We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
Presentation is loading. Please wait.
Published byChrystal Patrick
Modified over 5 years ago
Copyright © Peter Cappello Logical Inferences Goals for propositional logic 1.Introduce notion of a valid argument & rules of inference. 2.Use inference rules to build correct arguments.
Copyright © Peter Cappello What is a rule of inference? A rule of inference allows us to specify which conclusions may be inferred from assertions known, assumed, or previously established. A tautology is a propositional function that is true for all values of the propositional variables (e.g., p ~p).
Copyright © Peter Cappello Modus ponens A rule of inference is a tautological implication. Modus ponens: ( p (p q) ) q
Copyright © Peter Cappello Modus ponens: An example Suppose the following 2 statements are true: If it is 11am in Miami then it is 8am in Santa Barbara. It is 11am in Miami. By modus ponens, we infer that it is 8am in Santa Barbara.
Copyright © Peter Cappello Other rules of inference Other tautological implications include: (Is there a finite number of rules of inference?) p (p q) (p q) p [~q (p q)] ~p [(p q) ~p] q [(p q) (q r)] (p r) hypothetical syllogism [(p q) (r s) (p r) ] (q s) [(p q) (r s) (~q ~s) ] (~p ~r) [ (p q) (~p r) ] (q r ) resolution
Copyright © Peter Cappello Common fallacies 3 fallacies are common: Affirming the converse: [(p q) q] p If Socrates is a man then Socrates is mortal. Socrates is mortal. Therefore, Socrates is a man.
Copyright © Peter Cappello Common fallacies... Assuming the antecedent: [(p q) ~p] ~q If Socrates is a man then Socrates is mortal. Socrates is not a man. Therefore, Socrates is not mortal.
Copyright © Peter Cappello Common fallacies... Non sequitur: p q Socrates is a man. Therefore, Socrates is mortal. The following is valid: If Socrates is a man then Socrates is mortal. Socrates is a man. Therefore, Socrates is mortal. The argument’s form is what matters.
Copyright © Peter Cappello Examples of arguments Given an argument whose form isn’t obvious: Decompose the argument into premise assertions Connect the premises according to the argument Check to see that the inference is valid. Example argument: If a baby is hungry, it cries. If a baby is not mad, it doesn’t cry. If a baby is mad, it has a red face. Therefore, if a baby is hungry, it has a red face.
Copyright © Peter Cappello ( (h c) (~m ~c) (m r) ) (h r) r m c h
Copyright © Peter Cappello Examples of arguments... Argument: McCain will be elected if and only if California votes for him. If California keeps its air base, McCain will be elected. Therefore, McCain will be elected. Assertions: m: McCain will be elected c: California votes for McCain b: California keeps its air base Argument: [(m c) (b m)] m (valid?)
-- in other words, logic is
Rules of Inference Rosen 1.5.
Discrete Mathematics University of Jazeera College of Information Technology & Design Khulood Ghazal Mathematical Reasoning Methods of Proof.
Hypotheticals: The If/Then Form Hypothetical arguments are usually more obvious than categorical ones. A hypothetical argument has an “if/then” pattern.
Rules of Inferences Section 1.5. Definitions Argument: is a sequence of propositions (premises) that end with a proposition called conclusion. Valid Argument:
1 Section 1.5 Rules of Inference. 2 Definitions Theorem: a statement that can be shown to be true Proof: demonstration of truth of theorem –consists of.
The Foundations: Logic and Proofs
Higher / Int.2 Philosophy 5. ” All are lunatics, but he who can analyze his delusion is called a philosopher.” Ambrose Bierce “ Those who lack the courage.
1 Introduction to Abstract Mathematics Valid AND Invalid Arguments 2.3 Instructor: Hayk Melikya
CSE115/ENGR160 Discrete Mathematics 01/26/12 Ming-Hsuan Yang UC Merced 1.
Valid Arguments An argument is a sequence of propositions. All but the final proposition are called premises. The last statement is the conclusion. The.
Chapter 1 The Logic of Compound Statements. Section 1.3 Valid & Invalid Arguments.
CS128 – Discrete Mathematics for Computer Science
Uses for Truth Tables Determine the truth conditions for any compound statementDetermine the truth conditions for any compound statement Determine whether.
Logic 3 Tautological Implications and Tautological Equivalences
Essential Deduction Techniques of Constructing Formal Expressions and Evaluating Attempts to Create Valid Arguments.
Essential Deduction Techniques of Constructing Formal Expressions Evaluating Attempts to Create Valid Arguments.
Proof by Deduction. Deductions and Formal Proofs A deduction is a sequence of logic statements, each of which is known or assumed to be true A formal.
Logical and Rule-Based Reasoning Part I. Logical Models and Reasoning Big Question: Do people think logically?
Fall 2002CMSC Discrete Structures1 Let’s proceed to… Mathematical Reasoning.
© 2020 SlidePlayer.com Inc. All rights reserved.