A BRIEF INTRODUCTION TO MODAL LOGIC Introduction Consider
... Schema K For any wffs α and β, we will assume that (α =⇒ β) =⇒ (α =⇒ β). Our proof system will also have two rules of inference. Definition 2.7. A ‘rule of inference’ is an ordered pair (Γ, α), where Γ is a set of wffs and α is a single wff. If the propositions of Γ are theorems of the system, so ...
... Schema K For any wffs α and β, we will assume that (α =⇒ β) =⇒ (α =⇒ β). Our proof system will also have two rules of inference. Definition 2.7. A ‘rule of inference’ is an ordered pair (Γ, α), where Γ is a set of wffs and α is a single wff. If the propositions of Γ are theorems of the system, so ...
Lesson 1
... This apple is an agaric. ---------------------------------------------------------------------Hence This apple has a strong toxic effect. The argument is valid. But the conclusion is evidently not true (false). Hence, at least one premise is false (obviously the second). Circumstances according to ...
... This apple is an agaric. ---------------------------------------------------------------------Hence This apple has a strong toxic effect. The argument is valid. But the conclusion is evidently not true (false). Hence, at least one premise is false (obviously the second). Circumstances according to ...
Analysis of the paraconsistency in some logics
... 1. We will say that a theory Γ is contradictory, with respect to ¬, if there exists a formula A such that Γ ` A y Γ ` ¬A; 2. We say that a theory Γ is trivial if ∀A : Γ ` A; 3. We say that a theory is explosive if, when adding to it any couple of contradictory formulas, the theory becomes trivial; 4 ...
... 1. We will say that a theory Γ is contradictory, with respect to ¬, if there exists a formula A such that Γ ` A y Γ ` ¬A; 2. We say that a theory Γ is trivial if ∀A : Γ ` A; 3. We say that a theory is explosive if, when adding to it any couple of contradictory formulas, the theory becomes trivial; 4 ...
The (re-)emergence of representationalism in semantics Ruth Kempson
... Technically, the lexicon contains solely predicate letters, individual constants, individual variables, and brackets, as the operators are introduced by the relevant syntactic rules, hence defined syncategorematically. However we approximate here, to bring out the parallelism with natural language. ...
... Technically, the lexicon contains solely predicate letters, individual constants, individual variables, and brackets, as the operators are introduced by the relevant syntactic rules, hence defined syncategorematically. However we approximate here, to bring out the parallelism with natural language. ...
MODULE I
... 2) ((PQ)(P┐Q)) ((P┐Q)(┐P┐Q)) 3) ( P┐(QR)( PQ)┐R)P) Disjunctive normal form of a given formula is not unique. They are equivalent. Conjunctive normal form (DNF) A formula which is equivalent to a given formula and which consists of product of elementary sum is known as CNF. 1 ...
... 2) ((PQ)(P┐Q)) ((P┐Q)(┐P┐Q)) 3) ( P┐(QR)( PQ)┐R)P) Disjunctive normal form of a given formula is not unique. They are equivalent. Conjunctive normal form (DNF) A formula which is equivalent to a given formula and which consists of product of elementary sum is known as CNF. 1 ...
(pdf)
... given a set of atomic or prime formulae, which intuitively can be thought of as possessing truth values (true or false). Here the prime formulae will be propositional variables, as well as a constant > for truth. It is also common to include a constant for falsity, and possibly non-logical constants ...
... given a set of atomic or prime formulae, which intuitively can be thought of as possessing truth values (true or false). Here the prime formulae will be propositional variables, as well as a constant > for truth. It is also common to include a constant for falsity, and possibly non-logical constants ...
Between Truth and Falsity
... it is impossible for A to be false or indeterminate. Hence it is valid. (But in fact there are no valid formulas in K3) Test for unexceptionability Assume B. If this leads to a contradiction, then the formula must be always either true or indeterminate. Test for contradictoriness Assume formula is ...
... it is impossible for A to be false or indeterminate. Hence it is valid. (But in fact there are no valid formulas in K3) Test for unexceptionability Assume B. If this leads to a contradiction, then the formula must be always either true or indeterminate. Test for contradictoriness Assume formula is ...
