Proof theory of witnessed G¨odel logic: a
... calculi for Gödel logic in [2, 21, 22] are ∃-analytic and so are all calculi in the various formalisms that are presented as “well-behaved” (or called analytic). Having provided a precise definition of the notion of ∃-analyticity, we show that a large class of logics, including Gw , cannot have suc ...
... calculi for Gödel logic in [2, 21, 22] are ∃-analytic and so are all calculi in the various formalisms that are presented as “well-behaved” (or called analytic). Having provided a precise definition of the notion of ∃-analyticity, we show that a large class of logics, including Gw , cannot have suc ...
The Continuum Hypothesis H. Vic Dannon September 2007
... commonly accepted postulates of set theory are consistent, then the addition of the negation of the hypothesis does not result in inconsistency [1]. Cohen’s result was interpreted to mean that there is another set theory that utilizes the negation of the Continuum Hypothesis. However, sets with card ...
... commonly accepted postulates of set theory are consistent, then the addition of the negation of the hypothesis does not result in inconsistency [1]. Cohen’s result was interpreted to mean that there is another set theory that utilizes the negation of the Continuum Hypothesis. However, sets with card ...
Modular Construction of Complete Coalgebraic Logics
... property which ensures that combined logics have the Hennessy-Milner property w.r.t. behavioural equivalence, that is, the logical equivalence of states coincides with behavioural equivalence. Since this property is present in all of the basic constructs and is preserved by each combination of const ...
... property which ensures that combined logics have the Hennessy-Milner property w.r.t. behavioural equivalence, that is, the logical equivalence of states coincides with behavioural equivalence. Since this property is present in all of the basic constructs and is preserved by each combination of const ...
PDF
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
Advanced Logic —
... • the non-logical vocabulary consists of two propositional symbols, p and q; • the logical vocabulary consists of just the symbol ∧. We might then define a well-formed string as follows: • Let p and q be atoms; thus they are well-formed string. • If we have two well formed string, say ϕ and χ, then ...
... • the non-logical vocabulary consists of two propositional symbols, p and q; • the logical vocabulary consists of just the symbol ∧. We might then define a well-formed string as follows: • Let p and q be atoms; thus they are well-formed string. • If we have two well formed string, say ϕ and χ, then ...
Notes on Discrete Mathematics CS 202: Fall 2013 James Aspnes 2014-10-24 21:23
... Naive set theory . . . . . . . . . . . . . . . . . . . . . Operations on sets . . . . . . . . . . . . . . . . . . . . Proving things about sets . . . . . . . . . . . . . . . . Axiomatic set theory . . . . . . . . . . . . . . . . . . . Cartesian products, relations, and functions . . . . . . 3.5.1 Ex ...
... Naive set theory . . . . . . . . . . . . . . . . . . . . . Operations on sets . . . . . . . . . . . . . . . . . . . . Proving things about sets . . . . . . . . . . . . . . . . Axiomatic set theory . . . . . . . . . . . . . . . . . . . Cartesian products, relations, and functions . . . . . . 3.5.1 Ex ...
Notes on Discrete Mathematics
... Operations on sets . . . . . . . . . . . . . . . . . . . . . . . . 49 Proving things about sets . . . . . . . . . . . . . . . . . . . . 50 ...
... Operations on sets . . . . . . . . . . . . . . . . . . . . . . . . 49 Proving things about sets . . . . . . . . . . . . . . . . . . . . 50 ...
A Mathematical Introduction to Modal Logic
... Nevertheless, we will try our best not to lose our basic intuition by making occasional remarks to the philosophical and applied considerations. The intended course is a short one, and these notes will cover only the basics. For this reason, based on a subjective judgement, I left out several import ...
... Nevertheless, we will try our best not to lose our basic intuition by making occasional remarks to the philosophical and applied considerations. The intended course is a short one, and these notes will cover only the basics. For this reason, based on a subjective judgement, I left out several import ...
Modal Reasoning
... Bisimulations have two major uses; we consider tree unraveling first, then model contraction. Definition: Tree Unraveling Every modal M, s has a bisimulation with a rooted tree-like model constructed as follows. The worlds in the tree unraveling are all finite paths of worlds in M starting with s an ...
