Discrete Mathematics
... A WFF A is a substitution instance of another formula B if A is formed from B by substituting formulas for variables in B under condition that the same formula is substituted for the same variable each time that variable is occurred. • Ex. B: p→(j ∧p), A: (r→s)→(j ∧(r→s)) ...
... A WFF A is a substitution instance of another formula B if A is formed from B by substituting formulas for variables in B under condition that the same formula is substituted for the same variable each time that variable is occurred. • Ex. B: p→(j ∧p), A: (r→s)→(j ∧(r→s)) ...
Modular Construction of Complete Coalgebraic Logics
... interest, all of which can be written as such combinations. This paper derives logics and proof systems for these probabilistic system types, using similar combinations on the logical level. Apart from making this kind of compositionality explicit, the coalgebraic approach also allows a uniform defi ...
... interest, all of which can be written as such combinations. This paper derives logics and proof systems for these probabilistic system types, using similar combinations on the logical level. Apart from making this kind of compositionality explicit, the coalgebraic approach also allows a uniform defi ...
The Foundations
... A language(Vocabulary) for expressing them. A concise notation(Syntax) for writing them. A methodology for objectively reasoning about their truth or falsity (Semantics and Axiomatics). It is the foundation for expressing formal proofs in all branches of mathematics. ...
... A language(Vocabulary) for expressing them. A concise notation(Syntax) for writing them. A methodology for objectively reasoning about their truth or falsity (Semantics and Axiomatics). It is the foundation for expressing formal proofs in all branches of mathematics. ...
CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction
... were able to conduct an argument of the type: if this formula has a proof the only way to construct it is from such and such formulas by the means of one of the inference rules, and that formula can be found automatically. We will see now, that one can’t apply the above argument to the proof search ...
... were able to conduct an argument of the type: if this formula has a proof the only way to construct it is from such and such formulas by the means of one of the inference rules, and that formula can be found automatically. We will see now, that one can’t apply the above argument to the proof search ...
Reductio ad Absurdum Argumentation in Normal Logic
... current standard 2-valued semantics for Normal Logic Programs— the Stable Models Semantics [11] — has been around for almost 20 years now, and it is generally accepted as the de facto standard 2-valued semantics for NLPs. This thoroughly studied semantics, however, lacks some important properties am ...
... current standard 2-valued semantics for Normal Logic Programs— the Stable Models Semantics [11] — has been around for almost 20 years now, and it is generally accepted as the de facto standard 2-valued semantics for NLPs. This thoroughly studied semantics, however, lacks some important properties am ...
Finite satisfiability for guarded fixpoint logic
... in diverse fields of computer science. One of the most powerful extensions to date, guarded fixpoint logic (μGF) was introduced by Grädel and Walukiewicz in [6], who showed that the satisfiability problem of guarded fixpoint logic is computationally no more complex than for the guarded fragment: 2ExpTim ...
... in diverse fields of computer science. One of the most powerful extensions to date, guarded fixpoint logic (μGF) was introduced by Grädel and Walukiewicz in [6], who showed that the satisfiability problem of guarded fixpoint logic is computationally no more complex than for the guarded fragment: 2ExpTim ...
PLATONISM IN MODERN MATHEMATICS A University Thesis
... questions of knowledge, science, happiness, and politics, and distinguishes between knowledge and opinion. From Plato’s original theory of forms, the idea is a universal form. In modern logic, Plato’s ‘idea’ is synonymous with the ‘universal quantifier.’ In modern mathematics and logic, both univers ...
... questions of knowledge, science, happiness, and politics, and distinguishes between knowledge and opinion. From Plato’s original theory of forms, the idea is a universal form. In modern logic, Plato’s ‘idea’ is synonymous with the ‘universal quantifier.’ In modern mathematics and logic, both univers ...
Chapter 1 Logic
... each”. An example of using a universal quantifier is: “for all integers n, the integer n(n + 1) is even”. We could take a first step towards a symbolic representation of this statement by writing “∀n, n(n+1) is even”, and specifying that the universe of n is the integers. (This statement is true.) T ...
