A Propositional Modal Logic for the Liar Paradox Martin Dowd
... forms it has puzzled logicians and philosophers of natural language since the time of the Greeks. Within the last decade, the tools of mathematical logic have been brought to bear on this paradox. It is fair to say that a model which is satisfactory mathematically has been devised. Whether the issue ...
... forms it has puzzled logicians and philosophers of natural language since the time of the Greeks. Within the last decade, the tools of mathematical logic have been brought to bear on this paradox. It is fair to say that a model which is satisfactory mathematically has been devised. Whether the issue ...
The strong completeness of the tableau method 1 The strong
... some < in the well-ordering of the formulas of L+) has been left out of , this can only mean that the corresponding set { } was inconsistent, hence so would be { } itself. On the other hand, the consistency of alone is guaranteed by condition (2). And finally, again by the ...
... some < in the well-ordering of the formulas of L+) has been left out of , this can only mean that the corresponding set { } was inconsistent, hence so would be { } itself. On the other hand, the consistency of alone is guaranteed by condition (2). And finally, again by the ...
Predicate_calculus
... Jump to: navigation, search A formal axiomatic theory; a calculus intended for the description of logical laws (cf. Logical law) that are true for any non-empty domain of objects with arbitrary predicates (i.e. properties and relations) given on these objects. In order to formulate the predicate cal ...
... Jump to: navigation, search A formal axiomatic theory; a calculus intended for the description of logical laws (cf. Logical law) that are true for any non-empty domain of objects with arbitrary predicates (i.e. properties and relations) given on these objects. In order to formulate the predicate cal ...
A Logic of Explicit Knowledge - Lehman College
... Now we drop the operator K from the language, and introduce a family of explicit reasons instead— I’ll use t as a typical one. Following [1, 2] I’ll write t:X to indicate that t applies to X—read it as “X is known for reason t.” Formally, if t is a reason and X is a formula, t:X is a formula. Of cou ...
... Now we drop the operator K from the language, and introduce a family of explicit reasons instead— I’ll use t as a typical one. Following [1, 2] I’ll write t:X to indicate that t applies to X—read it as “X is known for reason t.” Formally, if t is a reason and X is a formula, t:X is a formula. Of cou ...
Lecture 3
... Therefore by (the contrapositives of) the two parts of F1, we conclude that he is able to prevent evil, and he is willing to prevent evil. By the implication of F0, we therefore ‘know’ that Superman does prevent evil. But this contradicts F2. Since we have arrived at a contradiction, ...
... Therefore by (the contrapositives of) the two parts of F1, we conclude that he is able to prevent evil, and he is willing to prevent evil. By the implication of F0, we therefore ‘know’ that Superman does prevent evil. But this contradicts F2. Since we have arrived at a contradiction, ...
A Proof of Cut-Elimination Theorem for U Logic.
... is finding a common base for BPL and B. To make the two systems more comparable, Ardeshir and Vaezian in [1], introduced a modified version of mentioned axiomatization, and called it GBPC*. They also excluded connective ← from B and called the new system B’. They justified this action, by mentioning ...
... is finding a common base for BPL and B. To make the two systems more comparable, Ardeshir and Vaezian in [1], introduced a modified version of mentioned axiomatization, and called it GBPC*. They also excluded connective ← from B and called the new system B’. They justified this action, by mentioning ...
Logic and Resolution
... More expressive than propositional logic Distinguished from propositional logic by its use of quantifiers Each interpretation of first-order logic includes a domain of discourse over which the quantifiers range Additionally, it covers predicates Used to represent either a property or a relation betw ...
... More expressive than propositional logic Distinguished from propositional logic by its use of quantifiers Each interpretation of first-order logic includes a domain of discourse over which the quantifiers range Additionally, it covers predicates Used to represent either a property or a relation betw ...
Propositional Logic
... is false, P(2) is true,…. • Examples of predicates: – Domain ASCII characters - IsAlpha(x) : TRUE iff x is an alphabetical character. – Domain floating point numbers - IsInt(x): TRUE iff x is an integer. – Domain integers: Prime(x) - TRUE if x is ...
... is false, P(2) is true,…. • Examples of predicates: – Domain ASCII characters - IsAlpha(x) : TRUE iff x is an alphabetical character. – Domain floating point numbers - IsInt(x): TRUE iff x is an integer. – Domain integers: Prime(x) - TRUE if x is ...
