Proofs in theories
... These course notes are organized in four parts. In Chapters 1, 2, and 3, we shall present the basic notions of proof, theory and model used in these course notes. When presenting the notion of proof we emphasize the notion of constructivity and that of cut. When we present the notion of theory, we e ...
... These course notes are organized in four parts. In Chapters 1, 2, and 3, we shall present the basic notions of proof, theory and model used in these course notes. When presenting the notion of proof we emphasize the notion of constructivity and that of cut. When we present the notion of theory, we e ...
Henkin`s Method and the Completeness Theorem
... In fact, a stronger version of this result is also true. For each first-order theory T and a sentence ϕ (in the language of T ), we have that T ` ϕ iff T |= ϕ. Thus, each first-order theory T (pick your favorite one!) is sound and complete in the sense that everything that we can derive from T is tr ...
... In fact, a stronger version of this result is also true. For each first-order theory T and a sentence ϕ (in the language of T ), we have that T ` ϕ iff T |= ϕ. Thus, each first-order theory T (pick your favorite one!) is sound and complete in the sense that everything that we can derive from T is tr ...
Proof theory for modal logic
... to establish decidability and consistency results through a combinatorial analysis of derivations. This analysis is made possible by the conversion, called normalization, of a derivation to a normal form that does not contain redundant parts and that satisfies the subformula property, a basic requi ...
... to establish decidability and consistency results through a combinatorial analysis of derivations. This analysis is made possible by the conversion, called normalization, of a derivation to a normal form that does not contain redundant parts and that satisfies the subformula property, a basic requi ...
Weyl`s Predicative Classical Mathematics as a Logic
... as described below. An LTT consists of a type theory augmented with a separate, primitive mechanism for forming and proving propositions. We introduce a new syntactic class of formulas, and new judgement forms for a formula being a well-formed proposition, and for a proposition being provable from g ...
... as described below. An LTT consists of a type theory augmented with a separate, primitive mechanism for forming and proving propositions. We introduce a new syntactic class of formulas, and new judgement forms for a formula being a well-formed proposition, and for a proposition being provable from g ...
CATEGORICAL MODELS OF FIRST
... It seems intuitive that under this interpretation classes of logically equivalent propositions are no longer identified: a proof of ψ ∧ ψ is not a proof of ψ. What is not immediately obvious is how we might formalize this notion; in order to do so, we must first know when two proofs are identical. T ...
... It seems intuitive that under this interpretation classes of logically equivalent propositions are no longer identified: a proof of ψ ∧ ψ is not a proof of ψ. What is not immediately obvious is how we might formalize this notion; in order to do so, we must first know when two proofs are identical. T ...
Deep Sequent Systems for Modal Logic
... approaches. It allows to capture a wide class of modal logics and does so systematically. In many important cases it yields systems which are natural and easy to use, which have good structural properties like contractionadmissibility and invertibility of all rules, and which give rise to decision p ...
... approaches. It allows to capture a wide class of modal logics and does so systematically. In many important cases it yields systems which are natural and easy to use, which have good structural properties like contractionadmissibility and invertibility of all rules, and which give rise to decision p ...
Intuitionistic Logic - Institute for Logic, Language and Computation
... Much more than formalism and Platonism, intuitionism is in principle normative. Formalism and Platonism may propose a foundation for existing mathematics, a reduction to logic (or set theory) in the case of Platonism, or a consistency proof in the case of formalism. Intuitionism in its stricter form ...
... Much more than formalism and Platonism, intuitionism is in principle normative. Formalism and Platonism may propose a foundation for existing mathematics, a reduction to logic (or set theory) in the case of Platonism, or a consistency proof in the case of formalism. Intuitionism in its stricter form ...
THE SEMANTICS OF MODAL PREDICATE LOGIC II. MODAL
... approach, treating constants and variables in the same way, but assuming a more sophisticated notion of a modal individual and identity-at-a-world. It remains unsatisfactory having to choose between these competing semantics. Moreover, it would be nice if the difference between these semantics was b ...
... approach, treating constants and variables in the same way, but assuming a more sophisticated notion of a modal individual and identity-at-a-world. It remains unsatisfactory having to choose between these competing semantics. Moreover, it would be nice if the difference between these semantics was b ...
Modal Logic - Web Services Overview
... Syntax of Modal Logic (□ and ◊) Formulae in (propositional) Modal Logic ML: • The Language of ML contains the Language of Propositional Calculus, i.e. if P is a formula in Propositional Calculus, then P is a formula in ML. • If and are formulae in ML, then ...
