ” the existence of a model (a mathematical structure)