Logic học các lý thuyết hệ E - Logic Học UEL

lOMoARcPSD| 69237702 An argument is a group of statements, one or more of which (the premises) are claimed to provide support for, or reasons to believe, one of the others (the conclusion). A statement is a sentence that is either true or false — in other words, typically a declarative sentence or a sentence component that could stand as a declarative sentence. conclusion indicators are: therefore, accordingly, entails that, wherefore, we may conclude, hence, thus, it must be that, it follows that, consequently, for this reason implies that, we may infer, so, as a result • premise indicators are: since, as indicated by, because, for, in that, may be inferred from, as, given that, seeing that, for the reason that, in as much as, owing to “inference” is used interchangeably with “argument.” “proposition” and “statement” are used interchangeably. Nonarguments Indicator words (“hence,” “since,” etc.) Simple Noninferential Passages (những đoạn văn đơn giản không suy diễn) - warning - piece of advice - statement of belief or opinion - loosely associated statements: tuyên bố lỏng lẻo - report Expository passages Illustrations Explanations Conditional statements Deductive argument: diễn dịch (kết luận đầu) - An argument based on mathematics - An argument from definition - A categorical syllogism - A hypothetical syllogism - A disjunctive syllogism Inductive argument: quy nạp (kết luận cuối) - A prediction - An argument from analogy - A generalization - An argument from authority - An argument based on signs - A causal inference Argument Forms: Proving Invalidity Counterexample method Extended Arguments lOMoARcPSD| 69237702 Categorical proposition A (Universal Affirmative): All S are P E (Universal No S are P Negative): Some S are P I (Particular Affirmative): Some S are not P O (Particular Negative): Obversion thay đổi phẩm chất Contraposition: thay đổi chủ ngữ + phẩm chất Conversion đổi vị trí a và b E and its converse are logically equivalent I and its converse are logically equivalent A and its converse are logically unrelated as to truth value O and its converse are logically unrelated as to truth value lOMoARcPSD| 69237702 Traditional Square of Opposite (Aristotelian standpoint) Contradictory: mâu thuẫn đối lập Contrary: mâu thuẫn ngụ ý sai Subcontrary: mâu thuẫn phụ Subalternation: sự xen kẽ phụ Categorical syllogism • The major term is defined as the predicate of the conclusion (vị ngữ kết luận) • The minor term is defined as the subject of the conclusion (chủ ngữ kết luận) • The middle term is the term that occurs once in each premise, and does not occur in the conclusion. Rule 1: The Middle Term Must Be Distributed at Least Once. Rule 2: If a Term Is Distributed in the Conclusion, Then It Must Be Distributed in a Premise If the major term (P) is distributed in the conclusion but not in the premises, the syllogism commits the fallacy of the illicit major. If the minor term (S) is distributed in the conclusion but not in the premises, the syllogism commits the fallacy of the illicit minor. Rule 3: Two Negative Premises Are Not Allowed Rule 4: A Negative Premise Requires a Negative Conclusion, and a Negative Conclusion Requires a Negative Premise. (tiền đề phủ định cần kết luận phủ định, và ngược lại) Rule 5: If Both Premises Are Universal (phổ quát “All, No”), the Conclusion Cannot Be Particular (“Some”) (for Boolean standpoint only) Aristotelian Standpoint in Categorical Syllogisms lOMoARcPSD| 69237702 Enthymemes (thiếu tiền đề hoặc kết luận) missing a premise or a conclusion Sorites the argument flows directly from the premises to the conclusion. (bỏ qua trung gian từ tiền đề -> kết luận) 1 Rewrite the sorites in the standard form 2 Introduce the intermediate conclusions 3 Check the validity of each syllogism 4. Since all conclusions are drawn validly, the sorites is valid. Propositional logic - introduces special symbols to simplify the expression of statements and arguments and expose their structure. • In the two previous chapters, the fundamental elements were terms. In propositional logic, however, the fundamental elements are whole statements (or propositions). Statements are represented by letters, and these letters are then combined by means of the operators to form more-complex