(P Q). - Snistnote
... derivation is called a deduction or formal proof. In a formal proof, every rule of inference that is used at any stage in the derivation is acknowledged. ...
... derivation is called a deduction or formal proof. In a formal proof, every rule of inference that is used at any stage in the derivation is acknowledged. ...
Friendly Logics, Fall 2015, Homework 1
... (a) Prove that the statement of the Soundness Theorem (3.1) is equivalent to the following statement: for any ⌃ if ⌃ is satisfiable then it is consistent. (b) Prove that the statement of the Compleness Theorem (3.3) is equivalent to the following statement: for any ⌃ if ⌃ is consistent then it is sa ...
... (a) Prove that the statement of the Soundness Theorem (3.1) is equivalent to the following statement: for any ⌃ if ⌃ is satisfiable then it is consistent. (b) Prove that the statement of the Compleness Theorem (3.3) is equivalent to the following statement: for any ⌃ if ⌃ is consistent then it is sa ...
First-Order Logic
... A substitution σ is a mapping from variables to terms, written as σ : {x1 7→ t1 , . . . , xn 7→ tn } such that n ≥ 0 and xi 6= xj for all i, j = 1..n with i 6= j. The set dom(σ) = {x1 , . . . , xn } is called the domain of σ. The set cod(σ) = {t1 , . . . , tn } is called the codomain of σ. The set o ...
... A substitution σ is a mapping from variables to terms, written as σ : {x1 7→ t1 , . . . , xn 7→ tn } such that n ≥ 0 and xi 6= xj for all i, j = 1..n with i 6= j. The set dom(σ) = {x1 , . . . , xn } is called the domain of σ. The set cod(σ) = {t1 , . . . , tn } is called the codomain of σ. The set o ...
ON A MINIMAL SYSTEM OF ARISTOTLE`S SYLLOGISTIC Introduction
... The inductive definition of the notion of truth in the model MS for an arbitrary formula α is the same as for ML . Let S be the set of all models MS (models based on I S with different sets B and functions f and g). We shall understand that a formula is valid in S iff it is true in all models from S ...
... The inductive definition of the notion of truth in the model MS for an arbitrary formula α is the same as for ML . Let S be the set of all models MS (models based on I S with different sets B and functions f and g). We shall understand that a formula is valid in S iff it is true in all models from S ...
A HIGHER-ORDER FINE-GRAINED LOGIC FOR INTENSIONAL
... Abstract. This paper describes a higher-order logic with fine-grained intensionality (FIL). Unlike traditional Montogovian type theory, intensionality is treated as basic, rather than derived through possible worlds. This allows for fine-grained intensionality without impossible worlds. Possible wor ...
... Abstract. This paper describes a higher-order logic with fine-grained intensionality (FIL). Unlike traditional Montogovian type theory, intensionality is treated as basic, rather than derived through possible worlds. This allows for fine-grained intensionality without impossible worlds. Possible wor ...
Discrete Mathematics - Lecture 4: Propositional Logic and Predicate
... Using Propositional Logic for designing proofs A mathematical statement comprises of a premise (or assumptions). And when the assumptions are satisfied the statement deduces something. If A is the set of assumptions and B is the deduction then a mathematical statement is of the form ...
... Using Propositional Logic for designing proofs A mathematical statement comprises of a premise (or assumptions). And when the assumptions are satisfied the statement deduces something. If A is the set of assumptions and B is the deduction then a mathematical statement is of the form ...
Syllogistic Logic Sample Quiz Page 1
... I’ll do poorly in logic. I’m sure of this because of the following facts. First, I don’t do LogiCola. Second, I don’t read the book. Third, I spend my time playing Tetris. Assuming that I spend my time playing Tetris and I don’t do LogiCola, then, of course, if I don’t read the book then I’ll do poo ...
... I’ll do poorly in logic. I’m sure of this because of the following facts. First, I don’t do LogiCola. Second, I don’t read the book. Third, I spend my time playing Tetris. Assuming that I spend my time playing Tetris and I don’t do LogiCola, then, of course, if I don’t read the book then I’ll do poo ...
Partial Correctness Specification
... it is not necessary for the execution of C to terminate when started in a state satisfying P It is only required that if the execution terminates, then Q holds {X = 1} WHILE T DO X := X {Y = 2} – this specification is true! ...
... it is not necessary for the execution of C to terminate when started in a state satisfying P It is only required that if the execution terminates, then Q holds {X = 1} WHILE T DO X := X {Y = 2} – this specification is true! ...