Logic - Mathematical Institute SANU
... constants for operations and variables for numbers. The usual logical constants are the connectives and, or, if, if and only if and not, which are studied in propositional logic, and the quantifier expressions for every and for some, and identity (i.e. the relational expression equals), which, toget ...
... constants for operations and variables for numbers. The usual logical constants are the connectives and, or, if, if and only if and not, which are studied in propositional logic, and the quantifier expressions for every and for some, and identity (i.e. the relational expression equals), which, toget ...
Basic Logic - Progetto e
... Propositional logic is a useful tool for reasoning, but it is limited because it cannot see inside propositions and take advantage of relationships among inner elements. For example, we may assume that ...
... Propositional logic is a useful tool for reasoning, but it is limited because it cannot see inside propositions and take advantage of relationships among inner elements. For example, we may assume that ...
mathematical logic: constructive and non
... Much work has been done, especially by Péter since 1932, on special classes of computable functions, for which classes proofs are known t h a t all the computation procedures always terminate. To Church's thesis itself, the only suggested counterexamples involve 'computation procedures' in which the ...
... Much work has been done, especially by Péter since 1932, on special classes of computable functions, for which classes proofs are known t h a t all the computation procedures always terminate. To Church's thesis itself, the only suggested counterexamples involve 'computation procedures' in which the ...
slides
... formulas φ, ψ, if φ ⊨ ψ, then there is a first-order formula χ with BindPatt(χ) ⊆ BindPatt(φ) ∩ BindPatt(ψ) and φ ⊨ χ ⊨ ψ. Moreover the formula χ in question can effectively constructed from a proof of φ ⊨ ψ (in a suitable proof system). ...
... formulas φ, ψ, if φ ⊨ ψ, then there is a first-order formula χ with BindPatt(χ) ⊆ BindPatt(φ) ∩ BindPatt(ψ) and φ ⊨ χ ⊨ ψ. Moreover the formula χ in question can effectively constructed from a proof of φ ⊨ ψ (in a suitable proof system). ...
Propositional/First
... • Logic is a great knowledge representation language for many AI problems • Propositional logic is the simple foundation and fine for some AI problems • First order logic (FOL) is much more expressive as a KR language and more commonly used in AI • There are many variations: horn logic, higher order ...
... • Logic is a great knowledge representation language for many AI problems • Propositional logic is the simple foundation and fine for some AI problems • First order logic (FOL) is much more expressive as a KR language and more commonly used in AI • There are many variations: horn logic, higher order ...
slides
... A formula G is said to be a logical consequence of formulas F1 , F2 , . . . , Fn , notation F1 , . . . , Fn |= G , iff, for all interpretations I, if I |= F1 and . . . and I |= Fn then I |= G . Don’t get confused! The symbol |= is used in two different ways: I |= F F1 , . . . , Fn |= G In the first ...
... A formula G is said to be a logical consequence of formulas F1 , F2 , . . . , Fn , notation F1 , . . . , Fn |= G , iff, for all interpretations I, if I |= F1 and . . . and I |= Fn then I |= G . Don’t get confused! The symbol |= is used in two different ways: I |= F F1 , . . . , Fn |= G In the first ...
CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness
... I | L = I. This means that we have to define cI 0 for all c ∈ C. By the definition, cI 0 ∈ M , so this also means that we have to assign the elements of M to all constants c ∈ C in such a way that the resulting expansion is a model for all sentences from SHenkin . The quantifier axioms Q1, Q2 are fi ...
... I | L = I. This means that we have to define cI 0 for all c ∈ C. By the definition, cI 0 ∈ M , so this also means that we have to assign the elements of M to all constants c ∈ C in such a way that the resulting expansion is a model for all sentences from SHenkin . The quantifier axioms Q1, Q2 are fi ...
on fuzzy intuitionistic logic
... they m a y be t r u e 'in different ways'. By accepting different t r u t h values, we also break t h e true-false-dualism of classical logic. If we know t h e degree of t r u t h of a sentence we do not necessarily know t h e degree of falsehood of the sentence. In Fuzzy Intuitionistic Logic a half ...
... they m a y be t r u e 'in different ways'. By accepting different t r u t h values, we also break t h e true-false-dualism of classical logic. If we know t h e degree of t r u t h of a sentence we do not necessarily know t h e degree of falsehood of the sentence. In Fuzzy Intuitionistic Logic a half ...