symbolic representations. Simple statement is one that does not contain any other statement as a component. • Example: “Fast foods tend to be unhealthy.” • We usually denote a simple statement by an uppercase letter. Compound statement is one that contains at least one simple statement as a component. • Example: “It is not the case that the Taliban supports educating women.” • In order to express a compound statement by symbols, we use logical operators: negation, conjunction, disjunction, implication, and equivalence. The five Logical Operations and their translations Remark: • Some textbooks use the ampersand (&), in place of the dot; the arrow, (→), in place of the horseshoe; the double arrow, (↔), in place of the triple bar. • Whenever more than two letters appear in a translated statement, we must use parentheses, brackets, or braces to indicate the proper range of the operators. • The statement ∼ T is called a negation. • The statement D · C is called a conjunctive statement (liên hợp) (or a conjunction), and the components D and C are called conjuncts. • The statement P ∨ E is called a disjunctive statement (phân ly) (or a disjunction), and the components P and E are called disjuncts. • The statement N ⊃ F is called a conditional statement (or a conditional), and it expresses the relation of material implication (ngụ ý). Its components are called the antecedent (N) and consequent (F). • The statement B ≡ R is called a biconditional statement (song điều kiện) (or a biconditional), and it expresses the relation of material equivalence (tương đương). Main operator is the operator that has as its scope everything else in the statement. • If there are no parentheses (dấu hoặc đơn) in the statement, the main operator will either be the only operator or, if there is more than one, it will be the operator that is not a tilde (dấu ngã). lOMoARcPSD| 69237702 • If there are parentheses ( ), brackets [ ], or braces { } in the statement, the main operator will be the operator that lies outside all parentheses, brackets, and braces; if there is more than one such operator, the main operator will be the one that is not a tilde. The name of a statement depends on the main operator of that statement. Example The statement (E ∨ G)· ∼ (G ∨ H) is a conjunction. The main operator is a dot. Well-formed formulas (WFFs) is a syntactically correct arrangement of symbols. • Statements cannot be combined without an operator occurring between them. Not WFFs: AB, A(B ⊃ C),(A ⊃ B)(B ∨ C) WFFs: A ⊃ B, A∨ ∼ B, A · (B ≡ C) • A tilde cannot immediately follow a statement, but it can immediately precede any statement (except when it immediately follows a statement). Not WFFs: A ∼, A ∼ B,(A · B) ∼ WFFs: ∼ A, ∼ (A · B), ∼∼ A • A tilde cannot immediately precede any other operator. Not WFFs: ∼ ·A, A ∼ ∨B • A dot, wedge, horseshoe, or triple bar must go immediately between statements. Not WFFs: ·A, A∨, A ⊃⊃ B WFFs: A · B, A ⊃∼ B, A ∨ (B · C) • Parentheses, brackets, and braces must be inserted to prevent ambiguity (mơ hồ). Not WFFs: A ⊃ B ∨ C, A · B ≡ C WFFs: A ⊃ (B ∨ C),(A · B) ⊃ C Truth Functions is any compound proposition whose truth value is completely determined by the truth values of its components. Definitions of the Logical Operators • are presented in terms of statement variables, which are lowercase letters (p, q, r, s) that can stand for any statement. For example, the statement variable p could stand for the statements A, A ⊃ B, B ∨ C, and so on. • A statement form is an arrangement of statement variables and operators such that the uniform substitution of statements in place of the variables results in a statement. For example, ∼ p and p ⊃ q are statement forms. A truth table is an arrangement of truth values that shows in every possible case how the truth value of a compound statement is determined by the truth values of its components. Truth table is used to define the meaning of the five logical operators: Negation: ∼ p is true Conjunction: p · q is Disjunction: p ∨ q is Conditional: p ⊃ q is Biconditional: p ≡ q only when p is false. true only when both false only when both false only when p is is true only when p p and q are true. and q have the same p and q are false. true and q is false truth value. Computing the Truth Value of Longer Propositions