HISTORY OF LOGIC
... – Russell’s Paradox: If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be removed. However, if it does not list itself, then it should be added to itself. ...
... – Russell’s Paradox: If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be removed. However, if it does not list itself, then it should be added to itself. ...
Infinitistic Rules of Proof and Their Semantics
... from M is definable in M. If T is a set of formulas, by Cn (T) we mean the least set of formulas containing T and axioms of logic and which is closed with respect to modus ponens (we assume that logic is so axiomatized, that the only rule of inference is modus ponens). By ryl we mean the set of Gode ...
... from M is definable in M. If T is a set of formulas, by Cn (T) we mean the least set of formulas containing T and axioms of logic and which is closed with respect to modus ponens (we assume that logic is so axiomatized, that the only rule of inference is modus ponens). By ryl we mean the set of Gode ...
Discrete Mathematics
... The negation of a proposition P, written ¬ P, is a proposition. The conjunction (and) of two propositions, written P ∧ Q, is a proposition. The disjunction (or) of two propositions, written P ∨ Q, is a proposition. The conditional statement (implies), written P −→ Q, is a proposition. The Boolean va ...
... The negation of a proposition P, written ¬ P, is a proposition. The conjunction (and) of two propositions, written P ∧ Q, is a proposition. The disjunction (or) of two propositions, written P ∨ Q, is a proposition. The conditional statement (implies), written P −→ Q, is a proposition. The Boolean va ...
comments on the logic of constructible falsity (strong negation)
... given the constructive derivability of excluded middle for atomic (and other decidable) formulas of arithmetic, the addition of D to either intuitionistic of Nelson arithmetic, the would cause it to collapse into classical arithmetic. Görnemann’s result suggests the conjecture that a classical mode ...
... given the constructive derivability of excluded middle for atomic (and other decidable) formulas of arithmetic, the addition of D to either intuitionistic of Nelson arithmetic, the would cause it to collapse into classical arithmetic. Görnemann’s result suggests the conjecture that a classical mode ...
Is the principle of contradiction a consequence of ? Jean
... student, or even a professor, of the University of Oxbridge he will probably not be able to present such a proof. (S)he may even claim that there is no such a proof because it is false. The examination of the proof of PROPOSITION IV is interesting for various reasons. It is related to the question o ...
... student, or even a professor, of the University of Oxbridge he will probably not be able to present such a proof. (S)he may even claim that there is no such a proof because it is false. The examination of the proof of PROPOSITION IV is interesting for various reasons. It is related to the question o ...
Logical Consequence by Patricia Blanchette Basic Question (BQ
... Because of the expressive power of higher order logics, they can say things about a formal system S which are either true or false of that system, but which can not be derived using the deductive apparatus of that system. This is the basic insight captured in Gödel’s Incompleteness Theorem, but it i ...
... Because of the expressive power of higher order logics, they can say things about a formal system S which are either true or false of that system, but which can not be derived using the deductive apparatus of that system. This is the basic insight captured in Gödel’s Incompleteness Theorem, but it i ...
Unification in Propositional Logic
... Pitts’ theorem has many meanings: prooftheoretically, it means that second order IPC can be interpreted in IPC, model-theoretically it means that the theory of Heyting algebras has a model completion and categorically that the opposite of the category of finitely presented Heyting algebras is a Heyt ...
... Pitts’ theorem has many meanings: prooftheoretically, it means that second order IPC can be interpreted in IPC, model-theoretically it means that the theory of Heyting algebras has a model completion and categorically that the opposite of the category of finitely presented Heyting algebras is a Heyt ...
pdf
... processing.) We are learning that skill in a very realistic setting by showing how to translate informal mathematics into symbolic logic. We are also showing how to expose implicit knowledge and make it explicit. Rather than showing how to translate into symbolic logic statements made in the newspap ...
... processing.) We are learning that skill in a very realistic setting by showing how to translate informal mathematics into symbolic logic. We are also showing how to expose implicit knowledge and make it explicit. Rather than showing how to translate into symbolic logic statements made in the newspap ...
A systematic proof theory for several modal logics
... where the two rules not labelled with ↑ or ↓ are self-dual. 2. The entire up-fragment , ie. the rules labelled with ↑, is admissible. This is shown, in op. cit. and discussed in more detail below, by means of a translation from cut-free proofs of the sequent calculus into proofs of system KS, that i ...