CPS130, Lecture 1: Introduction to Algorithms
... true. By construction of S, we have p(k) is true for all k < m0, including m0-1. Therefore by the second part of the hypothesis for either (I1) or (I2), p(m0) must be true. This is a contradiction ( p(m0 ) p(m0 ) ) and there is no such set S. if (I1) then (W) and if (I2) then (W): Clearly a subse ...
... true. By construction of S, we have p(k) is true for all k < m0, including m0-1. Therefore by the second part of the hypothesis for either (I1) or (I2), p(m0) must be true. This is a contradiction ( p(m0 ) p(m0 ) ) and there is no such set S. if (I1) then (W) and if (I2) then (W): Clearly a subse ...
Document
... An argument has the form ∀x (P(x ) → Q(x )), where the domain of discourse is D To show that this implication is not true in the domain D, it must be shown that there exists some x in D such that (P(x ) → Q(x )) is not true This means that there exists some x in D such that P(x) is true but Q(x) ...
... An argument has the form ∀x (P(x ) → Q(x )), where the domain of discourse is D To show that this implication is not true in the domain D, it must be shown that there exists some x in D such that (P(x ) → Q(x )) is not true This means that there exists some x in D such that P(x) is true but Q(x) ...
Unification in Propositional Logic
... Digression. A very remarkable (absolutely non trivial) fact is Pitts’ theorem, that can be reformulated by saying that [σ ∗(A∗)]∼∞ is always of the kind B ∗, for some B (in modal logic, Pitts’ theorem holds for GL, Grz, but not for K4, S4). Here [σ ∗(A∗)]∼∞ is the closure of σ ∗(A∗) under bisimulat ...
... Digression. A very remarkable (absolutely non trivial) fact is Pitts’ theorem, that can be reformulated by saying that [σ ∗(A∗)]∼∞ is always of the kind B ∗, for some B (in modal logic, Pitts’ theorem holds for GL, Grz, but not for K4, S4). Here [σ ∗(A∗)]∼∞ is the closure of σ ∗(A∗) under bisimulat ...
Gödel`s Theorems
... characteristic function is recursive. A set S of formulas is called recursive (elementary, recursively enumerable), if pSq := { pAq | A ∈ S } is recursive (elementary, recursively enumerable). Clearly the sets StabL of stability axioms and EqL of L-equality axioms are elementary. Now let L be an ele ...
... characteristic function is recursive. A set S of formulas is called recursive (elementary, recursively enumerable), if pSq := { pAq | A ∈ S } is recursive (elementary, recursively enumerable). Clearly the sets StabL of stability axioms and EqL of L-equality axioms are elementary. Now let L be an ele ...
On Equivalent Transformations of Infinitary Formulas under the
... By Γ I we denote the set {GI | G ∈ Γ }; (Γ ⇒ F )I stands for Γ I ⇒ F I . Lemma 2. For any sequent S and any interpretation I, if S is a theorem of the basic system then so is S I . Proof. Consider the property of sequents: “S I is a theorem of the basic system.” To prove the lemma, it suffices to sh ...
... By Γ I we denote the set {GI | G ∈ Γ }; (Γ ⇒ F )I stands for Γ I ⇒ F I . Lemma 2. For any sequent S and any interpretation I, if S is a theorem of the basic system then so is S I . Proof. Consider the property of sequents: “S I is a theorem of the basic system.” To prove the lemma, it suffices to sh ...
CPSC 2105 Lecture 6 - Edward Bosworth, Ph.D.
... Algebraically, this function is denoted f(X) = X’ or f(X) = X . The notation X’ is done for typesetting convenience only; the notation The evaluation of the function is simple: ...
... Algebraically, this function is denoted f(X) = X’ or f(X) = X . The notation X’ is done for typesetting convenience only; the notation The evaluation of the function is simple: ...
Techniques for proving the completeness of a proof system
... Sometimes it is easier to show the truth of a formula than to derive the formula. The completeness result shows that nothing is missing in a proof system. The completeness result formalizes what a proof ...
... Sometimes it is easier to show the truth of a formula than to derive the formula. The completeness result shows that nothing is missing in a proof system. The completeness result formalizes what a proof ...