... Syntax of Modal Logic (□ and ◊) Formulae in (propositional) Modal Logic ML: • The Language of ML contains the Language of Propositional Calculus, i.e. if P is a formula in Propositional Calculus, then P is a formula in ML. • If and are formulae in ML, then ...
Fine`s Theorem on First-Order Complete Modal Logics
... discreteness property that between any two points there are only finitely many other points. Earlier, Kripke [42] had observed that Dummet’s formula is not preserved by the Jónsson–Tarski representation of modal algebras. This is an algebraic formulation of the non-canonicity of this formula. The a ...
... discreteness property that between any two points there are only finitely many other points. Earlier, Kripke [42] had observed that Dummet’s formula is not preserved by the Jónsson–Tarski representation of modal algebras. This is an algebraic formulation of the non-canonicity of this formula. The a ...
brouwer`s intuitionism as a self-interpreted mathematical theory
... based on a corpus of intuitionistic principles and concepts. Brouwer never used axioms in BIA and he tried to justify all his principles on conceptual grounds. All the above theories are constructive, i.e., they contain, tacitly or not, some constructive principles which guide the execution of proof ...
... based on a corpus of intuitionistic principles and concepts. Brouwer never used axioms in BIA and he tried to justify all his principles on conceptual grounds. All the above theories are constructive, i.e., they contain, tacitly or not, some constructive principles which guide the execution of proof ...
Proof by Induction
... proofs by contradiction are usually easier to write, direct proofs are almost always easier to read. So as a service to our audience (and our grade), let’s transform our minimal-counterexample proof into a direct proof. Let’s first rewrite the indirect proof slightly, to make the structure more appa ...
... proofs by contradiction are usually easier to write, direct proofs are almost always easier to read. So as a service to our audience (and our grade), let’s transform our minimal-counterexample proof into a direct proof. Let’s first rewrite the indirect proof slightly, to make the structure more appa ...
Proof by Induction
... A proof by induction for the proposition “P(n) for every positive integer n” is nothing but a direct proof of the more complex proposition “(P(1) ∧ P(2) ∧ · · · ∧ P(n − 1)) → P(n) for every positive integer n”. Because it’s a direct proof, it must start by considering an arbitrary positive integer, ...
... A proof by induction for the proposition “P(n) for every positive integer n” is nothing but a direct proof of the more complex proposition “(P(1) ∧ P(2) ∧ · · · ∧ P(n − 1)) → P(n) for every positive integer n”. Because it’s a direct proof, it must start by considering an arbitrary positive integer, ...
Set Theory for Computer Science (pdf )
... sets as completed objects in their own right. Mathematicians were familiar with properties such as being a natural number, or being irrational, but it was rare to think of say the collection of rational numbers as itself an object. (There were exceptions. From Euclid mathematicians were used to thin ...
... sets as completed objects in their own right. Mathematicians were familiar with properties such as being a natural number, or being irrational, but it was rare to think of say the collection of rational numbers as itself an object. (There were exceptions. From Euclid mathematicians were used to thin ...
Introduction to Mathematical Logic lecture notes
... the convention that ¬ binds more strongly than the binary connectives). When wanting to prove that all formulae have a certain property, we usually use “proof by induction on the construction of the formula”: Theorem 1.1.1 (Proof by induction on the structure). Let X is a property that a formula may ...
... the convention that ¬ binds more strongly than the binary connectives). When wanting to prove that all formulae have a certain property, we usually use “proof by induction on the construction of the formula”: Theorem 1.1.1 (Proof by induction on the structure). Let X is a property that a formula may ...
Chapter 2 Propositional Logic
... this “circularity” is benign, because the definition is recursive. A recursive (or “inductive”) definition of a concept F contains a circular-seeming clause, often called the “inductive” clause, which specifies that if such-and-such objects are F , then so-and-so objects are also F . But a recursive ...
... this “circularity” is benign, because the definition is recursive. A recursive (or “inductive”) definition of a concept F contains a circular-seeming clause, often called the “inductive” clause, which specifies that if such-and-such objects are F , then so-and-so objects are also F . But a recursive ...
pdf
... Hilbert says nothing about what the “things” are. Axioms. An axiom is a proposition about the objects in question which we do not attempt to prove but rather which we accept as given. One of Euclid’s axioms, for example, was “It shall be possible to draw a straight line joining any two points.” Aris ...
