- Abbreviated truth tables check an argument's validity by assuming the premises are true and conclusion is false, working backwards to assign true/false values to statements.
- For the argument "If A then B, A. Therefore B", assuming A is true and B is false leads to a contradiction, showing the argument is valid.
- More complex arguments may involve choices when multiple assignments could satisfy premises/conclusion. Finding a contradiction means that choice leads to a valid argument; an assignment without contradiction means the argument is invalid.
Truth tables complete and p1 of short methodNat Karablina
The document discusses truth tables and how to use partial truth tables (also called the short method) to evaluate sentences. Some key points:
- Truth tables determine the truth values of sentences based on the truth values of their parts and connectives like negation, conjunction, disjunction, conditional, biconditional.
- To show a sentence is a tautology or contradiction requires a full truth table, but only one line is needed to show it is not a tautology/contradiction.
- The short method constructs a partial truth table to evaluate sentences more efficiently in some cases, like showing a sentence is contingent by having one line with true and one with false.
- For tasks like
The document discusses inductive and deductive reasoning. Inductive reasoning involves forming general conclusions from specific observations, while deductive reasoning draws specific conclusions from general statements. Examples are given of inductive arguments building from specific cases to a general rule, and deductive arguments applying a general premise to specific cases. The key features of deductive reasoning, including conditional statements and the five types of if-then logical structures (conditional, converse, counter example, inverse, and contrapositive), are also explained through examples.
This document introduces some basic concepts in propositional logic. It defines propositional logic as the study of how simple propositions combine to form more complex propositions. It discusses statements as descriptions that can be true or false, and provides examples. It also introduces logical connectives like negation, conjunction, disjunction, implication and biconditional, and shows how they combine atomic propositions into compound propositions. Truth tables are provided to illustrate the truth values of compound propositions formed with different connectives.
Logical Operators in Brief with examplesMujtaBa Khan
This document defines and explains different types of logical operators: negation, conjunctions, disjunctions, conditionals, and bi-conditionals. It provides examples of each operator and includes their truth tables showing how the operators evaluate statements as true or false based on the truth values of the individual statements.
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
1. The document discusses mathematical logic and proofs. It introduces logic operators such as NOT, AND and OR and how they are used to construct truth tables and logical formulas.
2. Conditional statements like "if P then Q" are explained along with their contrapositive and negation. Logical equivalences between statements are important.
3. The concept of an argument is introduced, where valid arguments are those where the conclusion follows logically from the assumptions. Specific argument forms like modus ponens and modus tollens are discussed.
- Abbreviated truth tables check an argument's validity by assuming the premises are true and conclusion is false, working backwards to assign true/false values to statements.
- For the argument "If A then B, A. Therefore B", assuming A is true and B is false leads to a contradiction, showing the argument is valid.
- More complex arguments may involve choices when multiple assignments could satisfy premises/conclusion. Finding a contradiction means that choice leads to a valid argument; an assignment without contradiction means the argument is invalid.
Truth tables complete and p1 of short methodNat Karablina
The document discusses truth tables and how to use partial truth tables (also called the short method) to evaluate sentences. Some key points:
- Truth tables determine the truth values of sentences based on the truth values of their parts and connectives like negation, conjunction, disjunction, conditional, biconditional.
- To show a sentence is a tautology or contradiction requires a full truth table, but only one line is needed to show it is not a tautology/contradiction.
- The short method constructs a partial truth table to evaluate sentences more efficiently in some cases, like showing a sentence is contingent by having one line with true and one with false.
- For tasks like
The document discusses inductive and deductive reasoning. Inductive reasoning involves forming general conclusions from specific observations, while deductive reasoning draws specific conclusions from general statements. Examples are given of inductive arguments building from specific cases to a general rule, and deductive arguments applying a general premise to specific cases. The key features of deductive reasoning, including conditional statements and the five types of if-then logical structures (conditional, converse, counter example, inverse, and contrapositive), are also explained through examples.
This document introduces some basic concepts in propositional logic. It defines propositional logic as the study of how simple propositions combine to form more complex propositions. It discusses statements as descriptions that can be true or false, and provides examples. It also introduces logical connectives like negation, conjunction, disjunction, implication and biconditional, and shows how they combine atomic propositions into compound propositions. Truth tables are provided to illustrate the truth values of compound propositions formed with different connectives.