... Bisimulations have two major uses; we consider tree unraveling first, then model contraction. Definition: Tree Unraveling Every modal M, s has a bisimulation with a rooted tree-like model constructed as follows. The worlds in the tree unraveling are all finite paths of worlds in M starting with s an ...
A Taste of Categorical Logic — Tutorial Notes
... is a sequent expressing that the sum of two odd natural numbers is an even natural number. However that is not really the case. The sequent we wrote is just a piece of syntax and the intuitive description we have given is suggested by the suggestive names we have used for predicate symbols (odd, eve ...
... is a sequent expressing that the sum of two odd natural numbers is an even natural number. However that is not really the case. The sequent we wrote is just a piece of syntax and the intuitive description we have given is suggested by the suggestive names we have used for predicate symbols (odd, eve ...
The logic and mathematics of occasion sentences
... which such problems do not exist. This is the basis of the programme initiated by Russell and continued by Quine, who dubbed it the programme of ‘elimination of particulars’ (Quine 1960). This programme, which underlies virtually all the work done in present-day model-theoretic or ‘formal’ semantics ...
... which such problems do not exist. This is the basis of the programme initiated by Russell and continued by Quine, who dubbed it the programme of ‘elimination of particulars’ (Quine 1960). This programme, which underlies virtually all the work done in present-day model-theoretic or ‘formal’ semantics ...
Completeness theorems and lambda
... where we use only arithmetical comprehension It is standard that second-order arithmetic with arithmetical comprehension is conservative over Peano arithmetic From this, it is possible to deduce that the functions of type N → N representable in the system F0 are exactly the functions provably total ...
... where we use only arithmetical comprehension It is standard that second-order arithmetic with arithmetical comprehension is conservative over Peano arithmetic From this, it is possible to deduce that the functions of type N → N representable in the system F0 are exactly the functions provably total ...
CUED PhD and MPhil Thesis Classes
... presented as sequent systems, and we use some basic proof-theoretic properties of them (cut elimination, subformula property). We prove some versions of these properties for theories. An interpolation lemma, proved here, plays an essential role. We also consider algebras corresponding to these logic ...
... presented as sequent systems, and we use some basic proof-theoretic properties of them (cut elimination, subformula property). We prove some versions of these properties for theories. An interpolation lemma, proved here, plays an essential role. We also consider algebras corresponding to these logic ...
full text (.pdf)
... logic, foreshadowing Kripke’s [1963; 1965] formulation of similar state-based semantics for these logics (see [Artemov 2001]). Kripke models also form the basis of the standard semantics of DL (see [Harel et al. 2000]), although as mentioned, DL does not realize the intuitionistic nature of partial ...
... logic, foreshadowing Kripke’s [1963; 1965] formulation of similar state-based semantics for these logics (see [Artemov 2001]). Kripke models also form the basis of the standard semantics of DL (see [Harel et al. 2000]), although as mentioned, DL does not realize the intuitionistic nature of partial ...
Structural Multi-type Sequent Calculus for Inquisitive Logic
... An easy inductive proof shows that classical formulas χ are flat (also called truth conditional); that is, for every state S , (Flatness Property) S |= χ iff {v} |= χ for any v ∈ S iff v(χ) = 1 for any v ∈ S . Well-formed formulas φ of inquisitive logic (InqL) are given by expanding the language of ...
... An easy inductive proof shows that classical formulas χ are flat (also called truth conditional); that is, for every state S , (Flatness Property) S |= χ iff {v} |= χ for any v ∈ S iff v(χ) = 1 for any v ∈ S . Well-formed formulas φ of inquisitive logic (InqL) are given by expanding the language of ...
full text (.pdf)
... with every inductive proof that the proof is a valid application of the induction principle. We emphasize that we are not claiming to introduce any new coinductive proof principles. The foundations of coinduction underlying our approach are well known. Rather, our purpose is only to present an infor ...
... with every inductive proof that the proof is a valid application of the induction principle. We emphasize that we are not claiming to introduce any new coinductive proof principles. The foundations of coinduction underlying our approach are well known. Rather, our purpose is only to present an infor ...
Chapter 6: The Deductive Characterization of Logic
... critical subordinate lemma called Lindenbaum’s Lemma. This lemma claims that every deductivelyconsistent subset can be extended to a maximal deductively-consistent set. In order to understand this lemma, we must understand the term ‘maximal’, which is a general set theoretic term, defined as follows ...
