Canonicity and representable relation algebras
... Goldblatt (1975–89) on canonicity of modal logics/varieties of BAOs. Fact 2 (Goldblatt) If K is an elementary class of atom structures, then the variety HSP{Cm S : S ∈ K} generated by K is canonical. Proof. [of theorem 1] By definition, RRA is generated by K = {At Re(U ) : U a set}. But K is element ...
... Goldblatt (1975–89) on canonicity of modal logics/varieties of BAOs. Fact 2 (Goldblatt) If K is an elementary class of atom structures, then the variety HSP{Cm S : S ∈ K} generated by K is canonical. Proof. [of theorem 1] By definition, RRA is generated by K = {At Re(U ) : U a set}. But K is element ...
Logical Methods in Computer Science Vol. 8(4:19)2012, pp. 1–28 Submitted Oct. 27, 2011
... least r; similarly, Mra φ states that the rate is at most r. In this respect, our logic is similar to the Aumann’s system [Aum99b] developed for Harsanyi type spaces [Har67]. In spite of their syntactic similarities, CML and PML are very different. In the probabilistic case axiomatized by Zhou in hi ...
... least r; similarly, Mra φ states that the rate is at most r. In this respect, our logic is similar to the Aumann’s system [Aum99b] developed for Harsanyi type spaces [Har67]. In spite of their syntactic similarities, CML and PML are very different. In the probabilistic case axiomatized by Zhou in hi ...
PDF
... – Assume r wouldn’t exist for some n – Then for all r: r 2 > n or (r+1)2≤n – Consider r1 = min{r|r 2 > n}-1 – Then r12≤n and (r1+1)2 > n – a contradiction Unnecessary indirect approach - it masks a constructive argument ...
... – Assume r wouldn’t exist for some n – Then for all r: r 2 > n or (r+1)2≤n – Consider r1 = min{r|r 2 > n}-1 – Then r12≤n and (r1+1)2 > n – a contradiction Unnecessary indirect approach - it masks a constructive argument ...
Modal fixpoint logic: some model theoretic questions
... of recursive principle. At the end of the 1970s, Amir Pnueli [Pnu77] argued that linear temporal logic (LTL), which is obtained by restricting to models based on the natural numbers and by adding the “until” operator to modal logic, could be a useful formalism in that respect. Since then, other temp ...
... of recursive principle. At the end of the 1970s, Amir Pnueli [Pnu77] argued that linear temporal logic (LTL), which is obtained by restricting to models based on the natural numbers and by adding the “until” operator to modal logic, could be a useful formalism in that respect. Since then, other temp ...
PPT - UBC Department of CPSC Undergraduates
... not faster than itself for problem size n.” i N, n N, n > i ~Faster(a, a, n) Consider an arbitrary (positive integer) i. Let n = ??. (Must be > i; so, at least i+1.) So, we need to prove: “a is not faster than itself for problem size ?? (for an arbitrary positive integer i)” ...
... not faster than itself for problem size n.” i N, n N, n > i ~Faster(a, a, n) Consider an arbitrary (positive integer) i. Let n = ??. (Must be > i; so, at least i+1.) So, we need to prove: “a is not faster than itself for problem size ?? (for an arbitrary positive integer i)” ...
Specifying and Verifying Fault-Tolerant Systems
... the earlier syntax for the same operator, (iii) single square brackets have replaced ...
... the earlier syntax for the same operator, (iii) single square brackets have replaced ...
KURT GÖDEL - National Academy of Sciences
... N is a member of M. For example, the set {0, 1, 2} with the three members shown has the following 8 (= 23) subsets: {0, 1, 2}, {1, 2}, {0, 2}, {0, 1}, {0}, {1}, {2}, { }. Cantor showed that there is no way of pairing all the sets of natural numbers (that is, all the subsets of the natural numbers) w ...
... N is a member of M. For example, the set {0, 1, 2} with the three members shown has the following 8 (= 23) subsets: {0, 1, 2}, {1, 2}, {0, 2}, {0, 1}, {0}, {1}, {2}, { }. Cantor showed that there is no way of pairing all the sets of natural numbers (that is, all the subsets of the natural numbers) w ...