Jean Van Heijenoort`s View of Modern Logic
... the proposition into subject and predicate had been replaced by its analysis into function and argument(s). A preliminary accomplishment was the propositional calculus, with a truth-functional definition of the connectives, including the conditional. Of cardinal importance was the realization that, ...
... the proposition into subject and predicate had been replaced by its analysis into function and argument(s). A preliminary accomplishment was the propositional calculus, with a truth-functional definition of the connectives, including the conditional. Of cardinal importance was the realization that, ...
Propositional Logic Syntax of Propositional Logic
... – reasoning from a true conclusion to premises that may have caused the conclusion ...
... – reasoning from a true conclusion to premises that may have caused the conclusion ...
MATHEMATICAL LOGIC CLASS NOTE 1. Propositional logic A
... Proposition 3.1. If s1 , s2 are maps from V to a model M , and agree on all free variables in a formula φ, then M |= φ[s1 ] iff M |= φ[s2 ]. The proposition is proved using routine induction on formulas. By the proposition, without ambiguity we often write M |= φ[a1 , ..., an ] instead of M |= φ[s] w ...
... Proposition 3.1. If s1 , s2 are maps from V to a model M , and agree on all free variables in a formula φ, then M |= φ[s1 ] iff M |= φ[s2 ]. The proposition is proved using routine induction on formulas. By the proposition, without ambiguity we often write M |= φ[a1 , ..., an ] instead of M |= φ[s] w ...
A simple proof of Parsons` theorem
... where we are identifying the terms with their interpretations in M. Note that all elements c, d1 , d2 , . . . are members of the above subset because the variables vj appear in the enumeration of terms. It is also clear that the above subset defines a substructure M∗ of M. Using the fact that U is a ...
... where we are identifying the terms with their interpretations in M. Note that all elements c, d1 , d2 , . . . are members of the above subset because the variables vj appear in the enumeration of terms. It is also clear that the above subset defines a substructure M∗ of M. Using the fact that U is a ...
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 ...
Aristotle`s particularisation
... interpretation—in a classical logic which appeals to Aristotle’s particularisation— by what he believed to be “a purely formal and much weaker assumption” of ω-consistency; introduction of which, presumably, would not invite inconsistency in any formal reasoning that appeals only syntactically to co ...
... interpretation—in a classical logic which appeals to Aristotle’s particularisation— by what he believed to be “a purely formal and much weaker assumption” of ω-consistency; introduction of which, presumably, would not invite inconsistency in any formal reasoning that appeals only syntactically to co ...
Propositional Logic
... Definition 2.1 Let Op = {¬, ∧, ∨, ⇒, ⇔ , ( , )} be the set of logical operators and Σ a set of symbols. The sets Op, Σ and {t, f } are pairwise disjoint. Σ is called the signature and its elements are the proposition variables. The set of propositional logic formulas is now recursively defined: • t ...
... Definition 2.1 Let Op = {¬, ∧, ∨, ⇒, ⇔ , ( , )} be the set of logical operators and Σ a set of symbols. The sets Op, Σ and {t, f } are pairwise disjoint. Σ is called the signature and its elements are the proposition variables. The set of propositional logic formulas is now recursively defined: • t ...
full text (.pdf)
... We consider two related decision problems: given a rule of the form (1), (i) is it relationally valid? That is, is it true in all relational models? (ii) is it derivable in PHL? The paper Kozen 2000] considered problem (i) only. We show that both of these problems are PSPACE -hard by a single reduc ...
... We consider two related decision problems: given a rule of the form (1), (i) is it relationally valid? That is, is it true in all relational models? (ii) is it derivable in PHL? The paper Kozen 2000] considered problem (i) only. We show that both of these problems are PSPACE -hard by a single reduc ...
10a
... Propositional logic • Logical constants: true, false • Propositional symbols: P, Q,... (aka atomic sentences) • Wrapping parentheses: ( … ) • Sentences are combined by connectives: and [conjunction] or [disjunction] implies [implication / conditional] is equivalent [biconditional] not [ne ...
... Propositional logic • Logical constants: true, false • Propositional symbols: P, Q,... (aka atomic sentences) • Wrapping parentheses: ( … ) • Sentences are combined by connectives: and [conjunction] or [disjunction] implies [implication / conditional] is equivalent [biconditional] not [ne ...