Back to Basics: Revisiting the Incompleteness
... standard interpretation of LA is of course the one which gives the builtin non-logical expressions the obvious arithmetical interpretation, and takes the domain for the numerical quantifiers to be the naturals. To say an LA -sentence is true is just to say that it is true on the standard interpretat ...
... standard interpretation of LA is of course the one which gives the builtin non-logical expressions the obvious arithmetical interpretation, and takes the domain for the numerical quantifiers to be the naturals. To say an LA -sentence is true is just to say that it is true on the standard interpretat ...
Reading 2 - UConn Logic Group
... fragments of Int, classical logic, etc. ([21], [88], [89]). Abstract computational and functional semantics for Int which did not address the issue of the original BHK semantics for Int were also studied in [71], [94] and many other papers (cf. [18], [22], [106]). Kuznetsov-Muravitsky-Goldblatt sema ...
... fragments of Int, classical logic, etc. ([21], [88], [89]). Abstract computational and functional semantics for Int which did not address the issue of the original BHK semantics for Int were also studied in [71], [94] and many other papers (cf. [18], [22], [106]). Kuznetsov-Muravitsky-Goldblatt sema ...
Carnap and Quine on the analytic-synthetic - Philsci
... In the next section I start by clarifying the analytic-synthetic distinction to be employed in the historical analysis, and to eliminate some bothersome confusions. In the third section, I will discuss the statements of first order logic, and argue that throughout Quine’s work, first order logic sen ...
... In the next section I start by clarifying the analytic-synthetic distinction to be employed in the historical analysis, and to eliminate some bothersome confusions. In the third section, I will discuss the statements of first order logic, and argue that throughout Quine’s work, first order logic sen ...
SLD-Resolution And Logic Programming (PROLOG)
... We have the following normal form lemma. Lemma 9.2.1 If a set S of clauses is GCN F ! -provable, then a GCN F ! proof without weakenings and in which all the axioms contain only literals can be constructed. Proof : Since G! is complete, S → has a G! -proof T . By lemma 6.3.1 restricted to propositio ...
... We have the following normal form lemma. Lemma 9.2.1 If a set S of clauses is GCN F ! -provable, then a GCN F ! proof without weakenings and in which all the axioms contain only literals can be constructed. Proof : Since G! is complete, S → has a G! -proof T . By lemma 6.3.1 restricted to propositio ...
Notes on the Science of Logic
... 4. Easy set theory. Absolutely required. See NAL:§9A. We use “easy set theory” or “EST” to reference this material. 5. Elementary theory of functions and relations. Ditto as for elementary set theory; see NAL:§9B. 6. In addition, we explain certain set-theoretical ideas on an as-needed basis in thes ...
... 4. Easy set theory. Absolutely required. See NAL:§9A. We use “easy set theory” or “EST” to reference this material. 5. Elementary theory of functions and relations. Ditto as for elementary set theory; see NAL:§9B. 6. In addition, we explain certain set-theoretical ideas on an as-needed basis in thes ...
Continuous Markovian Logic – From Complete ∗ Luca Cardelli
... M, m φ. φ is valid, denoted by φ, if ¬φ is not satisfiable. ...
... M, m φ. φ is valid, denoted by φ, if ¬φ is not satisfiable. ...
AGM Postulates in Arbitrary Logics: Initial Results and - FORTH-ICS
... all of them hold in the more general class we consider. More specifically, there are logics in the class we consider in which no AGM-compliant operators exist. We will develop results allowing us to check whether the AGM theory makes sense for certain classes of logics; if it does, then it would be ...
... all of them hold in the more general class we consider. More specifically, there are logics in the class we consider in which no AGM-compliant operators exist. We will develop results allowing us to check whether the AGM theory makes sense for certain classes of logics; if it does, then it would be ...
Logic and Proof
... Although the patterns of language addressed by Aristotle’s theory of reasoning are limited, we have him to thank for a crucial insight: we can classify valid patterns of inference by their logical form, while abstracting away specific content. It is this fundamental observation that underlies the en ...
... Although the patterns of language addressed by Aristotle’s theory of reasoning are limited, we have him to thank for a crucial insight: we can classify valid patterns of inference by their logical form, while abstracting away specific content. It is this fundamental observation that underlies the en ...
Computability and Incompleteness
... Hilbert had a longstanding interest in foundational issues. He was a leading exponent of the new Cantor-Dedekind methods in mathematics, but, at the same time, was sensitive to foundational worries. By the early 1920’s he had developed a detailed program to address the foundational crisis. The idea ...
... Hilbert had a longstanding interest in foundational issues. He was a leading exponent of the new Cantor-Dedekind methods in mathematics, but, at the same time, was sensitive to foundational worries. By the early 1920’s he had developed a detailed program to address the foundational crisis. The idea ...
