Identity in modal logic theorem proving
... and methods are applications of what it is legal to do within the proof theory. (In Whitehead ~ Russell, this amounts to finding substitution instances of formulas for propositional variables in the axioms, and applying Modus Ponens). Were one directly constructing proofs in Smullyan [14] tableaux s ...
... and methods are applications of what it is legal to do within the proof theory. (In Whitehead ~ Russell, this amounts to finding substitution instances of formulas for propositional variables in the axioms, and applying Modus Ponens). Were one directly constructing proofs in Smullyan [14] tableaux s ...
Completeness and Decidability of a Fragment of Duration Calculus
... state is not more than one-twentieth of the elapsed time. One can design the Gas Burner as a real-time automaton depicted in Fig. 1 which expresses that any leak state must be detected and stopped within one second, and that leak must be separated by at least 30s. A natural way to express the behavi ...
... state is not more than one-twentieth of the elapsed time. One can design the Gas Burner as a real-time automaton depicted in Fig. 1 which expresses that any leak state must be detected and stopped within one second, and that leak must be separated by at least 30s. A natural way to express the behavi ...
vmcai - of Philipp Ruemmer
... branch, the close-ll rule yields the interpolant false, which is carried through by not-left. The rule or-left-l takes the interpolants of its two subproofs and generates false ∨ p(d). This is the final interpolant, since the last rule andleft propagates interpolants without applying modifications. ...
... branch, the close-ll rule yields the interpolant false, which is carried through by not-left. The rule or-left-l takes the interpolants of its two subproofs and generates false ∨ p(d). This is the final interpolant, since the last rule andleft propagates interpolants without applying modifications. ...
Can Modalities Save Naive Set Theory?
... For another example, we can understand in a special way as determinately. To motivate this idea, consider an analogy to philosophical discussions of truth. Those who wish to preserve classical logic in the face of the liar paradox sometimes introduce an operator to be read “determinately”, which dis ...
... For another example, we can understand in a special way as determinately. To motivate this idea, consider an analogy to philosophical discussions of truth. Those who wish to preserve classical logic in the face of the liar paradox sometimes introduce an operator to be read “determinately”, which dis ...
The Complete Proof Theory of Hybrid Systems
... ensure soundness by checking it locally per proof rule. More intriguingly, however, our logical setting also enables us to ask the converse: is the proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s incompleteness theorem shows that hybrid systems do not have a sou ...
... ensure soundness by checking it locally per proof rule. More intriguingly, however, our logical setting also enables us to ask the converse: is the proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s incompleteness theorem shows that hybrid systems do not have a sou ...
Document
... This last is not a serious restriction as we can always replace other constants by terms h(c) using a fresh function symbol h. ...
... This last is not a serious restriction as we can always replace other constants by terms h(c) using a fresh function symbol h. ...
Sets, Numbers, and Logic
... The empty set is a subset of every other set, but there is only one empty set — the set with no integers is the same as the one with no apples. Out of the empty set, man learned (only in the 20th century, actually) to construct N = {1, 2, 3, . . . , n, . . .} : the natural numbers. These have been a ...
... The empty set is a subset of every other set, but there is only one empty set — the set with no integers is the same as the one with no apples. Out of the empty set, man learned (only in the 20th century, actually) to construct N = {1, 2, 3, . . . , n, . . .} : the natural numbers. These have been a ...
Properties of binary transitive closure logics over trees
... to define a linear order on the nodes of a tree. The order resembles depth-first left-to-right traversal of a tree. A linear order is a powerful concept that can be used defining additional properties. For example, it is used to count the number of nodes in a tree modulo a given natural number. An i ...
... to define a linear order on the nodes of a tree. The order resembles depth-first left-to-right traversal of a tree. A linear order is a powerful concept that can be used defining additional properties. For example, it is used to count the number of nodes in a tree modulo a given natural number. An i ...
The Complete Proof Theory of Hybrid Systems
... More intriguingly, however, our logical setting also enables us to ask the converse: is the proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s incompleteness theorem shows, however, that hybrid systems do not have a sound and complete calculus that is effective, be ...
... More intriguingly, however, our logical setting also enables us to ask the converse: is the proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s incompleteness theorem shows, however, that hybrid systems do not have a sound and complete calculus that is effective, be ...
An Introduction to Discrete Mathematics: how to
... not yet aquired a strong inclination to follow her own lines of thought. Partly for that reason — and also with the aim of training the student for success in exams — exercises are usually supplied at the ends of each section or, at least, at the end of each chapter. You should spend much more time ...