Logical Operators in Brief with examplesMujtaBa Khan
This document defines and explains different types of logical operators: negation, conjunctions, disjunctions, conditionals, and bi-conditionals. It provides examples of each operator and includes their truth tables showing how the operators evaluate statements as true or false based on the truth values of the individual statements.
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
1. The document discusses mathematical logic and proofs. It introduces logic operators such as NOT, AND and OR and how they are used to construct truth tables and logical formulas.
2. Conditional statements like "if P then Q" are explained along with their contrapositive and negation. Logical equivalences between statements are important.
3. The concept of an argument is introduced, where valid arguments are those where the conclusion follows logically from the assumptions. Specific argument forms like modus ponens and modus tollens are discussed.
This document defines and explains logical concepts such as simple and compound statements, truth tables, logical operators like negation and conjunction, and argument validity. It discusses translating statements into symbolic logic using variables, determining statement truth values from truth tables, and classifying statements as tautologies, contradictions, or contingencies. Common valid argument forms like modus ponens and modus tollens are also defined.
Inductive reasoning uses examples and observations to reach a conclusion, called a conjecture. A conjecture is either always true or false. While examples can support a conjecture, they cannot prove it. A counterexample can demonstrate that a conjecture is false.
The document discusses categorical propositions and their logical properties. It covers:
- The standard form of categorical propositions involving quantifiers like "all" and "some."
- The quality and quantity of propositions.
- Letter names assigned to proposition types.
- The distribution of terms and existential import of propositions.
- Systems for representing propositions diagrammatically, like Venn diagrams and the square of opposition.
- Conversions between propositions through operations like conversion, obversion, and contraposition.
- Testing proposition validity and logical fallacies from Boolean and Aristotelian perspectives.
The document discusses syllogisms, which are logical arguments with two premises and a conclusion. It defines the key components of a syllogism, including terms, validity, categorical propositions, and the four figures or patterns that a syllogism can take. Rules for syllogisms are also outlined, such as that the middle term must be distributed at least once and premises and conclusions must align in terms of positive and negative forms.
- A categorical proposition relates two classes or categories, asserting whether all, part, or none of one class is included in or excluded from the other class.
- There are four standard forms of categorical propositions: All, No, Some, Some...not.
- A categorical syllogism is a formal deductive argument with three terms - major, minor, and middle - and three statements following rules about term distribution and relationship between premises and conclusion.
The document discusses the differences between deductive and inductive reasoning. Deductive reasoning involves drawing a specific conclusion from general statements using the form of "if p then q". Inductive reasoning involves drawing a general conclusion based on patterns or trends in specific observations, where the conclusions are not certain. Examples of each type of reasoning are provided using Venn diagrams and number/date sequences.
This document covers a lecture on compound propositions and logical operators in discrete structures. It defines logical operators such as negation, conjunction, disjunction, exclusive or, implication, and biconditional. It provides truth tables for each operator and examples of how to write compound propositions using the operators. De Morgan's laws and their applications are discussed. The concepts of tautology, contradiction, logical equivalence and various laws of logic are also introduced.
The document discusses various concepts in inductive and deductive reasoning including:
- Writing conjectures based on given information and finding examples/counterexamples
- Using Venn diagrams and truth tables to represent conjunctions, disjunctions, and conditionals
- The properties of conditionals including converse, inverse, and contrapositive
- Laws of logic like detachment and syllogism to make valid deductive arguments
- Postulates and properties related to geometry concepts like lines, planes, angles, and segments
This document discusses different types of logical syllogisms:
1. Hypothetical syllogism uses a conditional premise and categorical conclusions. There are three types of hypothetical propositions: conditional, disjunctive, and conjunctive.
2. Conditional syllogism uses a conditional premise and its valid forms are modus ponens and modus tollens.
3. Disjunctive syllogism uses a disjunctive premise and its valid forms are ponendo tollens and tollendo ponens.
4. Conjunctive syllogism uses a conjunctive premise stating two choices cannot be true together.
It provides examples and
Based from the book : "Logic Made Simple for Filipinos" by Florentino Timbreza here is the summary made into powerpoint of Lesson 12: The Categorical Syllogism.
