Some Principles of Logic
... • INDUCTION (inference of a general condition from a set of observed instances) • If the premises are true then the conclusion is probably but not necessarily true • The conclusion contains information not present, even implicitly, in the premises ...
... • INDUCTION (inference of a general condition from a set of observed instances) • If the premises are true then the conclusion is probably but not necessarily true • The conclusion contains information not present, even implicitly, in the premises ...
A constructive approach to nonstandard analysis*
... us here. As for Brouwer intuitionism [B] there is a first attempt by Vesley [31]. Moerdijk and Reyes [20] use topos theory to develop calculus with different kinds of infinitesimals. The logic used in the formal theories of their approach is intuitionistic, but the necessary properties of their mode ...
... us here. As for Brouwer intuitionism [B] there is a first attempt by Vesley [31]. Moerdijk and Reyes [20] use topos theory to develop calculus with different kinds of infinitesimals. The logic used in the formal theories of their approach is intuitionistic, but the necessary properties of their mode ...
3463: Mathematical Logic
... is applied to any configuration of the form αpaβ, or possibly αp if a is the blank symbol, and yields αbqβ. There are a few more cases to be considered for quintuples pabLq, but it is all quite simple. (1.7) Lemma If M is a Turing machine with initial state q0 , and x is an input string, then there ...
... is applied to any configuration of the form αpaβ, or possibly αp if a is the blank symbol, and yields αbqβ. There are a few more cases to be considered for quintuples pabLq, but it is all quite simple. (1.7) Lemma If M is a Turing machine with initial state q0 , and x is an input string, then there ...
Coordinate-free logic - Utrecht University Repository
... different than saying that there are ‘out there’ a less-than relation and a greaterthan relation. In my view, people who think there are really two such relations are misled by language. It seems hard to deny that 4’s being less than 6 is the very same fact as 6’s being greater than 4. In English an ...
... different than saying that there are ‘out there’ a less-than relation and a greaterthan relation. In my view, people who think there are really two such relations are misled by language. It seems hard to deny that 4’s being less than 6 is the very same fact as 6’s being greater than 4. In English an ...
The AND Operation - KFUPM Faculty List
... In other words, we can say that Zi is a function of the n input signals x1, x2, up to xn. Or we can write: Zi = Fi (x1, x2, ……, xn ) for i = 1, 2, 3, ….m The m output functions (Fi) are functions of binary signals and each produces a single binary output signal. Thus, these functions are binar ...
... In other words, we can say that Zi is a function of the n input signals x1, x2, up to xn. Or we can write: Zi = Fi (x1, x2, ……, xn ) for i = 1, 2, 3, ….m The m output functions (Fi) are functions of binary signals and each produces a single binary output signal. Thus, these functions are binar ...
Discrete Mathematics - Lecture 4: Propositional Logic and Predicate
... Lecture 4: Propositional Logic and Predicate Logic ...
... Lecture 4: Propositional Logic and Predicate Logic ...
Functional Programming
... Assignments are considered non-pure in functional programming because they can change the global state of the program and possibly influence function outcomes The value of a pure function only depends on its arguments (set! name x) re-assigns x to local or global name ...
... Assignments are considered non-pure in functional programming because they can change the global state of the program and possibly influence function outcomes The value of a pure function only depends on its arguments (set! name x) re-assigns x to local or global name ...
x - Stanford University
... Theorem: If R is transitive, then R-1 is transitive. Proof: Consider any a, b, and c such that aRb and bRc. Since R is transitive, we have aRc. Since aRb and bRc, we have bR-1a and cR-1b. Since we have aRc, we have cR-1a. Thus cR-1b, bR-1a, and cR-1a. ■ This proves ∀a. ∀b. ∀c. (aRb ∧ bRc → cR-1b ∧ b ...
... Theorem: If R is transitive, then R-1 is transitive. Proof: Consider any a, b, and c such that aRb and bRc. Since R is transitive, we have aRc. Since aRb and bRc, we have bR-1a and cR-1b. Since we have aRc, we have cR-1a. Thus cR-1b, bR-1a, and cR-1a. ■ This proves ∀a. ∀b. ∀c. (aRb ∧ bRc → cR-1b ∧ b ...
Hilbert Type Deductive System for Sentential Logic, Completeness
... (iii) β i is inferred via modus ponens from two previous wffs. Say they are γ → β i and γ. By the induction hypothesis, |– α → (γ → β i) and |– α → γ. The argument can be now brought to finish by showing: α → (γ → β i), α → γ |– α → β i. (This is not immediate, but easier than (ii); use axiom (ii) a ...
... (iii) β i is inferred via modus ponens from two previous wffs. Say they are γ → β i and γ. By the induction hypothesis, |– α → (γ → β i) and |– α → γ. The argument can be now brought to finish by showing: α → (γ → β i), α → γ |– α → β i. (This is not immediate, but easier than (ii); use axiom (ii) a ...
Interpreting and Applying Proof Theories for Modal Logic
... The identity axioms, the basic structural rules, the logical rules ¬L, ¬R, ∧L and ∧R, the modal rules L and R, plus the classical structural rules of weakening, contraction and cut, form the Display proof system for the basic normal modal logic K. In order to obtain display calculi for other modal ...
... The identity axioms, the basic structural rules, the logical rules ¬L, ¬R, ∧L and ∧R, the modal rules L and R, plus the classical structural rules of weakening, contraction and cut, form the Display proof system for the basic normal modal logic K. In order to obtain display calculi for other modal ...
Functional Programming
... Boolean values #t (true) and #f (false) Symbols, which are identifiers escaped with a single quote, e.g. 'y ...
... Boolean values #t (true) and #f (false) Symbols, which are identifiers escaped with a single quote, e.g. 'y ...
Logic Programming in Tabular Allegories
... computer science. In particular, the calculus of binary relations [37], whose main operations are intersection (∪), union (∩), relative complement \, inversion (_)o and relation composition (;) was shown by Tarski and Givant [40] to be a complete and adequate model for capturing all first-order logi ...
... computer science. In particular, the calculus of binary relations [37], whose main operations are intersection (∪), union (∩), relative complement \, inversion (_)o and relation composition (;) was shown by Tarski and Givant [40] to be a complete and adequate model for capturing all first-order logi ...
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.