Notes on Modal Logic - Stanford University
... The basic modal language is a generic formal language with unary operators that have been used to reason about situations involving modal notions. This language is defined as follows: Definition 1.1 (The Basic Modal Language) Let S = {p, q, r, . . .} be a set of sentence letters, or atomic propositi ...
... The basic modal language is a generic formal language with unary operators that have been used to reason about situations involving modal notions. This language is defined as follows: Definition 1.1 (The Basic Modal Language) Let S = {p, q, r, . . .} be a set of sentence letters, or atomic propositi ...
Formal deduction in propositional logic
... Comments • In (2) and (3) of the preceding proof, we used a finite subset Σ0 to replace Σ because Σ may be infinite and accordingly not available in (Tr). • Suppose Σ0 = C1 , . . . Cn . Then (3) consists of n steps Σ, ¬¬A ` C1 , ...
... Comments • In (2) and (3) of the preceding proof, we used a finite subset Σ0 to replace Σ because Σ may be infinite and accordingly not available in (Tr). • Suppose Σ0 = C1 , . . . Cn . Then (3) consists of n steps Σ, ¬¬A ` C1 , ...
A pragmatic dialogic interpretation of bi
... involves a choice between the disjuncts. Following Girard’s classification of connectives in linear logic [27], it is the additive form of intuitionistic disjunction that makes it an unsuitable candidate as a right adjoint of subtraction. The solution advocated in [11] is to take multiplicative disj ...
... involves a choice between the disjuncts. Following Girard’s classification of connectives in linear logic [27], it is the additive form of intuitionistic disjunction that makes it an unsuitable candidate as a right adjoint of subtraction. The solution advocated in [11] is to take multiplicative disj ...
Introduction to Logic
... be exchanged. The intuition might have been that they “essentially mean the same”. In a more abstract, and later formulation, one would say that “not to affect a proposition” is “not to change its truth value” – either both are were false or both are true. Thus one obtains the idea that Two statemen ...
... be exchanged. The intuition might have been that they “essentially mean the same”. In a more abstract, and later formulation, one would say that “not to affect a proposition” is “not to change its truth value” – either both are were false or both are true. Thus one obtains the idea that Two statemen ...
ICS 353: Design and Analysis of Algorithms
... • The quantifiers and have higher precedence than all logical operators from propositional calculus. • E.g., x P(x) Q(x) • means……………………….. • does not mean …………………… ...
... • The quantifiers and have higher precedence than all logical operators from propositional calculus. • E.g., x P(x) Q(x) • means……………………….. • does not mean …………………… ...
Modal Consequence Relations
... rules: h{p, q}, p ∧ qi and both h{p ∧ q}, pi and h{p ∧ q}, qi. In addition, if ` = `M for some logical matrix we say that ` has all constants if for each s ∈ T there exists a nullary term function s such that for all valuations v v(s) = s. (Note that since var(s) = ∅ the value of s does not depend a ...
... rules: h{p, q}, p ∧ qi and both h{p ∧ q}, pi and h{p ∧ q}, qi. In addition, if ` = `M for some logical matrix we say that ` has all constants if for each s ∈ T there exists a nullary term function s such that for all valuations v v(s) = s. (Note that since var(s) = ∅ the value of s does not depend a ...
chapter9
... 5. Eliminate existential quantification by introducing Skolem constants/functions (x)P(x) P(c) c is a Skolem constant (a brand-new constant symbol that is not used in any other sentence) (x)(y)P(x,y) becomes (x)P(x, F(x)) since is within the scope of a universally quantified variable, use a ...
... 5. Eliminate existential quantification by introducing Skolem constants/functions (x)P(x) P(c) c is a Skolem constant (a brand-new constant symbol that is not used in any other sentence) (x)(y)P(x,y) becomes (x)P(x, F(x)) since is within the scope of a universally quantified variable, use a ...
Formal systems of fuzzy logic and their fragments∗
... strong conjunction, lattice conjunction and disjunction, and the truth constant for falsity, there is a natural question: what about the other fragments? As we always want to keep implication in our language (because the implication-less fragments of our fuzzy logics are essentially classical, see [ ...
... strong conjunction, lattice conjunction and disjunction, and the truth constant for falsity, there is a natural question: what about the other fragments? As we always want to keep implication in our language (because the implication-less fragments of our fuzzy logics are essentially classical, see [ ...
Propositional logic - Cheriton School of Computer Science
... departure between schools of logical thought, and the choice we make fundamentally affects the properties of the resulting logic. If we believe that ¬φ means that φ is false, then we are classicists and our proof theory becomes a proof theory for classical logic. We will then handle negation in a wa ...
... departure between schools of logical thought, and the choice we make fundamentally affects the properties of the resulting logic. If we believe that ¬φ means that φ is false, then we are classicists and our proof theory becomes a proof theory for classical logic. We will then handle negation in a wa ...
