Propositional Logic and Methods of Inference
... Note that p → q is equivalent to ~p v q SEEM 5750 ...
... Note that p → q is equivalent to ~p v q SEEM 5750 ...
3.3 Inference
... the universe of even numbers. How did we know to start the proof that way? As usual, there is more than one possible answer to this question. In this case, our intuition was probably based on thinking about what an even number is, and realizing that the definition itself is essentiallly symbolic. (Yo ...
... the universe of even numbers. How did we know to start the proof that way? As usual, there is more than one possible answer to this question. In this case, our intuition was probably based on thinking about what an even number is, and realizing that the definition itself is essentiallly symbolic. (Yo ...
On Natural Deduction in Classical First-Order Logic: Curry
... The informal idea expressed by the associated reductions is to assume ∀α P and try to produce some complete proof of C out of u by reducing inside u. Whenever u needs the truth of an instance P[n/α] of the assumption ∀α P, it checks it, and if it is true, it replaces it by its canonical proof which ...
... The informal idea expressed by the associated reductions is to assume ∀α P and try to produce some complete proof of C out of u by reducing inside u. Whenever u needs the truth of an instance P[n/α] of the assumption ∀α P, it checks it, and if it is true, it replaces it by its canonical proof which ...
Logic and Resolution - Institute for Computing and Information
... others, pointed out that theorem proving could also be used for solving non-mathematical problems. This idea was elaborated by many authors. Well known is the early work on so-called question-answering systems by J.R. Slagle and the later work in this field by C.C. Green and B. Raphael. After some i ...
... others, pointed out that theorem proving could also be used for solving non-mathematical problems. This idea was elaborated by many authors. Well known is the early work on so-called question-answering systems by J.R. Slagle and the later work in this field by C.C. Green and B. Raphael. After some i ...
Minimal Epistemology: Beyond Terminal Philosophy to Truth
... their competitors that do not -- better in the sense of having survived so far, where they have failed, in the trial by prerequisites for truth. A hypothesis that has survived so well in an expanding body of evidence and argument is clearly reliable, so far as our experience goes, or we would have f ...
... their competitors that do not -- better in the sense of having survived so far, where they have failed, in the trial by prerequisites for truth. A hypothesis that has survived so well in an expanding body of evidence and argument is clearly reliable, so far as our experience goes, or we would have f ...
na.
... is a pseudo-model for m, where s and e are defined below and e· is the retiexlve and transitive closure of B. The relation s is defined by DIs Da itt there is a g-consistent and complete theory T1 such that D1 T1 n Cl (a,) and D2 = T{ n Cl ((I) where + is defined as in [LS]. The relation" is defined ...
... is a pseudo-model for m, where s and e are defined below and e· is the retiexlve and transitive closure of B. The relation s is defined by DIs Da itt there is a g-consistent and complete theory T1 such that D1 T1 n Cl (a,) and D2 = T{ n Cl ((I) where + is defined as in [LS]. The relation" is defined ...
Logical Consequence by Patricia Blanchette Basic Question (BQ
... Model theory is about truth and interpretation. Within model theory, we represent the concept of logical consequence by talking about model theoretic consequence. Very roughly speaking, we say that “B is a model theoretic consequence of A in formal system S” means that in S whenever B is true, A is ...
... Model theory is about truth and interpretation. Within model theory, we represent the concept of logical consequence by talking about model theoretic consequence. Very roughly speaking, we say that “B is a model theoretic consequence of A in formal system S” means that in S whenever B is true, A is ...
Logic, deontic. The study of principles of reasoning pertaining to
... independent and jointly consistent: Dr. Jones ought to administer anaesthesia if she operates; she ought not to if she doesn't; she has an obligation to operate, which she fails to meet. But attempts to represent these sentences within the standard system yield inconsistencies or redundancies. Accor ...
... independent and jointly consistent: Dr. Jones ought to administer anaesthesia if she operates; she ought not to if she doesn't; she has an obligation to operate, which she fails to meet. But attempts to represent these sentences within the standard system yield inconsistencies or redundancies. Accor ...
What is...Linear Logic? Introduction Jonathan Skowera
... relation between ⊗ and ⊕ is U ⊗ (V ⊕ W ) ∼ = U ⊗ V ⊕ U ⊗ W . The choice of symbols reflects this relation, but proving it requires a complete list of rules of inference, and we aren’t quite there yet. Implication Using the negative, implication can be defined in analogy with the classical case where ...
