Section 1: Propositional Logic
... • The first is the depth to which we explore the structure of statements. The study of the basic level of structure is called propositional logic. First order predicate logic, which is often called just predicate logic, studies structure on a deeper level. • The second direction is the nature of tru ...
... • The first is the depth to which we explore the structure of statements. The study of the basic level of structure is called propositional logic. First order predicate logic, which is often called just predicate logic, studies structure on a deeper level. • The second direction is the nature of tru ...
Proof Theory for Propositional Logic
... One of the most interesting issues in the philosophy of language concerns the notion of compositionality. It starts with a puzzle raised by Descartes.1 Given that the overwhelming majority of sentences you hear and speak have never been spoken before and will never be spoken before, how do you under ...
... One of the most interesting issues in the philosophy of language concerns the notion of compositionality. It starts with a puzzle raised by Descartes.1 Given that the overwhelming majority of sentences you hear and speak have never been spoken before and will never be spoken before, how do you under ...
Characterizations of stable model semantics for logic programs with
... 2003; Elkabani et al. 2004; Faber et al. 2004; Marek and Remmel 2004; Marek and Truszczynski 2004; Pelov 2004; Pelov and Truszczynski 2004; Calimeri et al . 2005; Elkabani et al. 2005; Ferraris 2005; Liu and Truszczynski 2005; Liu and Truszczynski 2006; Son et al. 2006; Liu et al. 2007; Pelov et al. ...
... 2003; Elkabani et al. 2004; Faber et al. 2004; Marek and Remmel 2004; Marek and Truszczynski 2004; Pelov 2004; Pelov and Truszczynski 2004; Calimeri et al . 2005; Elkabani et al. 2005; Ferraris 2005; Liu and Truszczynski 2005; Liu and Truszczynski 2006; Son et al. 2006; Liu et al. 2007; Pelov et al. ...
Consequence Operators for Defeasible - SeDiCI
... Arti¯cial Intelligence (AI) has long dealt with the issue of ¯nding a suitable formalization for commonsense reasoning. Defeasible argumentation has proven to be a successful approach in many respects, proving to be a con°uence point for many alternative logical frameworks. Di®erent formalisms have ...
... Arti¯cial Intelligence (AI) has long dealt with the issue of ¯nding a suitable formalization for commonsense reasoning. Defeasible argumentation has proven to be a successful approach in many respects, proving to be a con°uence point for many alternative logical frameworks. Di®erent formalisms have ...
Introduction to Modal and Temporal Logic
... Lemma 1 For any Kripke model hW, R, ϑi, any w ∈ W and any formula ϕ, either ϑ(w, ϕ) = t or else ϑ(w, ϕ) = f . Proof: Pick any Kripke model hW, R, ϑi, any w ∈ W , and any formula ϕ. Proceed by induction on the length l of ϕ. Base Case l = 1: If ϕ is an atomic formula p, either ϑ(w, p) = t or ϑ(w, p) ...
... Lemma 1 For any Kripke model hW, R, ϑi, any w ∈ W and any formula ϕ, either ϑ(w, ϕ) = t or else ϑ(w, ϕ) = f . Proof: Pick any Kripke model hW, R, ϑi, any w ∈ W , and any formula ϕ. Proceed by induction on the length l of ϕ. Base Case l = 1: If ϕ is an atomic formula p, either ϑ(w, p) = t or ϑ(w, p) ...
Notes on Mathematical Logic David W. Kueker
... which make all sentences in Γ true. Model Theory discusses the properties such classes of interpretations have. One important result of model theory for first-order languages is the Compactness Theorem, which states that if Mod(Γ) = ∅ then there must be some finite Γ0 ⊆ Γ with Mod(Γ0 ) = ∅. Part ?? ...
... which make all sentences in Γ true. Model Theory discusses the properties such classes of interpretations have. One important result of model theory for first-order languages is the Compactness Theorem, which states that if Mod(Γ) = ∅ then there must be some finite Γ0 ⊆ Γ with Mod(Γ0 ) = ∅. Part ?? ...
