Definition - Rogelio Davila
... (A1) (X (Y X)) (A2) ((X (Y Z)) ((X Y) (X Z))) (A3) ((X Y) ((X Y) X)) ...
... (A1) (X (Y X)) (A2) ((X (Y Z)) ((X Y) (X Z))) (A3) ((X Y) ((X Y) X)) ...
slides - National Taiwan University
... |= is about semantics, rather than syntax For Σ = ∅, we have ∅ |= τ , simply written |= τ . It says every truth assignment satisfies τ . In this case, τ is a tautology. ...
... |= is about semantics, rather than syntax For Σ = ∅, we have ∅ |= τ , simply written |= τ . It says every truth assignment satisfies τ . In this case, τ is a tautology. ...
MATH 312H–FOUNDATIONS
... B come into the play. The moral is that the two problems are logical equivalent, although to many of us it does not look alike. This is called the Watson test in phsychology. Mathematical theorems mostly have the form “A⇒B”. A is called the assumption (hypothesis, conditions) and B is called the cla ...
... B come into the play. The moral is that the two problems are logical equivalent, although to many of us it does not look alike. This is called the Watson test in phsychology. Mathematical theorems mostly have the form “A⇒B”. A is called the assumption (hypothesis, conditions) and B is called the cla ...
Document
... An argument in propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies that the conclusion is true. ...
... An argument in propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies that the conclusion is true. ...
A Partially Truth Functional Approach to
... truth assignments --- themselves, maps from S to {T,F} --- to [0,1] with unit sum. We define the assertibility of a statement to be the probability of its truth given that it has a truth value. In other words, relative to a given distribution Pr of probability, the assertibility of statement A is: Σ ...
... truth assignments --- themselves, maps from S to {T,F} --- to [0,1] with unit sum. We define the assertibility of a statement to be the probability of its truth given that it has a truth value. In other words, relative to a given distribution Pr of probability, the assertibility of statement A is: Σ ...
XR3a
... rational number. Assume that their sum is rational, i.e., q+r=s where s is a rational number. Then q = s-r. But by our previous proof the sum of two rational numbers must be rational, so we have an irrational number on the left equal to a rational number on the right. This is a contradiction. Theref ...
... rational number. Assume that their sum is rational, i.e., q+r=s where s is a rational number. Then q = s-r. But by our previous proof the sum of two rational numbers must be rational, so we have an irrational number on the left equal to a rational number on the right. This is a contradiction. Theref ...
Section I(e)
... This is an important result which many students find difficult to accept especially when proving propositions. They sometimes prove the converse, Q P [Q implies P], and think they have proven P Q [P implies Q]. You need to be very careful. Consider the example where P is the proposition ‘ x 5 ...
... This is an important result which many students find difficult to accept especially when proving propositions. They sometimes prove the converse, Q P [Q implies P], and think they have proven P Q [P implies Q]. You need to be very careful. Consider the example where P is the proposition ‘ x 5 ...
Notes on Propositional Logic
... In propositional logic, we would like to apply operators not only to atomic propositions, but also to the result of applying other operators. This means that our language of well-formed formulas in propositional logic should be inductively defined as follows. Definition 1. For a given set A of propo ...
... In propositional logic, we would like to apply operators not only to atomic propositions, but also to the result of applying other operators. This means that our language of well-formed formulas in propositional logic should be inductively defined as follows. Definition 1. For a given set A of propo ...
First-Order Logic
... Suppose that F is not valid, i.e., there is an interpretation I such that the above assumption holds. By following the semantic argument steps, one can show that each step preserves satisfiability. (For or-nodes, one new branch will be satisfiable.) This may require updating the current interpretati ...
... Suppose that F is not valid, i.e., there is an interpretation I such that the above assumption holds. By following the semantic argument steps, one can show that each step preserves satisfiability. (For or-nodes, one new branch will be satisfiable.) This may require updating the current interpretati ...
The Foundations: Logic and Proofs - UTH e
... Example: What are the types of A and B? Solution: Let p and q be the statements that A is a knight and B is a knight, respectively. So, then p represents the proposition that A is a knave and q that B is a knave. If A is a knight, then p is true. Since knights tell the truth, q must also be ...