Introduction to Mathematical Logic lecture notes
... from the previous ones (and sometimes, unfortunately, not even that). Here we will define formal proofs, or deductions, which will follow very strict rules that ensure that no mistake is possible. Throughout this course we will consider more than one logic, and therefore more than one deduction syste ...
... from the previous ones (and sometimes, unfortunately, not even that). Here we will define formal proofs, or deductions, which will follow very strict rules that ensure that no mistake is possible. Throughout this course we will consider more than one logic, and therefore more than one deduction syste ...
A Computationally-Discovered Simplification of the Ontological
... instance of Lemma 1 might be y = ıxF x → F y, which asserts that if the object y is identical to the x that is F , then y is F . Lemma 1 can then be used to prove Description Theorem 2, which asserts: if there is something that is the x such that φ, then it is such that φ. Intuitively, this tells us ...
... instance of Lemma 1 might be y = ıxF x → F y, which asserts that if the object y is identical to the x that is F , then y is F . Lemma 1 can then be used to prove Description Theorem 2, which asserts: if there is something that is the x such that φ, then it is such that φ. Intuitively, this tells us ...
A Computationally-Discovered Simplification of the Ontological
... instance of Lemma 1 might be y = ıxF x → F y, which asserts that if the object y is identical to the x that is F , then y is F . Lemma 1 can then be used to prove Description Theorem 2, which asserts: if there is something that is the x such that φ, then it is such that φ. Intuitively, this tells us ...
... instance of Lemma 1 might be y = ıxF x → F y, which asserts that if the object y is identical to the x that is F , then y is F . Lemma 1 can then be used to prove Description Theorem 2, which asserts: if there is something that is the x such that φ, then it is such that φ. Intuitively, this tells us ...
Independence logic and tuple existence atoms
... Definition R relation, ~x , ~y , ~z tuples of attributes. Then R |= ~x ~y | ~z if and only if, for all r , r 0 ∈ R such that r (~x ) = r 0 (~x ) there exists a r 00 ∈ R such that r 00 (~x ~y ) = r (~x ~y ) and r 00 (~x ~z ) = r (~x ~z ). Huge literature on the topic; If ~x ~y ~z contains all attri ...
... Definition R relation, ~x , ~y , ~z tuples of attributes. Then R |= ~x ~y | ~z if and only if, for all r , r 0 ∈ R such that r (~x ) = r 0 (~x ) there exists a r 00 ∈ R such that r 00 (~x ~y ) = r (~x ~y ) and r 00 (~x ~z ) = r (~x ~z ). Huge literature on the topic; If ~x ~y ~z contains all attri ...
PPT
... not faster than itself for problem size n.” i N, n N, n > i ~Faster(a, a, n) Consider an arbitrary (positive integer) i. Let n = ??. (Must be > i; so, at least i+1.) So, we need to prove: “a is not faster than itself for problem size ?? (for an arbitrary positive integer i)” ...
... not faster than itself for problem size n.” i N, n N, n > i ~Faster(a, a, n) Consider an arbitrary (positive integer) i. Let n = ??. (Must be > i; so, at least i+1.) So, we need to prove: “a is not faster than itself for problem size ?? (for an arbitrary positive integer i)” ...
PPT - UBC Department of CPSC Undergraduates
... not faster than itself for problem size n.” i N, n N, n > i ~Faster(a, a, n) Consider an arbitrary (positive integer) i. Let n = ??. (Must be > i; so, at least i+1.) So, we need to prove: “a is not faster than itself for problem size ?? (for an arbitrary positive integer i)” ...
... not faster than itself for problem size n.” i N, n N, n > i ~Faster(a, a, n) Consider an arbitrary (positive integer) i. Let n = ??. (Must be > i; so, at least i+1.) So, we need to prove: “a is not faster than itself for problem size ?? (for an arbitrary positive integer i)” ...
Argumentations and logic
... the basis of what we already know or do we need new information? 1. Settling Hypotheses by Argumentations. Argumentation is involved in settling hypotheses on the basis of what we already know. Every argumentation that deduces the hypothesis from premises already known to be true proves the hypothes ...
