Chapter 2, Logic
... Every A is either B or C, Some D are neither B nor C, Therefore some D are not A. (A = member of the golf club, B = tennis player, C = bridge player, D = member of the choir) However, the argument also fits the simpler pattern: All P is Q, Some R are not Q, therefore some R are not P, which also suf ...
... Every A is either B or C, Some D are neither B nor C, Therefore some D are not A. (A = member of the golf club, B = tennis player, C = bridge player, D = member of the choir) However, the argument also fits the simpler pattern: All P is Q, Some R are not Q, therefore some R are not P, which also suf ...
Introduction to Logic for Computer Science
... trying to symbolise the whole of mathematics could be disastrous as then it would become quite impossible to even read and understand mathematics, since what is presented usually as a one page proof could run into several pages. But at least in principle it can be done. Since the latter half of the ...
... trying to symbolise the whole of mathematics could be disastrous as then it would become quite impossible to even read and understand mathematics, since what is presented usually as a one page proof could run into several pages. But at least in principle it can be done. Since the latter half of the ...
1 The calculus of “predicates”
... property of mortality1. Such a reading is an example of an application of predicate logic – and another instance of the attempt to “force” natural language into the confines of formal logic. In mathematical logic any pretension to be dealing directly with natural language is immediately dropped and ...
... property of mortality1. Such a reading is an example of an application of predicate logic – and another instance of the attempt to “force” natural language into the confines of formal logic. In mathematical logic any pretension to be dealing directly with natural language is immediately dropped and ...
PREDICATE LOGIC
... St A only affects the free occurrence of the variable x. For example, Syx ∀ x P (x) is still ∀ x P (x), that is, the variable x is not free. However„ Syx (Q(x)∧∀ x P (x)) yields Q(y) ∧ ∀ x P (x). Hence, instantiation treats the variable x differently, depending on whether it is free or bound, even i ...
... St A only affects the free occurrence of the variable x. For example, Syx ∀ x P (x) is still ∀ x P (x), that is, the variable x is not free. However„ Syx (Q(x)∧∀ x P (x)) yields Q(y) ∧ ∀ x P (x). Hence, instantiation treats the variable x differently, depending on whether it is free or bound, even i ...
gödel`s completeness theorem with natural language formulas
... finite sequence of symbols taken from a countable or even finite alphabet. Let F1 , F2 , . . . be an enumeration of all formulas. Define a sequence A1 , A2 , . . . of formulas by recursion. At odd stages 1, 3, . . ., we ensure that every formula is decided by the sequence; at even stages 2, 4, 6, . ...
... finite sequence of symbols taken from a countable or even finite alphabet. Let F1 , F2 , . . . be an enumeration of all formulas. Define a sequence A1 , A2 , . . . of formulas by recursion. At odd stages 1, 3, . . ., we ensure that every formula is decided by the sequence; at even stages 2, 4, 6, . ...
Second-Order Logic of Paradox
... principles of Double Negation, and the classical De Morgan equivalences and their quantificational analogues. (As a result, every valid formula of classical logic is valid in LP as well.) What is missing are some of the traditional rules of inference, such as Modus Ponens and the principle of Ex Fal ...
... principles of Double Negation, and the classical De Morgan equivalences and their quantificational analogues. (As a result, every valid formula of classical logic is valid in LP as well.) What is missing are some of the traditional rules of inference, such as Modus Ponens and the principle of Ex Fal ...
Logic for Computer Science. Lecture Notes
... reasoning or, in other words, what are we going to talk about and what language are we going to use. The next step is to associate a precise meaning to basic notions of the language, in order to avoid ambiguities and misunderstandings. Finally we have to state clearly what kind of opinions (sentence ...
... reasoning or, in other words, what are we going to talk about and what language are we going to use. The next step is to associate a precise meaning to basic notions of the language, in order to avoid ambiguities and misunderstandings. Finally we have to state clearly what kind of opinions (sentence ...
proceedings version
... A here-and-there model (HT model) is made up of two sets of propositional variables H (‘here’) and T (‘there’) such that H ⊆ T . The logical language to talk about such models has connectives ⊥, ∧, ∨, and ⇒. The latter is interpreted in a non-classical way and is therefore different from the materia ...
... A here-and-there model (HT model) is made up of two sets of propositional variables H (‘here’) and T (‘there’) such that H ⊆ T . The logical language to talk about such models has connectives ⊥, ∧, ∨, and ⇒. The latter is interpreted in a non-classical way and is therefore different from the materia ...
Intuitionistic Logic
... that some proposition has as yet no proof, but it is not excluded that eventually a proof may be found. In formal logic there is a similar distinction: 6` A and ` ¬A. The Brouwerian counter examples are similar to the first case, strong counterexamples cannot always be expected. For example, althoug ...
... that some proposition has as yet no proof, but it is not excluded that eventually a proof may be found. In formal logic there is a similar distinction: 6` A and ` ¬A. The Brouwerian counter examples are similar to the first case, strong counterexamples cannot always be expected. For example, althoug ...
3.1.3 Subformulas
... Definition 3.8 Let F be a propositional formula. The set of subformulas of F is the smallest set S(F ) satisfying the following conditions: 1. F ∈ S(F ). 2. If ¬G ∈ S(F ) , then G ∈ S(F ). 3. If (G1 ◦ G2 ) ∈ S(F ) , then G1 , G2 ∈ S(F ). It will be shown in Exercise 3.4 that such a smallest set exis ...
... Definition 3.8 Let F be a propositional formula. The set of subformulas of F is the smallest set S(F ) satisfying the following conditions: 1. F ∈ S(F ). 2. If ¬G ∈ S(F ) , then G ∈ S(F ). 3. If (G1 ◦ G2 ) ∈ S(F ) , then G1 , G2 ∈ S(F ). It will be shown in Exercise 3.4 that such a smallest set exis ...