Chapter 2
... prove that the product of two even integers is also even, we can use knowledge about number theory. In particular, we could use the fact that an even integer is divisible by 2, or that an even integer m can be rewritten as 2k for some integer k. In this example, P(x, y) : x is an even integer y is ...
... prove that the product of two even integers is also even, we can use knowledge about number theory. In particular, we could use the fact that an even integer is divisible by 2, or that an even integer m can be rewritten as 2k for some integer k. In this example, P(x, y) : x is an even integer y is ...
Notes on the ACL2 Logic
... But what we are after is reasoning about programs, and while propositional logic will play an important role, we need more powerful logics. To see why, let’s simplify things for a moment and consider conjectures involving numbers and arithmetic operations. Consider the conjecture: 1. a+b = ba What d ...
... But what we are after is reasoning about programs, and while propositional logic will play an important role, we need more powerful logics. To see why, let’s simplify things for a moment and consider conjectures involving numbers and arithmetic operations. Consider the conjecture: 1. a+b = ba What d ...
lecture notes in logic - UCLA Department of Mathematics
... where P is a non-empty set and ≤ is a binary relation on P satisfying the following conditions: 1. For all x ∈ P , x ≤ x (reflexivity). 2. For all x, y, z ∈ P , if x ≤ y and y ≤ z, then x ≤ z (transitivity). 3. For all x, y ∈ P , if x ≤ y and y ≤ x, then x = y (antisymmetry). A linear ordering is a ...
... where P is a non-empty set and ≤ is a binary relation on P satisfying the following conditions: 1. For all x ∈ P , x ≤ x (reflexivity). 2. For all x, y, z ∈ P , if x ≤ y and y ≤ z, then x ≤ z (transitivity). 3. For all x, y ∈ P , if x ≤ y and y ≤ x, then x = y (antisymmetry). A linear ordering is a ...
Curry-Howard Isomorphism - Department of information engineering
... on the basis of Boolean algebras—and the soundness and completeness results are then proved. An informal proof semantics, the so-called BHKinterpretation, is also presented. Chapter 3 presents the simply typed λ-calculus and its most fundamental properties up to the subject reduction property and th ...
... on the basis of Boolean algebras—and the soundness and completeness results are then proved. An informal proof semantics, the so-called BHKinterpretation, is also presented. Chapter 3 presents the simply typed λ-calculus and its most fundamental properties up to the subject reduction property and th ...
Comparing sizes of sets
... So, by the pigeonhole principle, |A| ≤ |B|. Also f −1 : B → A is injective. [Do you follow this step?] So, again by the pigeonhole principle, |B| ≤ |A|. We can conclude that |A| = |B|. RL: Assume that |A| = |B|. Since A is finite, there is a bijection f : A → {1, ..., |A|}. And since B is also finit ...
... So, by the pigeonhole principle, |A| ≤ |B|. Also f −1 : B → A is injective. [Do you follow this step?] So, again by the pigeonhole principle, |B| ≤ |A|. We can conclude that |A| = |B|. RL: Assume that |A| = |B|. Since A is finite, there is a bijection f : A → {1, ..., |A|}. And since B is also finit ...
CERES for Propositional Proof Schemata
... notion of schematic sequent calculus proof. To the best of our knowledge, there does not yet exist a sequent calculus for propositional formula schemata, although cyclic proofs which are similar to our proof schemata have been considered in the literature [15, 11]. We assume a countably infinite set ...
... notion of schematic sequent calculus proof. To the best of our knowledge, there does not yet exist a sequent calculus for propositional formula schemata, although cyclic proofs which are similar to our proof schemata have been considered in the literature [15, 11]. We assume a countably infinite set ...
Completeness in modal logic - Lund University Publications
... Definition 2.1 (∧): Φ ∧ Ψ =df ∼(∼Φ ∨ ∼Ψ) Definition 2.2 (⊃): Φ ⊃ Ψ =df ∼Φ ∨ Ψ Definition 2.3 (≡): Φ ≡ Ψ =df (Φ ⊃ Ψ) ∧ (Ψ ⊃ Φ) ...
... Definition 2.1 (∧): Φ ∧ Ψ =df ∼(∼Φ ∨ ∼Ψ) Definition 2.2 (⊃): Φ ⊃ Ψ =df ∼Φ ∨ Ψ Definition 2.3 (≡): Φ ≡ Ψ =df (Φ ⊃ Ψ) ∧ (Ψ ⊃ Φ) ...
Decision procedures in Algebra and Logic
... Three binary operations. Quasigroups are listed here, despite their having 3 binary operations, because they are (nonassociative) magmas. Quasigroups feature 3 binary operations only because establishing the quasigroup cancellation property by means of identities alone requires two binary operations ...