... critical subordinate lemma called Lindenbaum’s Lemma. This lemma claims that every deductivelyconsistent subset can be extended to a maximal deductively-consistent set. In order to understand this lemma, we must understand the term ‘maximal’, which is a general set theoretic term, defined as follows ...
Master Thesis - Yoichi Hirai
... or it has not reached the intended receiver. We point out that this reasoning assumes the existence of a current state of the world. The notion of the current state implicitly assumes global clock within the use of the adjective “current”. In classical epistemic logic, the description of knowledge r ...
... or it has not reached the intended receiver. We point out that this reasoning assumes the existence of a current state of the world. The notion of the current state implicitly assumes global clock within the use of the adjective “current”. In classical epistemic logic, the description of knowledge r ...
Large cardinals and the Continuum Hypothesis
... The question regarding the size of the continuum – i.e. the number of the reals – is probably the most famous question in set theory. Its appeal comes from the fact that, apparently, everyone knows what a real number is and so the question concerning their quantity seems easy to understand. While th ...
... The question regarding the size of the continuum – i.e. the number of the reals – is probably the most famous question in set theory. Its appeal comes from the fact that, apparently, everyone knows what a real number is and so the question concerning their quantity seems easy to understand. While th ...
A proof
... A proof : is a valid argument that establishes the truth of a mathematical statement. There are two types of proofs : Formal proof : where all steps are supplied and the rules for each step in the argument are given Informal proof : where more than one rule of inference may be used in ...
... A proof : is a valid argument that establishes the truth of a mathematical statement. There are two types of proofs : Formal proof : where all steps are supplied and the rules for each step in the argument are given Informal proof : where more than one rule of inference may be used in ...
Mathematical Logic
... To save parentheses in quantified formulas, we use a mild form of the dot notation: a dot immediately after ∀x or ∃x makes the scope of that quantifier as large as possible, given the parentheses around. So ∀x.A → B means ∀x(A → B), not (∀xA) → B. We also save on parentheses by writing e.g. Rxyz, Rt ...
... To save parentheses in quantified formulas, we use a mild form of the dot notation: a dot immediately after ∀x or ∃x makes the scope of that quantifier as large as possible, given the parentheses around. So ∀x.A → B means ∀x(A → B), not (∀xA) → B. We also save on parentheses by writing e.g. Rxyz, Rt ...
Document
... …shew. But that’s not the end of the story. This was just the intuitive derivation of the formula, not the proof. LEMMA: The maximal number of intersection points of n lines in the plane is n(n-1)/2. Proof. Prove by induction. Base case: If n = 1, then there is only one line and therefore no interse ...
... …shew. But that’s not the end of the story. This was just the intuitive derivation of the formula, not the proof. LEMMA: The maximal number of intersection points of n lines in the plane is n(n-1)/2. Proof. Prove by induction. Base case: If n = 1, then there is only one line and therefore no interse ...
pdf
... a set S of pure formulas (i.e formulas without parameters). Such a tableau starts with an arbitrary element of S at its origin and is then constructed by applying either one of the four rules α, , γ, or δ, or by adding another element of S to the end of an open branch. The elements of S so added are ...
... a set S of pure formulas (i.e formulas without parameters). Such a tableau starts with an arbitrary element of S at its origin and is then constructed by applying either one of the four rules α, , γ, or δ, or by adding another element of S to the end of an open branch. The elements of S so added are ...
pdf
... world, Ki is a binary relation on S for each agent i = 1, . . . , n, and Ai is a function associating a set of sentences with each world in S, for i = 1, ..., n. Intuitively, if (s, t) ∈ Ki , then agent i considers world t possible at world s, while Ai (s) is the set of sentences that agent i is awa ...
... world, Ki is a binary relation on S for each agent i = 1, . . . , n, and Ai is a function associating a set of sentences with each world in S, for i = 1, ..., n. Intuitively, if (s, t) ∈ Ki , then agent i considers world t possible at world s, while Ai (s) is the set of sentences that agent i is awa ...
KURT GÖDEL - National Academy of Sciences
... (1978) by Quine gives an excellent overview in just under four pages. ...
... (1978) by Quine gives an excellent overview in just under four pages. ...
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita).The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set ""number"". The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the ""underlying logic"". The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.