Safety Metric Temporal Logic is Fully Decidable
... In this paper we are concerned in particular with Metric Temporal Logic (MTL), one of the most widely known real-time logics. MTL is a variant of Linear Temporal Logic in which the temporal operators are replaced by timeconstrained versions. For example, the formula [0,5] ϕ expresses that ϕ holds ...
... In this paper we are concerned in particular with Metric Temporal Logic (MTL), one of the most widely known real-time logics. MTL is a variant of Linear Temporal Logic in which the temporal operators are replaced by timeconstrained versions. For example, the formula [0,5] ϕ expresses that ϕ holds ...
How complicated is the set of stable models of a recursive logic
... ([Marek, Nerode and Remmel, 1990a]) that for every recursive program P , then the class S(P ) is a Π02 subset of the Cantor space, under a suitable coding of the Herbrand universe as ω. This converts questions about the complexity of stable models to questions in Cantor space or Baire space. So we a ...
... ([Marek, Nerode and Remmel, 1990a]) that for every recursive program P , then the class S(P ) is a Π02 subset of the Cantor space, under a suitable coding of the Herbrand universe as ω. This converts questions about the complexity of stable models to questions in Cantor space or Baire space. So we a ...
The Foundations
... their truth or falsity (Semantics and Axiomatics). It is the foundation for expressing formal proofs in all branches of mathematics. ...
... their truth or falsity (Semantics and Axiomatics). It is the foundation for expressing formal proofs in all branches of mathematics. ...
A Logical Framework for Default Reasoning
... ∆ is a set of formulae, called the set of possible hypotheses. Any ground instance of these can be used as a hypothesis if it is consistent. Definition 1 a scenario of F, ∆ is a set D ∪ F where D is a set of ground instances of elements of ∆ such that D ∪ F is consistent. Definition 2 If g is a clos ...
... ∆ is a set of formulae, called the set of possible hypotheses. Any ground instance of these can be used as a hypothesis if it is consistent. Definition 1 a scenario of F, ∆ is a set D ∪ F where D is a set of ground instances of elements of ∆ such that D ∪ F is consistent. Definition 2 If g is a clos ...
Propositional Logic
... of a theorem of Church, there is no procedure that will halt for every input formula and decide whether or not a formula is valid. There are many ways of proving the completeness of a proof system. Oddly, most proofs establishing completeness only show that if a formula A is valid, then there exists ...
... of a theorem of Church, there is no procedure that will halt for every input formula and decide whether or not a formula is valid. There are many ways of proving the completeness of a proof system. Oddly, most proofs establishing completeness only show that if a formula A is valid, then there exists ...
On the Construction of Analytic Sequent Calculi for Sub
... In what follows, we assume a propositional language for classical logic, that consists of a countably infinite set of atomic variables At = {p1 , p2 , . . .}, the binary connectives ∧, ∨ and ⊃, the unary connective ¬, and the nullary connectives > and ⊥. A sequent is a pair hΓ, ∆i (denoted by Γ ⇒ ∆) ...
... In what follows, we assume a propositional language for classical logic, that consists of a countably infinite set of atomic variables At = {p1 , p2 , . . .}, the binary connectives ∧, ∨ and ⊃, the unary connective ¬, and the nullary connectives > and ⊥. A sequent is a pair hΓ, ∆i (denoted by Γ ⇒ ∆) ...
Dialectica Interpretations A Categorical Analysis
... The work presented in this thesis is a contribution to the area of type theory and semantics for programming languages in that we develop and study new models for type theories and programming logics. It is also a contribution to the area of logic in computer science, in that our categorical analys ...
... The work presented in this thesis is a contribution to the area of type theory and semantics for programming languages in that we develop and study new models for type theories and programming logics. It is also a contribution to the area of logic in computer science, in that our categorical analys ...
Thesis Proposal: A Logical Foundation for Session-based
... it introduces potentially divergent computations, but allows me to showcase additionally interesting programs. To recover the connections with logic I restrict general recursive types to inductive and coinductive types and ensure nondivergence of computation through the introduction of syntactic res ...
... it introduces potentially divergent computations, but allows me to showcase additionally interesting programs. To recover the connections with logic I restrict general recursive types to inductive and coinductive types and ensure nondivergence of computation through the introduction of syntactic res ...