... Three binary operations. Quasigroups are listed here, despite their having 3 binary operations, because they are (nonassociative) magmas. Quasigroups feature 3 binary operations only because establishing the quasigroup cancellation property by means of identities alone requires two binary operations ...
A BOUND FOR DICKSON`S LEMMA 1. Introduction Consider the
... with and without usage of non-constructive (or “classical”) arguments. The original proof of Dickson [5] and the particularly nice one by Nash-Williams [11] (using minimal bad sequences) are non-constructive, and hence do not immediately provide a bound. But it is well known that by using some logic ...
... with and without usage of non-constructive (or “classical”) arguments. The original proof of Dickson [5] and the particularly nice one by Nash-Williams [11] (using minimal bad sequences) are non-constructive, and hence do not immediately provide a bound. But it is well known that by using some logic ...
GLukG logic and its application for non-monotonic reasoning
... to a designated value. The most simple example of a multivalued logic is classical logic where: D = {0, 1}, 1 is the unique designated value, and connectives are defined through the usual basic truth tables. If X is any logic, we write |=X α to denote that α is a tautology in the logic X. We say V t ...
... to a designated value. The most simple example of a multivalued logic is classical logic where: D = {0, 1}, 1 is the unique designated value, and connectives are defined through the usual basic truth tables. If X is any logic, we write |=X α to denote that α is a tautology in the logic X. We say V t ...
Barwise: Infinitary Logic and Admissible Sets
... potentially isomorphic structures are very similar to each other, but are not necessarily isomorphic. For example, any two infinite structures with the empty vocabulary are potentially isomorphic. While two potentially isomorphic structures may not be isomorphic, Barwise [9] and Nadel [54] showed th ...
... potentially isomorphic structures are very similar to each other, but are not necessarily isomorphic. For example, any two infinite structures with the empty vocabulary are potentially isomorphic. While two potentially isomorphic structures may not be isomorphic, Barwise [9] and Nadel [54] showed th ...
Structural Logical Relations
... over the structure of types, typing derivations, and proofs in the assertion logics. No definition of simultaneous substitutions is needed. No extension of the concept of logical relation to simultaneous substitutions is needed. And finally, no Kripke-style explicit contexts are needed in the defini ...
... over the structure of types, typing derivations, and proofs in the assertion logics. No definition of simultaneous substitutions is needed. No extension of the concept of logical relation to simultaneous substitutions is needed. And finally, no Kripke-style explicit contexts are needed in the defini ...
Foundations of Mathematics I Set Theory (only a draft)
... prove these inequalities we need to know the mathematical definitions of 0, 1, 2 and 3. Since these numbers are not mathematically defined yet, we cannot prove for the moment that 3 6∈ {0, 1, 2}. We will do this in the next part. For the moment we assume that every natural number is different from t ...
... prove these inequalities we need to know the mathematical definitions of 0, 1, 2 and 3. Since these numbers are not mathematically defined yet, we cannot prove for the moment that 3 6∈ {0, 1, 2}. We will do this in the next part. For the moment we assume that every natural number is different from t ...
Set theory and logic
... In Chapter 8 several axiomatic theories which fall within the realm of modern algebra are introduced. The primary purpose is to enable us to give self-contained characterizations in turn of the system of integers, of rational numbers, and, finally, of real numbers. This is clone in the last three se ...
... In Chapter 8 several axiomatic theories which fall within the realm of modern algebra are introduced. The primary purpose is to enable us to give self-contained characterizations in turn of the system of integers, of rational numbers, and, finally, of real numbers. This is clone in the last three se ...
Views: Compositional Reasoning for Concurrent Programs
... again have knowledge that variables agree with their types, but may make updates that change the types of variables. Threads’ views may be consistently composed only if they describe disjoint sets of variables, which each thread can be seen to own. Note that, since heap locations may be aliased by m ...
... again have knowledge that variables agree with their types, but may make updates that change the types of variables. Threads’ views may be consistently composed only if they describe disjoint sets of variables, which each thread can be seen to own. Note that, since heap locations may be aliased by m ...
The Computer Modelling of Mathematical Reasoning Alan Bundy
... their essential contribution. I call such descriptions, rational reconstructions. This does not imply that the original work was irrational – only that my reconstructions are rational. I apologise to any of the rationally reconstructed who feel mistreated. My excuse is that the reworking of research ...
... their essential contribution. I call such descriptions, rational reconstructions. This does not imply that the original work was irrational – only that my reconstructions are rational. I apologise to any of the rationally reconstructed who feel mistreated. My excuse is that the reworking of research ...
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.