... relation between ⊗ and ⊕ is U ⊗ (V ⊕ W ) ∼ = U ⊗ V ⊕ U ⊗ W . The choice of symbols reflects this relation, but proving it requires a complete list of rules of inference, and we aren’t quite there yet. Implication Using the negative, implication can be defined in analogy with the classical case where ...
Restricted notions of provability by induction
... Theorem 3.5 (Wilkie–Paris). The following are equivalent for a bounded formula ψ(x). (i) I∆0 + exp ` ∀x ψ(x). (ii) There is an inductive formula ϕ(x) such that PA− ` ∀x (ϕ(x) → ψ(x)). Proof. See Corollary 8.7 in Wilkie–Paris [25] or Theorem V.5.26 in Hájek– Pudlák [13]. As mentioned before, most ...
... Theorem 3.5 (Wilkie–Paris). The following are equivalent for a bounded formula ψ(x). (i) I∆0 + exp ` ∀x ψ(x). (ii) There is an inductive formula ϕ(x) such that PA− ` ∀x (ϕ(x) → ψ(x)). Proof. See Corollary 8.7 in Wilkie–Paris [25] or Theorem V.5.26 in Hájek– Pudlák [13]. As mentioned before, most ...
Standardization of Formulæ
... Skolem function of the form f (x1 , ..xn ), where: f is a fresh function symbol x1 , .., xn are the variables which are universally quantified before the quantifier to be removed ∀x∃y (p(x) → ¬q(y )) ∃x∀z(q(x, z) ∨ r (a, x)) ∃x∀y ∃z(p(x) ∧ q(y ) → r (f (z), x)) ...
... Skolem function of the form f (x1 , ..xn ), where: f is a fresh function symbol x1 , .., xn are the variables which are universally quantified before the quantifier to be removed ∀x∃y (p(x) → ¬q(y )) ∃x∀z(q(x, z) ∨ r (a, x)) ∃x∀y ∃z(p(x) ∧ q(y ) → r (f (z), x)) ...
Ambient Logic II.fm
... (νn) must be formally pushed outwards to encompass the new scope of n and maintain the scoping invariant; this procedure is called name extrusion. Processes are considered equivalent up to extrusion; that is, extrusion is not regarded as a computational step. Conversely, when a name is forgotten in ...
... (νn) must be formally pushed outwards to encompass the new scope of n and maintain the scoping invariant; this procedure is called name extrusion. Processes are considered equivalent up to extrusion; that is, extrusion is not regarded as a computational step. Conversely, when a name is forgotten in ...
Review - Gerry O nolan
... could the proposition 'All ravens are black.' It follows from the necessity of such assessments that there could never be some contingent presupposition, such as the uniformity of nature (23), the long run success rate of inductive inferences (22, 45), or Mill's law of universal causation (45) upon ...
... could the proposition 'All ravens are black.' It follows from the necessity of such assessments that there could never be some contingent presupposition, such as the uniformity of nature (23), the long run success rate of inductive inferences (22, 45), or Mill's law of universal causation (45) upon ...
Basic Logic - Progetto e
... propositions and take advantage of relationships among inner elements. For example, we may assume that “All men are mortal” and that “Socratis is a man” and hence conclude “Socratis is mortal” ...
... propositions and take advantage of relationships among inner elements. For example, we may assume that “All men are mortal” and that “Socratis is a man” and hence conclude “Socratis is mortal” ...
On Herbrand`s Theorem for Intuitionistic Logic
... to the deducibility of a Herbrand extension, and then the necessary Herbrand terms can be extracted from the cut-free proof. In fact, a similar idea is used in free-variable tableau methods [8], where quantifiers are dealt with separately from dealing with the propositional proof skeleton. Since fre ...
... to the deducibility of a Herbrand extension, and then the necessary Herbrand terms can be extracted from the cut-free proof. In fact, a similar idea is used in free-variable tableau methods [8], where quantifiers are dealt with separately from dealing with the propositional proof skeleton. Since fre ...
CHAPTER 1 The main subject of Mathematical Logic is
... • Inductive predicates are defined by their clauses and a least-fixedpoint (or induction) axiom. Their witnesses are generation trees. • Dually coductive predicates are defined by a single clause and a greatest-fixed-point (or coinduction) axiom. Their witnesses are destruction trees. • It could be ...