... Hilbert says nothing about what the “things” are. Axioms. An axiom is a proposition about the objects in question which we do not attempt to prove but rather which we accept as given. One of Euclid’s axioms, for example, was “It shall be possible to draw a straight line joining any two points.” Aris ...
Bounded Proofs and Step Frames - Università degli Studi di Milano
... satisfying this first-order condition. This part is not automatic, but we have some standard templates. For example, we define a procedure modifying the relation of a one-step frame so that the obtained frame is standard. In easy cases, e.g., for modal logics such as K, T, K4, S4, this frame is a fr ...
... satisfying this first-order condition. This part is not automatic, but we have some standard templates. For example, we define a procedure modifying the relation of a one-step frame so that the obtained frame is standard. In easy cases, e.g., for modal logics such as K, T, K4, S4, this frame is a fr ...
Chapter 9: Initial Theorems about Axiom System AS1
... if „α, and „α→β, then „β. The remaining occurrence of ‘→’ is under the predicate ‘„’; it is accordingly a proper noun referring to the conditional connective of the object language. When we translate ‘„’ as ‘is a theorem’, we obtain: if α is a theorem, and α→β is a theorem, then β is a theorem. The ...
... if „α, and „α→β, then „β. The remaining occurrence of ‘→’ is under the predicate ‘„’; it is accordingly a proper noun referring to the conditional connective of the object language. When we translate ‘„’ as ‘is a theorem’, we obtain: if α is a theorem, and α→β is a theorem, then β is a theorem. The ...
minimum models: reasoning and automation
... theories does not allow parametrisation. Isabelle has constructs for defining complex types, i.e. function types, product types and recursively defined abstract data types. Isabelle automatically generates induction principles for each declared recursive datatype. Also, Isabelle allows inductive and ...
... theories does not allow parametrisation. Isabelle has constructs for defining complex types, i.e. function types, product types and recursively defined abstract data types. Isabelle automatically generates induction principles for each declared recursive datatype. Also, Isabelle allows inductive and ...
Frege, Boolos, and Logical Objects
... [email protected] In [1884], Frege formulated some ‘abstraction’ principles that imply the existence of abstract objects in classical logic. The most well-known of these is: Hume’s Principle: The number of F s is identical to the number of Gs iff there is a one-to-one correspondence between the F s ...
... [email protected] In [1884], Frege formulated some ‘abstraction’ principles that imply the existence of abstract objects in classical logic. The most well-known of these is: Hume’s Principle: The number of F s is identical to the number of Gs iff there is a one-to-one correspondence between the F s ...
An Introduction to Mathematical Logic
... Example 2 • If = is the equality relation on a non-empty set A, then (A, =) is an equivalence structure. • For m, n ∈ Z let m ≡ n iff there is a k ∈ Z s.t. m − n = 5k. Then (Z, ≡) is an equivalence structure. Consider the following simple theorem on equivalence structures: Theorem 2 Let (A, ≈) be an ...
... Example 2 • If = is the equality relation on a non-empty set A, then (A, =) is an equivalence structure. • For m, n ∈ Z let m ≡ n iff there is a k ∈ Z s.t. m − n = 5k. Then (Z, ≡) is an equivalence structure. Consider the following simple theorem on equivalence structures: Theorem 2 Let (A, ≈) be an ...
Kripke completeness revisited
... which clearly is wrong. In the axiomatic formulation, the premiss of necessitation was a theorem. In a natural deduction system, one requires that A be derivable with no open assumptions. If one thinks of the analogy between necessity and universal quantification, it appears that the restriction is ...
... which clearly is wrong. In the axiomatic formulation, the premiss of necessitation was a theorem. In a natural deduction system, one requires that A be derivable with no open assumptions. If one thinks of the analogy between necessity and universal quantification, it appears that the restriction is ...
The Pure Calculus of Entailment Author(s): Alan Ross Anderson and
... the conclusion to be proved, and setting up subproofs by hyp until we find one with a variable as last step. Only then do we begin applying reit, rep, and -*E. Our description of HI* has been somewhat informal, and for the purpose of checking proofs it would be desirable to have a more rigorous form ...
... the conclusion to be proved, and setting up subproofs by hyp until we find one with a variable as last step. Only then do we begin applying reit, rep, and -*E. Our description of HI* has been somewhat informal, and for the purpose of checking proofs it would be desirable to have a more rigorous form ...
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.