... the basis of what we already know or do we need new information? 1. Settling Hypotheses by Argumentations. Argumentation is involved in settling hypotheses on the basis of what we already know. Every argumentation that deduces the hypothesis from premises already known to be true proves the hypothes ...
Counterfactuals
... body armor’, and ψ as ‘Jon would have lived’, we have both: φψ (φ ∧ φ0 ) ¬ψ Noting that the sentences above, in natural language, appear to be unproblematic together, our sketched analysis of counterfactuals should give the same result. Unfortunately, it does precisely the opposite. In order for ...
... body armor’, and ψ as ‘Jon would have lived’, we have both: φψ (φ ∧ φ0 ) ¬ψ Noting that the sentences above, in natural language, appear to be unproblematic together, our sketched analysis of counterfactuals should give the same result. Unfortunately, it does precisely the opposite. In order for ...
![arXiv:1512.05177v1 [cs.LO] 16 Dec 2015](http://s1.studyres.com/store/data/012886476_1-5eedb3c008e9a2ffa0e571a06be0b669-300x300.png)
CS243: Discrete Structures Mathematical Proof Techniques
... Proof: Assume n is odd. By definition of oddness, there must exist some integer k such that n = 2k + 1. Then, n 2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k ) + 1, which is odd. Thus, if n is odd, n 2 is also odd. ...
... Proof: Assume n is odd. By definition of oddness, there must exist some integer k such that n = 2k + 1. Then, n 2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k ) + 1, which is odd. Thus, if n is odd, n 2 is also odd. ...
The History of Categorical Logic
... to qualify it later [Mac Lane, 2002, 130], was meant to provide an autonomous framework for the concept of natural transformation, a concept whose generality, pervasiveness and usefulness had become clear to both of them during their collaboration on the clarification of an unsuspected link between ...
... to qualify it later [Mac Lane, 2002, 130], was meant to provide an autonomous framework for the concept of natural transformation, a concept whose generality, pervasiveness and usefulness had become clear to both of them during their collaboration on the clarification of an unsuspected link between ...
THE LOGIC OF QUANTIFIED STATEMENTS
... “some” or “all” and tell for how many elements a given predicate is true. • e.g., For some integer x, x is divisible by 5 • e.g., For all integer x, x is divisible by 5 • e.g., there exists two integer x, such that x is divisible by 5. • All above three are now propositions (i.e., they have truth va ...
... “some” or “all” and tell for how many elements a given predicate is true. • e.g., For some integer x, x is divisible by 5 • e.g., For all integer x, x is divisible by 5 • e.g., there exists two integer x, such that x is divisible by 5. • All above three are now propositions (i.e., they have truth va ...
Predicate logic definitions
... Warning: This is the informal semantics presented in Bergmann et al. Some important details dealt with by the formal semantics are left implicit. ...
... Warning: This is the informal semantics presented in Bergmann et al. Some important details dealt with by the formal semantics are left implicit. ...
Modus Ponens Defended
... preliminaries. Since my agenda in this section is only to defend modus ponens against McGee, I next want to sketch my preferred conception of logic on which this inference rule comes out valid. I will then argue that McGee’s attack, as it stands, does not undermine this way of seeing things. Inspire ...
... preliminaries. Since my agenda in this section is only to defend modus ponens against McGee, I next want to sketch my preferred conception of logic on which this inference rule comes out valid. I will then argue that McGee’s attack, as it stands, does not undermine this way of seeing things. Inspire ...
Cut-elimination for provability logics and some results in display logic
... suggests, the formal inference rules in this system mimic (formalise) the sort of deductive reasoning that is employed in practice. In order to study the properties of this system, Gentzen then constructed yet another proof-system called the sequent calculus. Gentzen’s Hauptsatz or main theorem for ...
... suggests, the formal inference rules in this system mimic (formalise) the sort of deductive reasoning that is employed in practice. In order to study the properties of this system, Gentzen then constructed yet another proof-system called the sequent calculus. Gentzen’s Hauptsatz or main theorem for ...