... not yet aquired a strong inclination to follow her own lines of thought. Partly for that reason — and also with the aim of training the student for success in exams — exercises are usually supplied at the ends of each section or, at least, at the end of each chapter. You should spend much more time ...
Day00a-Induction-proofs - Rose
... "has a lower bound" – Unlike integers, a set of rational numbers can have a lower bound but no smallest member: {1/3, 1/5, 1/7, 1/9, … } ...
... "has a lower bound" – Unlike integers, a set of rational numbers can have a lower bound but no smallest member: {1/3, 1/5, 1/7, 1/9, … } ...
1 The Natural Numbers
... In Chapter 1, we introduced 0 (aka ∅), its successor 1 = s(0) = 0∪{0} = {0}, 1’s successor 2 = s(1) = 1 ∪ {1} = {0, 1}, and 2’s successor 3 = s(2) = 2 ∪ {2} = {0, 1, 2}. This is how the first four natural numbers are usually modelled within set theory; it’s intuitively obvious that we could go on in ...
... In Chapter 1, we introduced 0 (aka ∅), its successor 1 = s(0) = 0∪{0} = {0}, 1’s successor 2 = s(1) = 1 ∪ {1} = {0, 1}, and 2’s successor 3 = s(2) = 2 ∪ {2} = {0, 1, 2}. This is how the first four natural numbers are usually modelled within set theory; it’s intuitively obvious that we could go on in ...
Logic and Computation Lecture notes Jeremy Avigad Assistant Professor, Philosophy
... With this analysis in hand, will can try to give an account of what it means for a statement ϕ to follow “logically” from a set of hypotheses Γ. One intuitive approach is to say that ϕ follows from Γ if whenever every statement in Γ is true, then so is ϕ. More precisely, we will say that ϕ is a logi ...
... With this analysis in hand, will can try to give an account of what it means for a statement ϕ to follow “logically” from a set of hypotheses Γ. One intuitive approach is to say that ϕ follows from Γ if whenever every statement in Γ is true, then so is ϕ. More precisely, we will say that ϕ is a logi ...
3. Mathematical Induction 3.1. First Principle of
... Here is another type of problem from number theory that is amenable to induction. Example 3.5.2. Prove: For every natural number n, n(n2 + 5) is a multiple of 6 (i.e. n(n2 + 5) equals 6 times some integer). Proof. Let P (n) be the statement n(n2 + 5) is a multiple of 6. 1. Basis Step, n = 0: 0(02 + ...
... Here is another type of problem from number theory that is amenable to induction. Example 3.5.2. Prove: For every natural number n, n(n2 + 5) is a multiple of 6 (i.e. n(n2 + 5) equals 6 times some integer). Proof. Let P (n) be the statement n(n2 + 5) is a multiple of 6. 1. Basis Step, n = 0: 0(02 + ...
Monadic Second Order Logic and Automata on Infinite Words
... MSOL0 [S] formulas φ, ψ := S(X, Y ) | X ⊆ Y | X ⊆ Qa | ¬ψ | ψ ∧ φ | ∃X.ψ I will show how to translate between MSOL0 [S] and MSOL[S]. We can then conclude that MSOL0 [S] defines the same languages as MSOL[S]. First we do the translation from MSOL[S] to MSOL0 [S]. The usual conditions on variable bind ...
... MSOL0 [S] formulas φ, ψ := S(X, Y ) | X ⊆ Y | X ⊆ Qa | ¬ψ | ψ ∧ φ | ∃X.ψ I will show how to translate between MSOL0 [S] and MSOL[S]. We can then conclude that MSOL0 [S] defines the same languages as MSOL[S]. First we do the translation from MSOL[S] to MSOL0 [S]. The usual conditions on variable bind ...
proofs in mathematics
... implies that, 1 = 0. But this disproves Statement 1, which states that no whole number is equal to its successor. From Statement 2, we get ...
... implies that, 1 = 0. But this disproves Statement 1, which states that no whole number is equal to its successor. From Statement 2, we get ...
appendix-1
... (ii) Make sure the axioms are consistent. We say a collection of axioms is inconsistent, if we can use one axiom to show that another axiom is not true. For example, consider the following two statements. We will show that they are inconsistent. Statement1: No whole number is equal to its successor. ...
... (ii) Make sure the axioms are consistent. We say a collection of axioms is inconsistent, if we can use one axiom to show that another axiom is not true. For example, consider the following two statements. We will show that they are inconsistent. Statement1: No whole number is equal to its successor. ...
