Slide Logic học C2 - Logic Học UEL

Introduction to LOGIC Chapter 2: Categorical Propositions (Mệnh đề đơn) (Ref.: Chapter 4) Hà Văn Hiếu Faculty of Economic Mathematics, University of Economics and Law Outline The Components of Categorical Propositions Logical Relationships Operations on categorical statements The Traditional Square of Opposition Translating Statements into Categorical Form The components of categorical propositions (mệnh đề đơn) ▶ Categorical propositions are statements that relate two classes (or categories). ▶ They consist of four main components: ▶ Subject term (S) - Chủ từ ▶ Predicate term (P) - Thuộc từ, vị từ ▶ Copula (linking verb, e.g., ”is” or ”are”) - Hệ từ ▶ Quantifier (e.g., “All”, “Some”, “No”) - Lượng từ ▶ “Some birds are flightless”. ▶ Subjective term (S): “birds”. ▶ Predicate term (P): “flightless”. ▶ Copula: “are” ▶ Quantifier: “Some”. The Four Standard Forms Categorical propositions are classified into four standard forms (4 dạng chuẩn): 1. A (Universal Affirmative, Mệnh đề khẳng định toàn thể): All S are P (Mọi/Tất cả S là P). 2. E (Universal Negative, MĐ phủ định toàn thể): No S are P (Mọi/Tất cả S không là P). 3. I (Particular Affirmative, MĐ khẳng định bộ phận): Some S are P (Một số S là P). 4. O (Particular Negative, MĐ phủ định bộ phận): Some S are not P (Một số S không là P). Examples: ▶ A: “All students are diligent.” ▶ E: “No cars are boats.” ▶ I: “Some trees are tall.” ▶ O: “Some birds are not carnivorous.” Exercises: 4.1.I (1, 2, 3) Venn diagrams of the four standard forms ▶ All S are P ≡ No members of S are outside P. ▶ No S are P ≡ No members of S are inside P. ▶ Some S are P ≡ At least one S exists that is a P. ▶ Some S are not P ≡ At least one S exists that is not a P. Shading = emptiness, X = existence The Modern Square of Opposition (Hình vuông logic hiện đại) ▶ Contradictory (mâu thuẫn): A vs O, E vs I. If a certain A proposition is given as true, the corresponding O proposition must be false. Similarly to E and I. We say that “A and O have the contradictory relation (mối quan hệ mâu thuẫn)”. ▶ No other inferences are possible. That is, given the truth value of an A or O proposition, nothing can be determined about the truth value of the corresponding E or I propositions. We say that “A and E have logically undetermined truth value”. Conversion (Phép đảo ngược) ▶ Conversion, obversion, and contraposition are operations that can be performed on a categorical proposition, resulting in a new statement that may or may not have the same meaning and truth value as the original statement. Example: ▶ Original proposition: “All cats are animals.” ▶ Its converse: “All animals are cats.” Figure: Shading = emptiness, X = existence Properties of conversion ▶ Two statements are said to be logically equivalent (tương đương logic) if they necessarily have the same truth value. ▶ E and its converse are logically equivalent. ▶ I and its converse are logically equivalent. ▶ A and its converse are logically unrelated (không liên quan về mặt logic) as to truth value. ▶ O and its converse are logically unrelated as to truth value. Example ▶ Original: “No dogs are cats”, and inverse: “No cats are dogs” are both true. ▶ Original: “All birds are animals”, and inverse: “All animals are birds”