F - Teaching-WIKI
... to claims that a statement is true, even tough its truth does not necessarily follow from the premises => Logic provides a way to talk about truth and correctness in a rigorous way, so that we can prove things, rather than make intelligent guesses and just hope they are correct ...
... to claims that a statement is true, even tough its truth does not necessarily follow from the premises => Logic provides a way to talk about truth and correctness in a rigorous way, so that we can prove things, rather than make intelligent guesses and just hope they are correct ...
Completeness theorems and lambda
... A suggestive way to interpret these results is the following We can explain the meaning of a statement ∀X.φ(X) without relying on quantification over all subsets of N by explaining how to prove φ(X) In this way we have an explanation of some impredicative quantifications ...
... A suggestive way to interpret these results is the following We can explain the meaning of a statement ∀X.φ(X) without relying on quantification over all subsets of N by explaining how to prove φ(X) In this way we have an explanation of some impredicative quantifications ...
Fraïssé`s conjecture in Pi^1_1-comprehension
... showed that WKL0 can prove block-bqos and barrier-bqos are the same thing. The equivalence between block-bqos and continuous-bqos is immediate from the translation between bad continuous functions and bad arrays. The notion of Borel-bqos was introduced by Simpson [Sim85]. His proof that they are the ...
... showed that WKL0 can prove block-bqos and barrier-bqos are the same thing. The equivalence between block-bqos and continuous-bqos is immediate from the translation between bad continuous functions and bad arrays. The notion of Borel-bqos was introduced by Simpson [Sim85]. His proof that they are the ...
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... from SC. This is in line with our earlier observation that WC is a ruling out task rather than an explanation task. From a logical point of view, one could argue that the two forms of classification are rather different ways of reasoning, even though procedurally they are very similar. One could eve ...
... from SC. This is in line with our earlier observation that WC is a ruling out task rather than an explanation task. From a logical point of view, one could argue that the two forms of classification are rather different ways of reasoning, even though procedurally they are very similar. One could eve ...
LOGIC AND p-RECOGNIZABLE SETS OF INTEGERS 1
... p, the sets recognizable in certain bases only, and the sets recognizable in no base. The first class is quite restricted, as it is limited to the rational sets of the monoid (Nm , +). When m = 1, it is precisely the ultimately periodic sets. It is rather easy to see that ultimately periodic sets ar ...
... p, the sets recognizable in certain bases only, and the sets recognizable in no base. The first class is quite restricted, as it is limited to the rational sets of the monoid (Nm , +). When m = 1, it is precisely the ultimately periodic sets. It is rather easy to see that ultimately periodic sets ar ...
THE MODEL CHECKING PROBLEM FOR INTUITIONISTIC
... 1. Introduction. Intuitionistic logic (see e.g. [10, 27]) is a part of classical logic that can be proven using constructive proofs–e.g. by proofs that do not use reductio ad absurdum. For example, the law of the excluded middle a ∨ ¬a and the weak law of the excluded middle ¬a ∨ ¬¬a do not have con ...
... 1. Introduction. Intuitionistic logic (see e.g. [10, 27]) is a part of classical logic that can be proven using constructive proofs–e.g. by proofs that do not use reductio ad absurdum. For example, the law of the excluded middle a ∨ ¬a and the weak law of the excluded middle ¬a ∨ ¬¬a do not have con ...
Hilbert`s investigations on the foundations of arithmetic (1935) Paul
... rational positive numbers 2, 3, 4,. . . as arisen [entstanden] from the process of counting and developed their laws of calculation [Rechnungsgesetze entwickelt]; then one arrives at the negative number by the requirement of the general execution [allgemeinen Ausführung ] of subtraction; one furth ...
... rational positive numbers 2, 3, 4,. . . as arisen [entstanden] from the process of counting and developed their laws of calculation [Rechnungsgesetze entwickelt]; then one arrives at the negative number by the requirement of the general execution [allgemeinen Ausführung ] of subtraction; one furth ...
Chapter 2 - Princeton University Press
... have stated their definitions and theorems with enough precision and clarity that any competent mathematician reading the work could expand it to a complete formalization if so desired. Formalizability is a requirement for mathematical publications in refereed research journals; formalizability give ...
... have stated their definitions and theorems with enough precision and clarity that any competent mathematician reading the work could expand it to a complete formalization if so desired. Formalizability is a requirement for mathematical publications in refereed research journals; formalizability give ...
