Concept Hierarchies from a Logical Point of View
... that takes each monadic predicate p ∈ Σ to a subset of U . Now observe that a formal context hU, Σ, i uniquely corresponds to an interpretation M of Σ, and vice versa: simply define M (p) = p⊳ = {x ∈ U | x p}. The notion of an interpretation gives us the notion of truth and model as well: a state ...
... that takes each monadic predicate p ∈ Σ to a subset of U . Now observe that a formal context hU, Σ, i uniquely corresponds to an interpretation M of Σ, and vice versa: simply define M (p) = p⊳ = {x ∈ U | x p}. The notion of an interpretation gives us the notion of truth and model as well: a state ...
Lecture 3
... • A term can be a constant, a variable or a function name applied to zero or more arguments e.g., add(X,Y). More complex terms can be built from a vocabulary of function symbols and variable symbols. Terms can be considered as simple strings. • Term rewriting is a computational method that is based ...
... • A term can be a constant, a variable or a function name applied to zero or more arguments e.g., add(X,Y). More complex terms can be built from a vocabulary of function symbols and variable symbols. Terms can be considered as simple strings. • Term rewriting is a computational method that is based ...
Modal Logic and Model Theory
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... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://dv1litvip.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an e ...
Logic (Mathematics 1BA1) Reminder: Sets of numbers Proof by
... leads to a contradiction (ie something false). Using the theorem proof by contradiction, we have proven by equivalence that p ≡ T. ...
... leads to a contradiction (ie something false). Using the theorem proof by contradiction, we have proven by equivalence that p ≡ T. ...
THE MODAL LOGIC OF INNER MODELS §1. Introduction. In [10, 11
... The modal theory S4.2 is the smallest class of formulas containing all substitution instances of the above axioms and closed under modus ponens and necessitation (in other words, the smallest normal modal logic containing the above axioms). As usual, a preorder is a set P with a reflexive and transi ...
... The modal theory S4.2 is the smallest class of formulas containing all substitution instances of the above axioms and closed under modus ponens and necessitation (in other words, the smallest normal modal logic containing the above axioms). As usual, a preorder is a set P with a reflexive and transi ...
Proof Theory in Type Theory
... The negative translation provides a general way to make constructive sense of some non effective reasoning. However this method has some limitations. It does not work in presence of the axiom of description/choice. In this note, we analyse the interaction of classical logic with generalised inductiv ...
... The negative translation provides a general way to make constructive sense of some non effective reasoning. However this method has some limitations. It does not work in presence of the axiom of description/choice. In this note, we analyse the interaction of classical logic with generalised inductiv ...
Chapter 1 Logic and Set Theory
... A proof in mathematics demonstrates the truth of certain statement. It is therefore natural to begin with a brief discussion of statements. A statement, or proposition, is the content of an assertion. It is either true or false, but cannot be both true and false at the same time. For example, the ex ...
... A proof in mathematics demonstrates the truth of certain statement. It is therefore natural to begin with a brief discussion of statements. A statement, or proposition, is the content of an assertion. It is either true or false, but cannot be both true and false at the same time. For example, the ex ...
A Proof of Cut-Elimination Theorem for U Logic.
... Basic Propositional Logic, BPL, was invented by Albert Visser in 1981 [5]. He wanted to interpret implication as formal provability. To protect his system against the liar paradox, modus ponens is weakened. His axiomatization of BPL uses natural deduction[3, p. 8]. The first sequent calculus for BPL ...
... Basic Propositional Logic, BPL, was invented by Albert Visser in 1981 [5]. He wanted to interpret implication as formal provability. To protect his system against the liar paradox, modus ponens is weakened. His axiomatization of BPL uses natural deduction[3, p. 8]. The first sequent calculus for BPL ...
CHAP02 Axioms of Set Theory
... With the first four axioms we can get sets of any finite size. For example: {∅, {∅} ∪ {∅, {{∅}} = {∅, {∅}, {{ ∅}}}. But all such sets will be finite. We need the Axiom of Infinity to get an infinite set and with the Axiom of Specification we can be sure that subclasses of sets are indeed subsets. Th ...
... With the first four axioms we can get sets of any finite size. For example: {∅, {∅} ∪ {∅, {{∅}} = {∅, {∅}, {{ ∅}}}. But all such sets will be finite. We need the Axiom of Infinity to get an infinite set and with the Axiom of Specification we can be sure that subclasses of sets are indeed subsets. Th ...
Pairing Functions and Gödel Numbers Pairing Functions and Gödel
... Pairing Functions and Gödel Numbers For each n, the function [a1, …, an] is clearly primitive recursive. Gödel numbering satisfies the following uniqueness property: Theorem 8.2: If [a1, …, an] = [b1, …, bn] then ai = bi for i = 1, …, n. This follows immediately from the fundamental theorem of arith ...
... Pairing Functions and Gödel Numbers For each n, the function [a1, …, an] is clearly primitive recursive. Gödel numbering satisfies the following uniqueness property: Theorem 8.2: If [a1, …, an] = [b1, …, bn] then ai = bi for i = 1, …, n. This follows immediately from the fundamental theorem of arith ...
Chapter 1 Logic and Set Theory
... be used to prove it. Rigorous proofs are used to verify that a given statement that appears intuitively true is indeed true. Ultimately, a mathematical proof is a convincing argument that starts from some premises, and logically deduces the desired conclusion. Most proofs do not mention the logical ...
... be used to prove it. Rigorous proofs are used to verify that a given statement that appears intuitively true is indeed true. Ultimately, a mathematical proof is a convincing argument that starts from some premises, and logically deduces the desired conclusion. Most proofs do not mention the logical ...
Extending modal logic
... Take any topological space, say the real line Interpret the ♢ modality as closure: the closure of a set of real number is obtained by adding limit points. ...
... Take any topological space, say the real line Interpret the ♢ modality as closure: the closure of a set of real number is obtained by adding limit points. ...
On the regular extension axiom and its variants
... The first interesting consequence of wREA is that the class of hereditarily countable sets, HC = H(ω ∪ {ω}), constitutes a set. In the Leeds-Manchester Proof Theory Seminar, Peter Aczel asked whether CZF is at least strong enough to show that HC is a set. This section is devoted to showing that this ...
... The first interesting consequence of wREA is that the class of hereditarily countable sets, HC = H(ω ∪ {ω}), constitutes a set. In the Leeds-Manchester Proof Theory Seminar, Peter Aczel asked whether CZF is at least strong enough to show that HC is a set. This section is devoted to showing that this ...
Truth, Conservativeness and Provability
... him: he should deny that (GR) should follow from his theory of truth and at the same time offer some non-truth-theoretic analysis of our epistemic obligations (cf. Ketland 2005). ...
... him: he should deny that (GR) should follow from his theory of truth and at the same time offer some non-truth-theoretic analysis of our epistemic obligations (cf. Ketland 2005). ...