First-order logic;
... each formula Ck is either an axiom in AX or follows from previous formulas using an inference rule in AX : I i.e., there is an inference rule A1 , . . . , An ` B such that Ai = Cji for some ji < N and B = CN . This is said to be a derivation or proof of CN . A derivation is a syntactic object: it’s ...
... each formula Ck is either an axiom in AX or follows from previous formulas using an inference rule in AX : I i.e., there is an inference rule A1 , . . . , An ` B such that Ai = Cji for some ji < N and B = CN . This is said to be a derivation or proof of CN . A derivation is a syntactic object: it’s ...
Lecture Notes 3
... equivalent, written P Q) iff they have the same truth value in all situations. If two sentences are logically equivalent, you can substitute one for the other. Identity Laws: P ^ T P; P v F P Domination Laws: P v T T; P ^ F F Idempotent Laws: P v P P; P ^ P P ...
... equivalent, written P Q) iff they have the same truth value in all situations. If two sentences are logically equivalent, you can substitute one for the other. Identity Laws: P ^ T P; P v F P Domination Laws: P v T T; P ^ F F Idempotent Laws: P v P P; P ^ P P ...
Chapter Nine - Queen of the South
... of other rules with a continual examination and reinterpretation as new horizons come into view. Unsolved problems remain an uncompromising challenge to the ever-questioning human mind seeking for rational formalistic solutions in its right-lobed brain for the novel intuitions in contemplation of it ...
... of other rules with a continual examination and reinterpretation as new horizons come into view. Unsolved problems remain an uncompromising challenge to the ever-questioning human mind seeking for rational formalistic solutions in its right-lobed brain for the novel intuitions in contemplation of it ...
Some Principles of Logic
... • Argument is a series of logically related propositions • All men are mortal • Socrates was a man ...
... • Argument is a series of logically related propositions • All men are mortal • Socrates was a man ...
1. Kripke`s semantics for modal logic
... Some philosophers have distinguished between essentialism, the belief in modality de re, and a mere advocacy of necessity, the belief in modality de dicto. Now, some people say: Let’s give you the concept of necessity. A much worse thing, something creating great additional problems, is whether we c ...
... Some philosophers have distinguished between essentialism, the belief in modality de re, and a mere advocacy of necessity, the belief in modality de dicto. Now, some people say: Let’s give you the concept of necessity. A much worse thing, something creating great additional problems, is whether we c ...
Chapter 1 - National Taiwan University
... only when both p and q are true and false otherwise. The proposition p ∧ q is also called the conjunction of p and q. Definition 3. Let p and q be propositions. The proposition “p or q” (write p ∨ q) is the proposition that is false only when both p and q are false and true otherwise. The propositio ...
... only when both p and q are true and false otherwise. The proposition p ∧ q is also called the conjunction of p and q. Definition 3. Let p and q be propositions. The proposition “p or q” (write p ∨ q) is the proposition that is false only when both p and q are false and true otherwise. The propositio ...
Chapter5
... The AND function can also be represented by a dot (.), so A AND B can also be written A.B - this is because the AND operator behaves in a similar way to the multiplication operator in arithmetic (see below). OR The output of the OR function is true if any of the inputs are true. That's to say that i ...
... The AND function can also be represented by a dot (.), so A AND B can also be written A.B - this is because the AND operator behaves in a similar way to the multiplication operator in arithmetic (see below). OR The output of the OR function is true if any of the inputs are true. That's to say that i ...
F - Teaching-WIKI
... • Given the truth values of all symbols in a sentence, it can be “evaluated” to determine its truth value (True or False) • A model for a KB is a “possible world” (assignment of truth values to propositional symbols) in which each sentence in the KB is True • A valid sentence or tautology is a sente ...
... • Given the truth values of all symbols in a sentence, it can be “evaluated” to determine its truth value (True or False) • A model for a KB is a “possible world” (assignment of truth values to propositional symbols) in which each sentence in the KB is True • A valid sentence or tautology is a sente ...
MUltseq: a Generic Prover for Sequents and Equations*
... logics. This means that it takes as input the rules of a many-valued sequent calculus as well as a many-sided sequent and searches – automatically or interactively – for a proof of the latter. For the sake of readability, the output of MUltseq is typeset as a LATEX document. Though the sequent rules ...
... logics. This means that it takes as input the rules of a many-valued sequent calculus as well as a many-sided sequent and searches – automatically or interactively – for a proof of the latter. For the sake of readability, the output of MUltseq is typeset as a LATEX document. Though the sequent rules ...
Document
... Formal proof • sequence of statements, ending in conclusion • statements preceding the conclusion are called premises • each statement is either an axiom, or is derived from previous premises using a rule of inference Informal proof • formal proofs are too tedious to read • humans don’t need that mu ...
... Formal proof • sequence of statements, ending in conclusion • statements preceding the conclusion are called premises • each statement is either an axiom, or is derived from previous premises using a rule of inference Informal proof • formal proofs are too tedious to read • humans don’t need that mu ...
CSI 2101 / Rules of Inference (§1.5)
... Formal proof • sequence of statements, ending in conclusion • statements preceding the conclusion are called premises • each statement is either an axiom, or is derived from previous premises using a rule of inference Informal proof • formal proofs are too tedious to read • humans don’t need that mu ...
... Formal proof • sequence of statements, ending in conclusion • statements preceding the conclusion are called premises • each statement is either an axiom, or is derived from previous premises using a rule of inference Informal proof • formal proofs are too tedious to read • humans don’t need that mu ...