Second-Order Logic of Paradox
... the familiar “truth tables” of Kleene’s (strong) 3-valued logic [9, §64], but whereas for Kleene (thinking of the “middle value” as truth-valuelessness) only the top value (True) is designated, for Priest the top two values are both designated. As Priest might say: a formula which is both true and f ...
... the familiar “truth tables” of Kleene’s (strong) 3-valued logic [9, §64], but whereas for Kleene (thinking of the “middle value” as truth-valuelessness) only the top value (True) is designated, for Priest the top two values are both designated. As Priest might say: a formula which is both true and f ...
Probability Captures the Logic of Scientific
... another individual a has both F and G would normally be taken to confirm that b also has G. This is a simple example of reasoning by analogy. The following theorem shows that our explication of confirmation agrees with this. (From here on, ‘a’ and ‘b’ stand for any distinct individual constants.) TH ...
... another individual a has both F and G would normally be taken to confirm that b also has G. This is a simple example of reasoning by analogy. The following theorem shows that our explication of confirmation agrees with this. (From here on, ‘a’ and ‘b’ stand for any distinct individual constants.) TH ...
pTopic8
... This form of definition is called a theory, whose subject is a sort, in this case, N. A model of a theory has some set of values, for its sort, and some functions, for its operators, that satisfy its axioms. The set NAT = {0,1,2, ...} is a model of N. The term algebra of a theory consists of the lan ...
... This form of definition is called a theory, whose subject is a sort, in this case, N. A model of a theory has some set of values, for its sort, and some functions, for its operators, that satisfy its axioms. The set NAT = {0,1,2, ...} is a model of N. The term algebra of a theory consists of the lan ...
First-Order Proof Theory of Arithmetic
... improves Gödel’s incompleteness theorem by showing that I∆0 + exp cannot prove the consistency of Q. The main prerequisite for reading this chapter is knowledge of the sequent calculus and cut-elimination, as contained in Chapter I of this volume. The proof theory of arithmetic is a major subfield ...
... improves Gödel’s incompleteness theorem by showing that I∆0 + exp cannot prove the consistency of Q. The main prerequisite for reading this chapter is knowledge of the sequent calculus and cut-elimination, as contained in Chapter I of this volume. The proof theory of arithmetic is a major subfield ...
Document
... P(n) are true. This means any elementary triangulation of an n'-vertex convex polygon, where 3 ≤ n' ≤ n, uses n'–3 lines. We prove P(n+1): any elementary triangulation of any (n+1)-vertex convex polygon uses n–2 lines. Let A be an arbitrary convex polygon with n+1 vertices. Pick any elementary trian ...
... P(n) are true. This means any elementary triangulation of an n'-vertex convex polygon, where 3 ≤ n' ≤ n, uses n'–3 lines. We prove P(n+1): any elementary triangulation of any (n+1)-vertex convex polygon uses n–2 lines. Let A be an arbitrary convex polygon with n+1 vertices. Pick any elementary trian ...
Induction
... 1. Base Case : Prove that the statement holds when n = 2 We are proving P(2). 2 itself is a prime number, so the prime factorization of 2 is 2. Trivially, the statement P(2) holds. 2. Induction Hypothesis : Assume that for all integers less than or equal to k, the statement holds. Note : In the prev ...
... 1. Base Case : Prove that the statement holds when n = 2 We are proving P(2). 2 itself is a prime number, so the prime factorization of 2 is 2. Trivially, the statement P(2) holds. 2. Induction Hypothesis : Assume that for all integers less than or equal to k, the statement holds. Note : In the prev ...
Object-Based Unawareness
... values to in the structures that we are about to construct in Section 2.3. Our language will be based on first order modal logic, and is different from the language used in the previous unawareness literature, which in turn is based on propositional modal logic. Introducing this new language is nece ...
... values to in the structures that we are about to construct in Section 2.3. Our language will be based on first order modal logic, and is different from the language used in the previous unawareness literature, which in turn is based on propositional modal logic. Introducing this new language is nece ...
p. 1 Math 490 Notes 4 We continue our examination of well
... empty set φ is a well-ordered set (vacuously), and the ordinal containing φ is naturally denoted 0 (zero). Now consider all well-ordered sets with exactly n elements for some n ∈ N. It should be easy to see that all such well-ordered sets are similar to each other, and thus they all belong to the sa ...
... empty set φ is a well-ordered set (vacuously), and the ordinal containing φ is naturally denoted 0 (zero). Now consider all well-ordered sets with exactly n elements for some n ∈ N. It should be easy to see that all such well-ordered sets are similar to each other, and thus they all belong to the sa ...
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.