T - STI Innsbruck
... to claims that a statement is true, even tough its truth does not necessarily follow from the premises => Logic provides a way to talk about truth and correctness in a rigorous way, so that we can prove things, rather than make intelligent guesses and just hope they are correct ...
... to claims that a statement is true, even tough its truth does not necessarily follow from the premises => Logic provides a way to talk about truth and correctness in a rigorous way, so that we can prove things, rather than make intelligent guesses and just hope they are correct ...
02_Artificial_Intelligence-PropositionalLogic
... to claims that a statement is true, even though its truth does not necessarily follow from the premises => Logic provides a way to talk about truth and correctness in a rigorous way, so that we can prove things, rather than make intelligent guesses and just hope they are correct ...
... to claims that a statement is true, even though its truth does not necessarily follow from the premises => Logic provides a way to talk about truth and correctness in a rigorous way, so that we can prove things, rather than make intelligent guesses and just hope they are correct ...
A Proof Theory for Generic Judgments: An extended abstract
... in logic programming can be seen as a (restricted) form of cut-free proof search. Cut and cut-elimination can then be used to reason directly about computation: for example, if A has a cut-free proof (that is, it can be computed) and we know that A ⊃ B can be proved (possibly with cuts), cuteliminat ...
... in logic programming can be seen as a (restricted) form of cut-free proof search. Cut and cut-elimination can then be used to reason directly about computation: for example, if A has a cut-free proof (that is, it can be computed) and we know that A ⊃ B can be proved (possibly with cuts), cuteliminat ...
F - Teaching-WIKI
... to claims that a statement is true, even tough its truth does not necessarily follow from the premises => Logic provides a way to talk about truth and correctness in a rigorous way, so that we can prove things, rather than make intelligent guesses and just hope they are correct ...
... to claims that a statement is true, even tough its truth does not necessarily follow from the premises => Logic provides a way to talk about truth and correctness in a rigorous way, so that we can prove things, rather than make intelligent guesses and just hope they are correct ...
T - STI Innsbruck
... to claims that a statement is true, even tough its truth does not necessarily follow from the premises => Logic provides a way to talk about truth and correctness in a rigorous way, so that we can prove things, rather than make intelligent guesses and just hope they are correct ...
... to claims that a statement is true, even tough its truth does not necessarily follow from the premises => Logic provides a way to talk about truth and correctness in a rigorous way, so that we can prove things, rather than make intelligent guesses and just hope they are correct ...
Fine`s Theorem on First-Order Complete Modal Logics
... Its underlying frame FL is the canonical frame of L, and L itself may be called a canonical logic if it is valid in FL .1 By the end of the 1960’s, numerous logics had been shown to be canonical, and hence complete for their Kripke-frame semantics, by showing that FL satisfies some first-order defin ...
... Its underlying frame FL is the canonical frame of L, and L itself may be called a canonical logic if it is valid in FL .1 By the end of the 1960’s, numerous logics had been shown to be canonical, and hence complete for their Kripke-frame semantics, by showing that FL satisfies some first-order defin ...
Local deduction, deductive interpolation and amalgamation in
... Logic, 141: 148-179, 2006. This is based on: N.Galatos, H. Ono, Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL, Studia Logica, 83:279-308, 2006. For the proof of Theorem 5.8, the following is needed: A. Wro\'nski, On a form of equational inter ...
... Logic, 141: 148-179, 2006. This is based on: N.Galatos, H. Ono, Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL, Studia Logica, 83:279-308, 2006. For the proof of Theorem 5.8, the following is needed: A. Wro\'nski, On a form of equational inter ...
Frege`s Other Program
... that, as far as is currently known, it is too weak as it does not entail Peano Arithmetic, but only weaker systems, in particular Robinson’s Q. Alternatively, unlike the approach just described, we can break the bond of logicism and extensionalism, rejecting one while maintaining the other. The most ...
... that, as far as is currently known, it is too weak as it does not entail Peano Arithmetic, but only weaker systems, in particular Robinson’s Q. Alternatively, unlike the approach just described, we can break the bond of logicism and extensionalism, rejecting one while maintaining the other. The most ...
Modal logic and the approximation induction principle
... in terms of observations. That is, a process semantics is captured by means of a sublogic of HennessyMilner logic; two states in an LTS are equivalent if and only if they make true exactly the same formulas in this sublogic. In particular, Hennessy-Milner logic itself characterizes bisimulation equi ...