It Includes:
Introduction to categorical syllogism
General Axioms of the Syllogism
Eight Syllogistic Rules
Figures and Moods of the Categorical Syllogism
Examples in these slides are our own, there were no examples derived from the book.
The document discusses syllogistic figures and principles of categorical syllogism. It explains:
1) Four syllogistic figures defined by the location of the middle term in the premises.
2) Five principles of categorical syllogism including reciprocal identity, reciprocal non-identity, dictum de omni, dictum de nullo, and contradiction.
3) Basic rules of categorical syllogism requiring three terms, limitations on negatives, and conclusions matching premises.
4) For a syllogism to be valid it must be correct in form and have true content.
1. Propositional logic (L1) has a simple syntax consisting of sentence letters like P and Q, and logical connectives like negation (¬), conjunction (∧), and disjunction (∨).
2. The semantics of L1 specifies the truth conditions of sentences based on the truth values (True/False) assigned to basic sentences under an interpretation.
3. The meanings of the logical connectives are given by their truth tables, which determine the truth value of complex sentences based on the truth values of their components. For example, a conjunction is only true if both conjuncts are true.
ON CATEGORICAL SYLLOGISM
LEGEND
U - universal / P - particular
A - affirmative / N - negative
3 PROPOSITIONS
- MAJOR Premise (first premise)
- MINOR Premise (second premise)
- Conclusion
Rule in constructing the argument:
UUU / UPP / PUP
AAA / ANA / NAN
ON THE TERMS (3 terms)
MAJOR term
- the Predicate of the Conclusion and is found in the Major premise (subject or predicate)
MINOR term
- the Subject of the Conclusion and is found in the Minor premise (subject or predicate)
MIDDLE term
- found in the major and minor premises but not in the conclusion
ON FIGURE (CATEGORICAL SYLLOGISM)
FIGURE-refers to the arrangement of MIDDLE term in the two premises in the syllogism
CONT.
ON MOODS (CATEGORICAL SYLLOGISM)
MOODS refers to the pattern of the types of proposition used in the syllogism.
This document discusses propositional logic and covers topics like propositions, common logical operators like negation and conjunction, proving the equivalence of logical formulas, constructing logical formulas based on truth tables, and simplifying logical formulas using laws like De Morgan's laws and distribution laws. Examples are provided for each topic to illustrate key concepts in propositional logic.
Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكروDr. Khaled Bakro
Discrete Mathematics chapter 2 covers propositional logic. A proposition is a statement that is either true or false. Propositional logic uses propositional variables and logical operators like negation, conjunction, disjunction, implication and biconditional. Truth tables are used to determine the truth value of compound propositions formed using these operators. Logical equivalences between compound propositions can be shown using truth tables or by applying equivalence rules.
This document discusses syllogisms in ordinary language. It begins by outlining objectives related to identifying ways arguments can deviate from standard form, reducing the number of terms in a syllogism, and translating categorical propositions. It then covers reducing terms to three, translating propositions into standard form, using parameters for uniform translation, identifying enthymemes and sorites, and disjunctive and hypothetical syllogisms. It concludes with discussing types of dilemmas and methods for responding to them.
The Foundations: Logic and Proofs: Propositional Logic, Applications of Propositional Logic, Propositional Equivalence, Predicates and Quantifiers, Nested Quantifiers, Rules of Inference, Introduction to Proofs, Proof Methods and Strategy.
The document discusses propositional logic including:
- Propositional logic uses propositions that can be either true or false and logical connectives to connect propositions.
- It introduces syntax of propositional logic including atomic and compound propositions.
- Logical connectives like negation, conjunction, disjunction, implication, and biconditional are explained along with their truth tables and significance.
- Other concepts discussed include precedence of connectives, logical equivalence, properties of operators, and limitations of propositional logic.
- Examples are provided to illustrate propositional logic concepts like truth tables, logical equivalence, and translating English statements to symbolic form.
The document discusses logic and propositions. It begins by defining a proposition as a statement that is either true or false. It then provides examples of propositions and non-propositions. The document also discusses arguments and their validity. An argument is valid if the premises guarantee the conclusion. It discusses logical operators like conjunction, disjunction, negation and implication. Truth tables are used to determine the truth values of compound propositions formed using logical operators. Laws of algebra are also discussed for propositional logic.
This book is written by LOIBANGUTI, BM, it is just an online copy provided for free. No part of this book mya be republished. but can be used and stored as a softcopy book, can be shared accordingly.