... • Inductive predicates are defined by their clauses and a least-fixedpoint (or induction) axiom. Their witnesses are generation trees. • Dually coductive predicates are defined by a single clause and a greatest-fixed-point (or coinduction) axiom. Their witnesses are destruction trees. • It could be ...
Propositional Logic: Normal Forms
... UNSAT: if false is marked, there exists a conjunct P1 ∧ . . . ∧ Pki → false of φ such that all Pi s are marked. If φ is satisfiable, by Corollary 1, this means (true → false) = false whenever φ is true. This is impossible, so φ is unsatisfiable. Reductio ad absurdum. SAT: if false is NOT marked, let ...
... UNSAT: if false is marked, there exists a conjunct P1 ∧ . . . ∧ Pki → false of φ such that all Pi s are marked. If φ is satisfiable, by Corollary 1, this means (true → false) = false whenever φ is true. This is impossible, so φ is unsatisfiable. Reductio ad absurdum. SAT: if false is NOT marked, let ...
(draft)
... that method cannot be used in the Constructive logic system. So the following classical logic proof rules are not found in Constructive logic: ¬¬φ φ ...
... that method cannot be used in the Constructive logic system. So the following classical logic proof rules are not found in Constructive logic: ¬¬φ φ ...
A logical basis for quantum evolution and entanglement
... category and each vertex a morphism with domain the tensor of the incoming edges and codomain the tensor of the outgoing edges. Evolution is described as happening between a special family of spacelike slices, which were referred to as locative slices. Locative slices differ from the maximal slices ...
... category and each vertex a morphism with domain the tensor of the incoming edges and codomain the tensor of the outgoing edges. Evolution is described as happening between a special family of spacelike slices, which were referred to as locative slices. Locative slices differ from the maximal slices ...
Multi-Agent Only
... I For a single agent possibilities are just worlds. I For many agents possibilities include other agents beliefs. ...
... I For a single agent possibilities are just worlds. I For many agents possibilities include other agents beliefs. ...
“Sometimes” and “Not Never” Revisited
... infinitely often, then eventually the process must actually be executed). This constraint can be expressed in linear time logic by the formula (- q lenabled) V -executed. However, by Theorem 2, it is not expressible in branching time logic. (2) In proving a program correct, it is often helpful to re ...
... infinitely often, then eventually the process must actually be executed). This constraint can be expressed in linear time logic by the formula (- q lenabled) V -executed. However, by Theorem 2, it is not expressible in branching time logic. (2) In proving a program correct, it is often helpful to re ...
Robot Morality and Review of classical logic.
... A set of premises logically entails a conclusion (written as |= ) if and only if every interpretation that satisfies the premises also satisfies the conclusion and every interpretation that satisfies the conclusion also satisfies the premises. If and only if is in definition, this is equality ...
... A set of premises logically entails a conclusion (written as |= ) if and only if every interpretation that satisfies the premises also satisfies the conclusion and every interpretation that satisfies the conclusion also satisfies the premises. If and only if is in definition, this is equality ...
Computing Default Extensions by Reductions on OR
... needed for the Modal Reduction Theorem. Our approach to formalizing default logic is complementary to the approach in (Lakemeyer and Levesque 2006), in which it is the logic of O R that is axiomatized. Although this is, from the point of view of theoremhood, a stronger system than the rewriting syst ...
... needed for the Modal Reduction Theorem. Our approach to formalizing default logic is complementary to the approach in (Lakemeyer and Levesque 2006), in which it is the logic of O R that is axiomatized. Although this is, from the point of view of theoremhood, a stronger system than the rewriting syst ...
PDF
... Let FO(Σ) be a first order language over signature Σ. Recall that the axioms for FO(Σ) are (universal) generalizations of wff’s belonging to one of the following six schemas: 1. A → (B → A) 2. (A → (B → C)) → ((A → B) → (A → C)) 3. ¬¬A → A 4. ∀x(A → B) → (∀xA → ∀xB), where x ∈ V 5. A → ∀xA, where x ...
... Let FO(Σ) be a first order language over signature Σ. Recall that the axioms for FO(Σ) are (universal) generalizations of wff’s belonging to one of the following six schemas: 1. A → (B → A) 2. (A → (B → C)) → ((A → B) → (A → C)) 3. ¬¬A → A 4. ∀x(A → B) → (∀xA → ∀xB), where x ∈ V 5. A → ∀xA, where x ...