... each”. An example of using a universal quantifier is: “for all integers n, the integer n(n + 1) is even”. We could take a first step towards a symbolic representation of this statement by writing “∀n, n(n+1) is even”, and specifying that the universe of n is the integers. (This statement is true.) T ...
Informal Proceedings of the 30th International Workshop on
... protocol for key exchange and then encryption with derived keys. For human users this is most visible as transport layer security (TLS) used by all web browsers. History has shown that developing such protocols is an error-prone process, and attacks have been found even after protocols were in wides ...
... protocol for key exchange and then encryption with derived keys. For human users this is most visible as transport layer security (TLS) used by all web browsers. History has shown that developing such protocols is an error-prone process, and attacks have been found even after protocols were in wides ...
If T is a consistent theory in the language of arithmetic, we say a set
... if a prime divides a product it divides one of its factors, and that if two numbers with no common prime factor both divide a number, then so does their product. (The reader may recognize these as results we took for granted in the proof of Lemma 16.5.) Once we have enough elementary lemmas, we can ...
... if a prime divides a product it divides one of its factors, and that if two numbers with no common prime factor both divide a number, then so does their product. (The reader may recognize these as results we took for granted in the proof of Lemma 16.5.) Once we have enough elementary lemmas, we can ...
page 113 THE AGM THEORY AND INCONSISTENT BELIEF
... beliefs from implicit beliefs which are derived from the explicit beliefs, or separating relevant beliefs from irrelevant beliefs. Based on this approach, several formal techniques have been developed in recent years to deal with inconsistent beliefs; for example, Chopra and Parikh (2000), Hansson a ...
... beliefs from implicit beliefs which are derived from the explicit beliefs, or separating relevant beliefs from irrelevant beliefs. Based on this approach, several formal techniques have been developed in recent years to deal with inconsistent beliefs; for example, Chopra and Parikh (2000), Hansson a ...
Fraïssé`s conjecture in Pi^1_1-comprehension
... antichains. The study of well-quasi-orderings has been of interest to reverse mathematicians for a long time because it seems to require stronger axioms than most other areas of mathematics. Many of the proofs use Π12 -CA0 , which is beyond the highest of the big five systems, Π11 -CA0 , where a lar ...
... antichains. The study of well-quasi-orderings has been of interest to reverse mathematicians for a long time because it seems to require stronger axioms than most other areas of mathematics. Many of the proofs use Π12 -CA0 , which is beyond the highest of the big five systems, Π11 -CA0 , where a lar ...
A Survey on Small Fragments of First-Order Logic over Finite
... McNaughton and Papert have considered definability in first-order logic for finite words and they showed that this coincides with the class of star-free languages [39]. Together with Schützenberger’s Theorem this gives decidability of the problem whether a regular language is definable in first-ord ...
... McNaughton and Papert have considered definability in first-order logic for finite words and they showed that this coincides with the class of star-free languages [39]. Together with Schützenberger’s Theorem this gives decidability of the problem whether a regular language is definable in first-ord ...
Beyond Quantifier-Free Interpolation in Extensions of Presburger
... Our interpolation procedure extracts an interpolant directly from a proof of A ⇒ C. Starting from a sound and complete proof system based on a sequent calculus, the proof rules are extended by labelled formulae and annotations that reduce, at the root of a closed proof, to interpolants. In earlier w ...
... Our interpolation procedure extracts an interpolant directly from a proof of A ⇒ C. Starting from a sound and complete proof system based on a sequent calculus, the proof rules are extended by labelled formulae and annotations that reduce, at the root of a closed proof, to interpolants. In earlier w ...
Proof analysis beyond geometric theories: from rule systems to
... The applicability of the method of proof analysis to logics characterized by a relational semantics has brought a wealth of applications to the proof theory of non-classican logics, including provability logic (Negri 2005), substructural logic (Negri 2008), intermediate logics (Dyckhoff and Negri 20 ...
... The applicability of the method of proof analysis to logics characterized by a relational semantics has brought a wealth of applications to the proof theory of non-classican logics, including provability logic (Negri 2005), substructural logic (Negri 2008), intermediate logics (Dyckhoff and Negri 20 ...