Intuitionistic modal logic made explicit
... sense and introduced epistemic, i.e., possible world, models for justification logics. These models have been further developed to modular models as we use them in this paper [5, 17]. This general reading of justification led to many applications in epistemic logic [3, 4, 7, 8, 9, 10, 14, 16, 18]. G ...
... sense and introduced epistemic, i.e., possible world, models for justification logics. These models have been further developed to modular models as we use them in this paper [5, 17]. This general reading of justification led to many applications in epistemic logic [3, 4, 7, 8, 9, 10, 14, 16, 18]. G ...
C311 First Class Objects
... First Class Objects Expressible as an anonymous literal value Storable in variables Storable in data structures Having an intrinsic identity (independent of any given name) Comparable for equality with other entities Passable as a parameter to a function Returnable as the result of a function call C ...
... First Class Objects Expressible as an anonymous literal value Storable in variables Storable in data structures Having an intrinsic identity (independent of any given name) Comparable for equality with other entities Passable as a parameter to a function Returnable as the result of a function call C ...
Lecture 23 Notes
... We will show how to define virtual constructive evidence for classical propositions using the refinement type of computational type theory to specify the classical computational content. The refinement type, {U nit|P }, is critical. If P is known by constructive evidence p, then the refinement type ...
... We will show how to define virtual constructive evidence for classical propositions using the refinement type of computational type theory to specify the classical computational content. The refinement type, {U nit|P }, is critical. If P is known by constructive evidence p, then the refinement type ...
.pdf
... An atomic sentence P a1 ..an is true under I if (ϕ(a1 ), ..ϕ(an )) ∈ ι(P ). In this manner, every interpretation induces an atomic valuation v0 (together with ϕ) and vice versa and from now on we will use whatever notion is more convenient. A formula A is called satisfiable if it is true under at l ...
... An atomic sentence P a1 ..an is true under I if (ϕ(a1 ), ..ϕ(an )) ∈ ι(P ). In this manner, every interpretation induces an atomic valuation v0 (together with ϕ) and vice versa and from now on we will use whatever notion is more convenient. A formula A is called satisfiable if it is true under at l ...
Constructive Set Theory and Brouwerian Principles1
... an n ∈ N such that (α, n) ∈ P must be able to calculate n from some finite ...
... an n ∈ N such that (α, n) ∈ P must be able to calculate n from some finite ...
Welcome - williamt.com
... • Boolean algebra allows formal expression, simplification, manipulation, minimization (G. Boole, 18050’s) • Niftily maps onto the physical world (usually voltage) ...
... • Boolean algebra allows formal expression, simplification, manipulation, minimization (G. Boole, 18050’s) • Niftily maps onto the physical world (usually voltage) ...
Reducing Propositional Theories in Equilibrium Logic to
... program reducts, in the style of [7,12], the new definition is also equivalent to that of equilibrium model. Consequently, to understand propositional theories, hence also embedded implications, in terms of answer sets one can apply equally well either equilibrium logic or the new reduct notion of [ ...
... program reducts, in the style of [7,12], the new definition is also equivalent to that of equilibrium model. Consequently, to understand propositional theories, hence also embedded implications, in terms of answer sets one can apply equally well either equilibrium logic or the new reduct notion of [ ...
The Diagonal Lemma Fails in Aristotelian Logic
... exist. However, the formulae in Table 2 are implausible translations of the natural language sentences. (Strawson, 1952, p. 173) So he proposed to take the term (∃x)Fx as a presupposition. It means that ~(Ex)Fx does not imply that A is false, but rather (Ex)Fx “is a necessary precondition not merely ...
... exist. However, the formulae in Table 2 are implausible translations of the natural language sentences. (Strawson, 1952, p. 173) So he proposed to take the term (∃x)Fx as a presupposition. It means that ~(Ex)Fx does not imply that A is false, but rather (Ex)Fx “is a necessary precondition not merely ...
Propositional Logic
... see that KB | Q. We call this syntactic process derivation and write KB Q. Such syntactic proof systems are called calculi. To ensure that a calculus does not generate errors, we define two fundamental properties of calculi. Definition 2.7 A calculus is called sound if every derived proposition f ...
... see that KB | Q. We call this syntactic process derivation and write KB Q. Such syntactic proof systems are called calculi. To ensure that a calculus does not generate errors, we define two fundamental properties of calculi. Definition 2.7 A calculus is called sound if every derived proposition f ...
First-order logic;
... A derivation (or proof ) in an axiom system AX is a sequence of formulas C1 , . . . , CN ; each formula Ck is either an axiom in AX or follows from previous formulas using an inference rule in AX : I i.e., there is an inference rule A1 , . . . , An ` B such that Ai = Cji for some ji < N and B = CN . ...
... A derivation (or proof ) in an axiom system AX is a sequence of formulas C1 , . . . , CN ; each formula Ck is either an axiom in AX or follows from previous formulas using an inference rule in AX : I i.e., there is an inference rule A1 , . . . , An ` B such that Ai = Cji for some ji < N and B = CN . ...
Curry–Howard correspondence
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambek correspondence.