... in terms of observations. That is, a process semantics is captured by means of a sublogic of HennessyMilner logic; two states in an LTS are equivalent if and only if they make true exactly the same formulas in this sublogic. In particular, Hennessy-Milner logic itself characterizes bisimulation equi ...
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 ...
On Action Logic
... of all binary relations on a set U ; now, product is relational product, 1 is the identity relation, and the remaining notions are defined as above. A Kleene algebra A is said to be *-continuous, if xa∗ y =l.u.b.{xan y : n ∈ ω}, for all x, y, a ∈ A. Clearly, the algebra of languages and the algebra ...
... of all binary relations on a set U ; now, product is relational product, 1 is the identity relation, and the remaining notions are defined as above. A Kleene algebra A is said to be *-continuous, if xa∗ y =l.u.b.{xan y : n ∈ ω}, for all x, y, a ∈ A. Clearly, the algebra of languages and the algebra ...
ON PRESERVING 1. Introduction The
... common underlying language) then they must agree on which sets are consistent and which are inconsistent. For consider, if conX (Γ) and Y preserves the X consistency predicate then conX (CY (Γ)). Suppose that Γ is not Y -consistent, then CY (Γ) = S. By [R] CX (CY (Γ)) = CX (S) = S which is to say th ...
... common underlying language) then they must agree on which sets are consistent and which are inconsistent. For consider, if conX (Γ) and Y preserves the X consistency predicate then conX (CY (Γ)). Suppose that Γ is not Y -consistent, then CY (Γ) = S. By [R] CX (CY (Γ)) = CX (S) = S which is to say th ...
Completeness - OSU Department of Mathematics
... • Whenever f is an n-ary function symbol h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )) for all a1 , . . . , an ∈ |A|. Notice that if = is in L, A and B respect equality and h is a homormorphism of A to B then h is 1-1 i.e. h is an embedding of A into B. When h is a homomorphism from A to ...
... • Whenever f is an n-ary function symbol h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )) for all a1 , . . . , an ∈ |A|. Notice that if = is in L, A and B respect equality and h is a homormorphism of A to B then h is 1-1 i.e. h is an embedding of A into B. When h is a homomorphism from A to ...
Sets with dependent elements: Elaborating on Castoriadis` notion of
... French language, the collection of one’s mental representations (or memories) etc., are in fact examples of collections whose (certain) members present a certain dependence to other members. We find this notion of ontological dependence interesting, and in this paper we shall focus on that and its i ...
... French language, the collection of one’s mental representations (or memories) etc., are in fact examples of collections whose (certain) members present a certain dependence to other members. We find this notion of ontological dependence interesting, and in this paper we shall focus on that and its i ...
The Science of Proof - University of Arizona Math
... In mathematical logic there is a distinction between syntax and semantics. Syntax is the study of the formal structure of the language. The formation of sentences and the structure of proofs are part of syntax. Semantics is about possible interpretations of sentences of the language and about the tr ...
... In mathematical logic there is a distinction between syntax and semantics. Syntax is the study of the formal structure of the language. The formation of sentences and the structure of proofs are part of syntax. Semantics is about possible interpretations of sentences of the language and about the tr ...
Introduction to Predicate Logic
... 2. If α is a constant, then [[α]] is specified by a function V (in the model M ) that assigns an individual object to each constant. [[α]]M,g = V (α) If P is a predicate, then [[P ]] is specified by a function V (in the model M ) that assigns a set-theoretic objects to each predicate. [[P ]]M,g = V ...
... 2. If α is a constant, then [[α]] is specified by a function V (in the model M ) that assigns an individual object to each constant. [[α]]M,g = V (α) If P is a predicate, then [[P ]] is specified by a function V (in the model M ) that assigns a set-theoretic objects to each predicate. [[P ]]M,g = V ...
Basic Metatheory for Propositional, Predicate, and Modal Logic
... issue here hinges on the connectives of L P . A set of connectives in an interpreted language (i.e., a language together with its semantics) for propositional logic is said to be adequate iff every truth function can be expressed by some formula of the language. The question, then, is whether the se ...
... issue here hinges on the connectives of L P . A set of connectives in an interpreted language (i.e., a language together with its semantics) for propositional logic is said to be adequate iff every truth function can be expressed by some formula of the language. The question, then, is whether the se ...