... Example: What are the types of A and B? Solution: Let p and q be the statements that A is a knight and B is a knight, respectively. So, then p represents the proposition that A is a knave and q that B is a knave. If A is a knight, then p is true. Since knights tell the truth, q must also be ...
Logic and Inferences
... sentence ϕ, may reply “yes”, “no”, or run forever. If I replies “yes” on input (Γ, ϕ), we say that Γ derives ϕ in I , Or, I derives ϕ from Γ, or, ϕ is deduced (derived) from Γ in I . and write ...
... sentence ϕ, may reply “yes”, “no”, or run forever. If I replies “yes” on input (Γ, ϕ), we say that Γ derives ϕ in I , Or, I derives ϕ from Γ, or, ϕ is deduced (derived) from Γ in I . and write ...
Exercise
... Proof by cases If n is an integer, then n(n+1)/2 is an integer • Case 1: n is even. or n = 2a, for some integer a So n(n+1)/2 = 2a*(n+1)/2 = a*(n+1), which is an integer. • Case 2: n is odd. n+1 is even, or n+1 = 2a, for an integer a So n(n+1)/2 = n*2a/2 = n*a, which is an integer. ...
... Proof by cases If n is an integer, then n(n+1)/2 is an integer • Case 1: n is even. or n = 2a, for some integer a So n(n+1)/2 = 2a*(n+1)/2 = a*(n+1), which is an integer. • Case 2: n is odd. n+1 is even, or n+1 = 2a, for an integer a So n(n+1)/2 = n*2a/2 = n*a, which is an integer. ...
Using Existential Graphs for Automated Theorem Proving
... • The obvious solution to the drawback of the short truth table method is to incorporate tools to systematically keep track of multiple options. • One method that does so is the truth tree method: – The truth tree method tries to systematically derive a contradiction from the assumption that certain ...
... • The obvious solution to the drawback of the short truth table method is to incorporate tools to systematically keep track of multiple options. • One method that does so is the truth tree method: – The truth tree method tries to systematically derive a contradiction from the assumption that certain ...
Symbolic Logic II
... KVI (φ) 6= 0 for each trivalent interpretation, and φ is a semantic consequence of Γ iff for every trivalent interpretation, I , if KVI (γ) 6= 0 for each γ ∈ Γ, then KVI (φ) 6= 0. Thus, to show that P → Q, Q → R 2LP P → R, we need to show that there is some interpretation in which KVI (P → Q) 6= 0, ...
... KVI (φ) 6= 0 for each trivalent interpretation, and φ is a semantic consequence of Γ iff for every trivalent interpretation, I , if KVI (γ) 6= 0 for each γ ∈ Γ, then KVI (φ) 6= 0. Thus, to show that P → Q, Q → R 2LP P → R, we need to show that there is some interpretation in which KVI (P → Q) 6= 0, ...
Argumentative Approaches to Reasoning with Maximal Consistency
... Dung-style semantics for abstract argumentation, and a computational approach that is based on extensions of Gentzentype proofs systems. This brings about a new perspective on reasoning with MCS which shows a strong link between the latter and argumentation systems, and which can be extended to rela ...
... Dung-style semantics for abstract argumentation, and a computational approach that is based on extensions of Gentzentype proofs systems. This brings about a new perspective on reasoning with MCS which shows a strong link between the latter and argumentation systems, and which can be extended to rela ...
We showed on Tuesday that Every relation in the arithmetical
... Every relation in the arithmetical hierarchy is definable in arithmetic. If R(~x ) is a Σn or Πn relation, then there is an expression α in first-order arithmetic (using 0, S, +, ·, exp, =, <, ∧, ∨, ¬, →, ↔) such that for all ...
... Every relation in the arithmetical hierarchy is definable in arithmetic. If R(~x ) is a Σn or Πn relation, then there is an expression α in first-order arithmetic (using 0, S, +, ·, exp, =, <, ∧, ∨, ¬, →, ↔) such that for all ...
FOR HIGHER-ORDER RELEVANT LOGIC
... a form of the converse Lindenbaum lemma, for we have shown-dual to the usual Lindenbaum lemma – that a certain complete though inconsistent theory has a normal subtheory: a lemma which, carefully characterized, may be shown to hold generally.) We have taken R2 to be a second-order version of RQ. Sim ...