Truth-Functional Propositional Logic
... to identify opposites: hot-cold, dry-wet, heavy-light, dark-light, full-empty, high-low, large-small, etc. In many cases, there is no natural (non-contradictory) contrary to a proposition because there is no unique plausible opposite or combination of opposites to the terms of the proposition. In th ...
... to identify opposites: hot-cold, dry-wet, heavy-light, dark-light, full-empty, high-low, large-small, etc. In many cases, there is no natural (non-contradictory) contrary to a proposition because there is no unique plausible opposite or combination of opposites to the terms of the proposition. In th ...
Chapter 1: The Foundations: Logic and Proofs
... •QP is the CONVERSE of P Q •¬ Q ¬ P is the CONTRAPOSITIVE of P Q •¬ P ¬ Q is the inverse of P Q •Example: Find the converse of the following statement: R: ‘Raining tomorrow is a sufficient condition for my not going to town.’ ...
... •QP is the CONVERSE of P Q •¬ Q ¬ P is the CONTRAPOSITIVE of P Q •¬ P ¬ Q is the inverse of P Q •Example: Find the converse of the following statement: R: ‘Raining tomorrow is a sufficient condition for my not going to town.’ ...
Propositional logic, I
... Inference rules: Typographical rules (they do not use meaning) to construct new sentences from a given group of sentences. Semantics: Interpretation rules that associate sentences in the language with statements in the domain of discourse. Forget meaning. Think of logic as a game. BEWARE: It is easy ...
... Inference rules: Typographical rules (they do not use meaning) to construct new sentences from a given group of sentences. Semantics: Interpretation rules that associate sentences in the language with statements in the domain of discourse. Forget meaning. Think of logic as a game. BEWARE: It is easy ...
Chapter 11: Other Logical Tools Syllogisms and Quantification
... before modern propositional logic. The philosopher Aristotle (384-322 B.C.) is credited with being the first person to attempt to distill patterns of valid and invalid inference from our everyday arguing and judging. Called classical deduction theory, Aristotelian logic focuses on what are called ca ...
... before modern propositional logic. The philosopher Aristotle (384-322 B.C.) is credited with being the first person to attempt to distill patterns of valid and invalid inference from our everyday arguing and judging. Called classical deduction theory, Aristotelian logic focuses on what are called ca ...
Notes
... 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 ...
Logic and Proof - Collaboratory for Advanced Computing and
... From K. H. Rosen, Discrete Mathematics and Its Applications (McGraw-Hill) Chapter 1 ...
... From K. H. Rosen, Discrete Mathematics and Its Applications (McGraw-Hill) Chapter 1 ...
From p
... Truth tables are also used to specify the functionality of hardware look-up tables (LUTs) in digital logic circuitry. For an n-input LUT, the truth table will have 2^n values (or rows in the above tabular format), completely specifying a boolean function for the LUT. By representing each boolean val ...
... Truth tables are also used to specify the functionality of hardware look-up tables (LUTs) in digital logic circuitry. For an n-input LUT, the truth table will have 2^n values (or rows in the above tabular format), completely specifying a boolean function for the LUT. By representing each boolean val ...
T - 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 ...
deductive system
... You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
... You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
Chapter 1, Part I: Propositional Logic
... 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 ...
Propositional logic, I (Lógica Proposicional, I)
... to combine symbols into grammatical sentences. Inference rules: Typographical rules (they do not use meaning) to construct new sentences from a given group of sentences. Semantics: Interpretation rules that associate sentences in the language with statements in the domain of discourse. Forget meanin ...
... to combine symbols into grammatical sentences. Inference rules: Typographical rules (they do not use meaning) to construct new sentences from a given group of sentences. Semantics: Interpretation rules that associate sentences in the language with statements in the domain of discourse. Forget meanin ...
Propositional and predicate logic - Computing Science
... [Q] How to formalize/validate our arguments? Argument = premises (propositions or statements) + conclusion To have confidence in the conclusion in your argument, the premises should be acceptable on their own merits or follow from other statements that are known to be true. [Q] Any logical forms for ...
... [Q] How to formalize/validate our arguments? Argument = premises (propositions or statements) + conclusion To have confidence in the conclusion in your argument, the premises should be acceptable on their own merits or follow from other statements that are known to be true. [Q] Any logical forms for ...
Chapter 2
... Objectives: Analyze statements in if-then form Write the converse, inverse and contrapositive of if-then statements Vocabulary: Implies symbol (→) Conditional statement – statement written in if-then form Hypothesis – phrase immediately following the word if in a conditional statement Conclusion – p ...
... Objectives: Analyze statements in if-then form Write the converse, inverse and contrapositive of if-then statements Vocabulary: Implies symbol (→) Conditional statement – statement written in if-then form Hypothesis – phrase immediately following the word if in a conditional statement Conclusion – p ...
Methods of Proofs Recall we discussed the following methods of
... A vacuous proof of an implication happens when the hypothesis of the implication is always false. Example 1: Prove that if x is a positive integer and x = -x, then x2 = x. An implication is trivially true when its conclusion is always true. A declared mathematical proposition whose truth value is un ...
... A vacuous proof of an implication happens when the hypothesis of the implication is always false. Example 1: Prove that if x is a positive integer and x = -x, then x2 = x. An implication is trivially true when its conclusion is always true. A declared mathematical proposition whose truth value is un ...