On the Notion of Coherence in Fuzzy Answer Set Semantics
... negation in the context of residuated logic programming is provided in terms of the notion of coherence as a generalization in the fuzzy framework of the concept of consistence. Then, fuzzy answer sets for general residuated logic programs are defined as a suitable generalization of the Gelfond-Lifs ...
... negation in the context of residuated logic programming is provided in terms of the notion of coherence as a generalization in the fuzzy framework of the concept of consistence. Then, fuzzy answer sets for general residuated logic programs are defined as a suitable generalization of the Gelfond-Lifs ...
Topological aspects of real-valued logic
... Theorem 1.0.3. Let S be a two-sorted metric signature, and let L be a countable fragment of Lω1 ,ω (S). Let T be an L-theory and let M = h M, V, . . . i be a model of T where M has density κ and V has density λ, with κ > λ ≥ ℵ0 . Then there is a model N = h N, W, . . . i ≡L M with N of density ℵ1 an ...
... Theorem 1.0.3. Let S be a two-sorted metric signature, and let L be a countable fragment of Lω1 ,ω (S). Let T be an L-theory and let M = h M, V, . . . i be a model of T where M has density κ and V has density λ, with κ > λ ≥ ℵ0 . Then there is a model N = h N, W, . . . i ≡L M with N of density ℵ1 an ...
Interpreting and Applying Proof Theories for Modal Logic
... (A ∧ B), which tells us that if A and B are true here, then at any accessible point there, A ∧ B holds. This can be reformulated as a claim that (A ∧ B) is true here, as the structural connective is rewritten as the object language connective . The identity axioms, the basic structural rules, ...
... (A ∧ B), which tells us that if A and B are true here, then at any accessible point there, A ∧ B holds. This can be reformulated as a claim that (A ∧ B) is true here, as the structural connective is rewritten as the object language connective . The identity axioms, the basic structural rules, ...
An Introduction to Proof Theory - UCSD Mathematics
... is a mapping from {T, F }k to {T, F } where we use T and F to represent True and False. The most frequently used examples of Boolean functions are the connectives > and ⊥ which are the 0-ary functions with values T and F , respectively; the binary connectives ∧, ∨, ⊃, ↔ and ⊕ for “and”, “or”, “if-th ...
... is a mapping from {T, F }k to {T, F } where we use T and F to represent True and False. The most frequently used examples of Boolean functions are the connectives > and ⊥ which are the 0-ary functions with values T and F , respectively; the binary connectives ∧, ∨, ⊃, ↔ and ⊕ for “and”, “or”, “if-th ...
1. Propositional Logic 1.1. Basic Definitions. Definition 1.1. The
... • If φ and ψ are formulas then so are (φ ∧ ψ), (φ ∨ ψ), (φ → ψ). We omit parentheses whenever they are not needed for clarity. We write ¬φ as an abbreviation for φ → ⊥. We occasionally use φ ↔ ψ as an abbreviation for (φ → ψ) ∧ (ψ → φ). To limit the number of inductive cases we need to write, we som ...
... • If φ and ψ are formulas then so are (φ ∧ ψ), (φ ∨ ψ), (φ → ψ). We omit parentheses whenever they are not needed for clarity. We write ¬φ as an abbreviation for φ → ⊥. We occasionally use φ ↔ ψ as an abbreviation for (φ → ψ) ∧ (ψ → φ). To limit the number of inductive cases we need to write, we som ...
Chapter X: Computational Complexity of Propositional Fuzzy Logics
... a given arity in N. The connectives with arity 0 are called constants. This chapter only considers languages with finitely many connectives of a positive arity (while there can be infinitely many constants). Given a countably infinite set of variables Var , using the connectives of L and parentheses ...
... a given arity in N. The connectives with arity 0 are called constants. This chapter only considers languages with finitely many connectives of a positive arity (while there can be infinitely many constants). Given a countably infinite set of variables Var , using the connectives of L and parentheses ...
Suszko`s Thesis, Inferential Many-Valuedness, and the
... Suszko's Reduction calls for an analysis not only because it seems to undermine many-valued logic, but also because it invites a re-consideration of the notion of a many-valued logic in particular and the notion of a logical system in general. According to Marcelo Tsuji [40, p. 308], "Suszkothought ...
... Suszko's Reduction calls for an analysis not only because it seems to undermine many-valued logic, but also because it invites a re-consideration of the notion of a many-valued logic in particular and the notion of a logical system in general. According to Marcelo Tsuji [40, p. 308], "Suszkothought ...
Linear Contextual Modal Type Theory
... any instantiation of F will need to mention x and y exactly once on two different rigid paths. Thus the left hand side and the right hand side of the equation above will differ in these two places. If we were to work in linear logic with >, the problem is also solvable by choosing the constant b . λ ...
... any instantiation of F will need to mention x and y exactly once on two different rigid paths. Thus the left hand side and the right hand side of the equation above will differ in these two places. If we were to work in linear logic with >, the problem is also solvable by choosing the constant b . λ ...