... a form of the converse Lindenbaum lemma, for we have shown-dual to the usual Lindenbaum lemma – that a certain complete though inconsistent theory has a normal subtheory: a lemma which, carefully characterized, may be shown to hold generally.) We have taken R2 to be a second-order version of RQ. Sim ...
Identity and Philosophical Problems of Symbolic Logic
... A predicate logic that forbids sentences that ascribe properties to properties themselves and restricts quantification to individual variables is a first-order predicate logic. ...
... A predicate logic that forbids sentences that ascribe properties to properties themselves and restricts quantification to individual variables is a first-order predicate logic. ...
PDF
... Intuitionists do not accept the law of double negation: P ↔ ¬¬P . They do believe that P → ¬¬P , that is, if P is true then it is not false; but they do not believe ¬¬P → P , that is, even if P is not false, then that does not automatically make it true. Similarly, intuitionists do not accept the la ...
... Intuitionists do not accept the law of double negation: P ↔ ¬¬P . They do believe that P → ¬¬P , that is, if P is true then it is not false; but they do not believe ¬¬P → P , that is, even if P is not false, then that does not automatically make it true. Similarly, intuitionists do not accept the la ...
Biconditional Statements
... q: the angles are congruent. • Conditional: p → q If the sides of a triangle are congruent, then the angles are congruent. • Converse: q → p If the angles are congruent, then the sides of the triangles are congruent. • Biconditional: p ↔ q The sides of a triangle are congruent if and only if the ang ...
... q: the angles are congruent. • Conditional: p → q If the sides of a triangle are congruent, then the angles are congruent. • Converse: q → p If the angles are congruent, then the sides of the triangles are congruent. • Biconditional: p ↔ q The sides of a triangle are congruent if and only if the ang ...
Lesson 12
... is often computationally expensive. Notice that even if inference is not complete it is desirable that it is sound. Propositional Logic and Predicate Logic each with Modus Ponens as their inference produce are sound but not complete. We shall see that we need further (sound) rules of inference to ac ...
... is often computationally expensive. Notice that even if inference is not complete it is desirable that it is sound. Propositional Logic and Predicate Logic each with Modus Ponens as their inference produce are sound but not complete. We shall see that we need further (sound) rules of inference to ac ...
PPT
... In fuzzy set theory any element can to be member of set with any uncertainty or confidence Is(a,A) = 0 or 1 or 0.5 or 0.126 or … from interval (0,1) This uncertainty is determined by membership function 0≤μA(a)≤1 ...
... In fuzzy set theory any element can to be member of set with any uncertainty or confidence Is(a,A) = 0 or 1 or 0.5 or 0.126 or … from interval (0,1) This uncertainty is determined by membership function 0≤μA(a)≤1 ...
Argumentative Approaches to Reasoning with Maximal Consistency Ofer Arieli Christian Straßer
... Dung’s semantics for abstract argumentation frameworks. Given a framework AF (Definition 1), a key issue in its understanding is the question what combinations of arguments (called extensions) can collectively be accepted from AF. According to Dung (1995), this is determined as follows: Definition 6 L ...
... Dung’s semantics for abstract argumentation frameworks. Given a framework AF (Definition 1), a key issue in its understanding is the question what combinations of arguments (called extensions) can collectively be accepted from AF. According to Dung (1995), this is determined as follows: Definition 6 L ...
(formal) logic? - Departamento de Informática
... intuitionistic logic, but the task is very difficult, so mathematicians use methods of classical logic (as proofs by contradiction). However the philosophy behind intuitionistic logic is appealing for a computer scientist. For an intuitionist, a mathematical object (such as the solution of an equation ...
... intuitionistic logic, but the task is very difficult, so mathematicians use methods of classical logic (as proofs by contradiction). However the philosophy behind intuitionistic logic is appealing for a computer scientist. For an intuitionist, a mathematical object (such as the solution of an equation ...
January 12
... Such limitations of Aristotelian logic give Frege criteria for the adequacy of a proper logical notation, or Begriffsschrift. Such a notation must be able to do 2 things: A. express all (and only) propositions, i.e., all and only things that are either true or false; and B. state all logical relati ...
... Such limitations of Aristotelian logic give Frege criteria for the adequacy of a proper logical notation, or Begriffsschrift. Such a notation must be able to do 2 things: A. express all (and only) propositions, i.e., all and only things that are either true or false; and B. state all logical relati ...