Sample Exam 1 - Moodle
... CSC 4-151 Discrete Mathematics for Computer Science Exam 1 May 7, 2017 ____________________ name For credit on these problems, you must show your work. On this exam, take the natural numbers to be N = {0,1,2,3, …}. 1. (6 pts.) State and prove one of DeMorgan’s Laws for propositional logic, using a t ...
... CSC 4-151 Discrete Mathematics for Computer Science Exam 1 May 7, 2017 ____________________ name For credit on these problems, you must show your work. On this exam, take the natural numbers to be N = {0,1,2,3, …}. 1. (6 pts.) State and prove one of DeMorgan’s Laws for propositional logic, using a t ...
CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness
... (P F, ⇒, ∪, ∩, ¬) a similar algebra of propositional formulas. We extend v to a homomorphism v ∗ : PF −→ B in a usual way, i.e. we put v ∗ (A) = v(A) for A ∈ P , and for any A, B ∈ P F, v ∗ (A ⇒ B) = v ∗ (A) ⇒ v ∗ (B), v ∗ (A ∪ B) = v ∗ (A) ∪ v ∗ (B), v ∗ (A ∩ B) = v ∗ (A) ∩ v ∗ (B), v ∗ (¬A) = ¬v ∗ ...
... (P F, ⇒, ∪, ∩, ¬) a similar algebra of propositional formulas. We extend v to a homomorphism v ∗ : PF −→ B in a usual way, i.e. we put v ∗ (A) = v(A) for A ∈ P , and for any A, B ∈ P F, v ∗ (A ⇒ B) = v ∗ (A) ⇒ v ∗ (B), v ∗ (A ∪ B) = v ∗ (A) ∪ v ∗ (B), v ∗ (A ∩ B) = v ∗ (A) ∩ v ∗ (B), v ∗ (¬A) = ¬v ∗ ...
Unit-1-B - WordPress.com
... It is mainly used for deriving a conclusion based on what one already knows. Logic is the study of correct reasoning. It provides rules to determine whether a given argument is valid or not. ...
... It is mainly used for deriving a conclusion based on what one already knows. Logic is the study of correct reasoning. It provides rules to determine whether a given argument is valid or not. ...
study guide.
... ∧ the negation of its encoding to the CNF, and apply DeMorgan’s law. • A resolution proof system is used to find a contradiction in a formula (and, similarly, to prove that a formula is a tautology by finding a contradiction in its negation). Resolution starts with a formula in a CNF form, and appli ...
... ∧ the negation of its encoding to the CNF, and apply DeMorgan’s law. • A resolution proof system is used to find a contradiction in a formula (and, similarly, to prove that a formula is a tautology by finding a contradiction in its negation). Resolution starts with a formula in a CNF form, and appli ...
An Introduction to Modal Logic VII The finite model property
... A normal modal logic Λ has the finite model property if and only if it has the finite frame property. Clearly the f.f.p. implies the f.m.p. On the other hand, suppose now that Λ has the f.m.p. and let ϕ∈ / Λ; by f.m.p., there is a finite model M where ϕ is not valid; consider M∼ , it is differentiat ...
... A normal modal logic Λ has the finite model property if and only if it has the finite frame property. Clearly the f.f.p. implies the f.m.p. On the other hand, suppose now that Λ has the f.m.p. and let ϕ∈ / Λ; by f.m.p., there is a finite model M where ϕ is not valid; consider M∼ , it is differentiat ...
3.1.3 Subformulas
... intended to establish a close connection between formulas and their meaning, i. e. between syntax and semantics. It will turn out that the meaning of a formula depends solely on the structure of the formula. ...
... intended to establish a close connection between formulas and their meaning, i. e. between syntax and semantics. It will turn out that the meaning of a formula depends solely on the structure of the formula. ...
Topological Completeness of First-Order Modal Logic
... associated to the possible-world structure [23,19,4]. In this article we provide a completeness proof for first-order S4 modal logic with respect to topologicalsheaf semantics of Awodey-Kishida [3], which combines the possible-world formulation of sheaf semantics with the topos-theoretic interpretat ...
... associated to the possible-world structure [23,19,4]. In this article we provide a completeness proof for first-order S4 modal logic with respect to topologicalsheaf semantics of Awodey-Kishida [3], which combines the possible-world formulation of sheaf semantics with the topos-theoretic interpretat ...