... where the two rules not labelled with ↑ or ↓ are self-dual. 2. The entire up-fragment , ie. the rules labelled with ↑, is admissible. This is shown, in op. cit. and discussed in more detail below, by means of a translation from cut-free proofs of the sequent calculus into proofs of system KS, that i ...
PDF
... D has n ’s, and so by induction, ` D[B/p] ↔ D[C/p], and therefore ` D[B/p] ↔ D[C/p] by 2. This means that ` A[B/p] ↔ A[C/p]. 4. ` A → B implies ` A → B Proof. By assumption, tautology ` (A → B) → (¬B → ¬A), and modus ponens, we get ` ¬B → ¬A. By 1, ` ¬B → ¬A. By another instance of the above ...
... D has n ’s, and so by induction, ` D[B/p] ↔ D[C/p], and therefore ` D[B/p] ↔ D[C/p] by 2. This means that ` A[B/p] ↔ A[C/p]. 4. ` A → B implies ` A → B Proof. By assumption, tautology ` (A → B) → (¬B → ¬A), and modus ponens, we get ` ¬B → ¬A. By 1, ` ¬B → ¬A. By another instance of the above ...
Exam 1 Solutions for Spring 2014
... 4. (10 points) A number n is a multiple of 3 if n = 3k for some integer k. Prove that if n2 is a multiple of 3, then n is a multiple of 3. Graded by Stacy Note: This question should have specified that n is an integer. To help compensate for this omission, the lowest score you can receive on this qu ...
... 4. (10 points) A number n is a multiple of 3 if n = 3k for some integer k. Prove that if n2 is a multiple of 3, then n is a multiple of 3. Graded by Stacy Note: This question should have specified that n is an integer. To help compensate for this omission, the lowest score you can receive on this qu ...
PDF
... 1. Strictly speaking, the language Li of PLi under this system is different from the language Lc of PLc under this system. In Li , the logical connectives consist of →, ¬, ∧, ∨, whereas in Lc , only → is used. The other connectives are introduced as abbreviational devices: ¬A is A →⊥, A ∨ B is ¬A → ...
... 1. Strictly speaking, the language Li of PLi under this system is different from the language Lc of PLc under this system. In Li , the logical connectives consist of →, ¬, ∧, ∨, whereas in Lc , only → is used. The other connectives are introduced as abbreviational devices: ¬A is A →⊥, A ∨ B is ¬A → ...
Predicate Logic Review
... (You may not be familiar with the set-theoretic notation we use here. It’s fairly simple. {1, 2} denotes the set containing 1 and 2. {x : x is F } denotes the set containing every x such that x is F , that is, the set of F s. Angle brackets indicate ordered sequences. So, 〈1, 2〉 is the sequence cons ...
... (You may not be familiar with the set-theoretic notation we use here. It’s fairly simple. {1, 2} denotes the set containing 1 and 2. {x : x is F } denotes the set containing every x such that x is F , that is, the set of F s. Angle brackets indicate ordered sequences. So, 〈1, 2〉 is the sequence cons ...
Predicate_calculus
... 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 calculus one must first fix an ...
... 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 calculus one must first fix an ...
Modal Logic
... for basic modal logic is quite general (although it can be further generalized as we will see later) and can be refined to yield the properties appropriate for the intended application. We will concentrate on three different applications: logic of necessity, temporal logic and logic of knowledge. T ...
... for basic modal logic is quite general (although it can be further generalized as we will see later) and can be refined to yield the properties appropriate for the intended application. We will concentrate on three different applications: logic of necessity, temporal logic and logic of knowledge. T ...
term 1 - Teaching-WIKI
... – A finite or countable set R of relation symbols or predicate symbols, with an integer n associated with symbol. This is called the arity of the symbol. – A finite or countable set F of function symbols. Again each function symbol has a specific arity. – A finite or countable set C of constant symb ...
... – A finite or countable set R of relation symbols or predicate symbols, with an integer n associated with symbol. This is called the arity of the symbol. – A finite or countable set F of function symbols. Again each function symbol has a specific arity. – A finite or countable set C of constant symb ...