Intuitionistic Logic - Institute for Logic, Language and Computation
... and associativity of conjunction and disjunction, both distributivity laws, and – (φ → ψ ∧ χ) ↔ (φ → ψ) ∧ (φ → χ), – (φ → χ) ∧ (ψ → χ) ↔ ((φ ∨ ψ) → χ)), – (φ → (ψ → χ)) ↔ (φ ∧ ψ) → χ. – (φ ∨ ψ) ∧ ¬φ → ψ) (needs ex falso!), – (φ → ψ) → ((ψ → χ) → (φ → χ)), – (φ → ψ) → (¬ψ → ¬φ) (the converse form of ...
... and associativity of conjunction and disjunction, both distributivity laws, and – (φ → ψ ∧ χ) ↔ (φ → ψ) ∧ (φ → χ), – (φ → χ) ∧ (ψ → χ) ↔ ((φ ∨ ψ) → χ)), – (φ → (ψ → χ)) ↔ (φ ∧ ψ) → χ. – (φ ∨ ψ) ∧ ¬φ → ψ) (needs ex falso!), – (φ → ψ) → ((ψ → χ) → (φ → χ)), – (φ → ψ) → (¬ψ → ¬φ) (the converse form of ...
An Introduction to Mathematical Logic
... In future we will use the following conventions for “metavariables”: “P ”,“Q”,“R” (with or without indices) denote predicates. “f ”,“g”,“h” (with or without indices) denote function signs. “c” (with or without indices) denote constants. “x”,“y”,“z” (with or without indices) denote variables. Remark ...
... In future we will use the following conventions for “metavariables”: “P ”,“Q”,“R” (with or without indices) denote predicates. “f ”,“g”,“h” (with or without indices) denote function signs. “c” (with or without indices) denote constants. “x”,“y”,“z” (with or without indices) denote variables. Remark ...
Modal Logic for Artificial Intelligence
... valid anymore. We call ‘or’ and ‘not’ logical constants. Together with ‘and’, ‘if . . . then’ and ‘if, and only if’, they are the logical constants of propositional logic (see section 1). A formal logic is a definition of valid argument forms, such as the one above. There are different methods for d ...
... valid anymore. We call ‘or’ and ‘not’ logical constants. Together with ‘and’, ‘if . . . then’ and ‘if, and only if’, they are the logical constants of propositional logic (see section 1). A formal logic is a definition of valid argument forms, such as the one above. There are different methods for d ...
Introduction to Logic
... without changing its value. In Aristotle this meant simply that the pairs he determined could be exchanged. The intuition might have been that they “essentially mean the same”. In a more abstract, and later formulation, one would say that “not to affect a proposition” is “not to change its truth val ...
... without changing its value. In Aristotle this meant simply that the pairs he determined could be exchanged. The intuition might have been that they “essentially mean the same”. In a more abstract, and later formulation, one would say that “not to affect a proposition” is “not to change its truth val ...
preliminary version
... Intuitionism. Proof checkers based on type theory, like for instance Coq, work with intuitionistic logic, sometimes also called constructive logic. This is the logic of the natural deduction proof system discussed so far. The intuition is that truth in intuitionistic logic corresponds to the existen ...
... Intuitionism. Proof checkers based on type theory, like for instance Coq, work with intuitionistic logic, sometimes also called constructive logic. This is the logic of the natural deduction proof system discussed so far. The intuition is that truth in intuitionistic logic corresponds to the existen ...
Partial Grounded Fixpoints
... grounded, we show that the A-well-founded fixpoint is always A-grounded. This last result illustrates that the wellfounded semantics does a great job at avoiding ungrounded models: the well-founded model is not just some grounded model, it is the least precise grounded model. A third reason is that ...
... grounded, we show that the A-well-founded fixpoint is always A-grounded. This last result illustrates that the wellfounded semantics does a great job at avoiding ungrounded models: the well-founded model is not just some grounded model, it is the least precise grounded model. A third reason is that ...
Notes on Classical Propositional Logic
... In fact a stronger result can be proved, which we leave to you as Exercise 5.1. So, we will make sure to choose axiom schemes whose instances are tautologies, and rules of derivation that are sound. Completeness is much harder, however. This will occupy the next several sections. Since it is more co ...
... In fact a stronger result can be proved, which we leave to you as Exercise 5.1. So, we will make sure to choose axiom schemes whose instances are tautologies, and rules of derivation that are sound. Completeness is much harder, however. This will occupy the next several sections. Since it is more co ...
Quantifiers
... • The truth-table method is a decision procedure for truthfunctional consequence. That is, for any and , the truthtable will systematically and correctly decide whether TF or not. • Because a decision procedure for truth-functional consequence exists, we say that truth-functional consequence ...
... • The truth-table method is a decision procedure for truthfunctional consequence. That is, for any and , the truthtable will systematically and correctly decide whether TF or not. • Because a decision procedure for truth-functional consequence exists, we say that truth-functional consequence ...