This document provides an overview of a course on discrete mathematics. The course objectives are to develop mathematical reasoning skills, learn how to analyze counting problems, and learn how to specify and analyze algorithms. Topics covered include formal logic, proofs, sets and combinatorics, algorithms and complexity, induction, counting, discrete probability, relations, graphs and trees, Boolean algebra, and finite-state and Turing machines. The primary textbook is Discrete Mathematics and Its Applications by Rosen. Chapter 1 covers propositional logic, including logical connectives like negation, conjunction, disjunction, implication, equivalence, and truth tables. Propositional equivalences like De Morgan's laws are also discussed. Bit operations on strings of Boolean values are introduced.
This document provides an overview of general philosophy of science and introduces some key concepts in logic that are relevant to philosophy of science. It discusses how general philosophy of science seeks to understand science across many disciplines rather than focusing on a single field. It also introduces deductive logic and different types of arguments, focusing on the distinction between valid and sound arguments. The document examines how logic has been used as a tool in philosophy of science but may have limitations, as scientific theories are not always logical deductions from evidence alone.
This document defines and explains logical concepts such as simple and compound statements, truth tables, logical operators like negation and conjunction, and argument validity. It discusses translating statements into symbolic logic using variables, determining statement truth values from truth tables, and classifying statements as tautologies, contradictions, or contingencies. Common valid argument forms like modus ponens and modus tollens are also defined.
Inductive reasoning uses examples and observations to reach a conclusion, called a conjecture. A conjecture is either always true or false. While examples can support a conjecture, they cannot prove it. A counterexample can demonstrate that a conjecture is false.
The document discusses categorical propositions and their logical properties. It covers:
- The standard form of categorical propositions involving quantifiers like "all" and "some."
- The quality and quantity of propositions.
- Letter names assigned to proposition types.
- The distribution of terms and existential import of propositions.
- Systems for representing propositions diagrammatically, like Venn diagrams and the square of opposition.
- Conversions between propositions through operations like conversion, obversion, and contraposition.
- Testing proposition validity and logical fallacies from Boolean and Aristotelian perspectives.
The document discusses syllogisms, which are logical arguments with two premises and a conclusion. It defines the key components of a syllogism, including terms, validity, categorical propositions, and the four figures or patterns that a syllogism can take. Rules for syllogisms are also outlined, such as that the middle term must be distributed at least once and premises and conclusions must align in terms of positive and negative forms.
- A categorical proposition relates two classes or categories, asserting whether all, part, or none of one class is included in or excluded from the other class.
- There are four standard forms of categorical propositions: All, No, Some, Some...not.
- A categorical syllogism is a formal deductive argument with three terms - major, minor, and middle - and three statements following rules about term distribution and relationship between premises and conclusion.
The document discusses the differences between deductive and inductive reasoning. Deductive reasoning involves drawing a specific conclusion from general statements using the form of "if p then q". Inductive reasoning involves drawing a general conclusion based on patterns or trends in specific observations, where the conclusions are not certain. Examples of each type of reasoning are provided using Venn diagrams and number/date sequences.
This document covers a lecture on compound propositions and logical operators in discrete structures. It defines logical operators such as negation, conjunction, disjunction, exclusive or, implication, and biconditional. It provides truth tables for each operator and examples of how to write compound propositions using the operators. De Morgan's laws and their applications are discussed. The concepts of tautology, contradiction, logical equivalence and various laws of logic are also introduced.
The document discusses various concepts in inductive and deductive reasoning including:
- Writing conjectures based on given information and finding examples/counterexamples
- Using Venn diagrams and truth tables to represent conjunctions, disjunctions, and conditionals
- The properties of conditionals including converse, inverse, and contrapositive
- Laws of logic like detachment and syllogism to make valid deductive arguments
- Postulates and properties related to geometry concepts like lines, planes, angles, and segments
This document discusses different types of logical syllogisms:
1. Hypothetical syllogism uses a conditional premise and categorical conclusions. There are three types of hypothetical propositions: conditional, disjunctive, and conjunctive.
2. Conditional syllogism uses a conditional premise and its valid forms are modus ponens and modus tollens.
3. Disjunctive syllogism uses a disjunctive premise and its valid forms are ponendo tollens and tollendo ponens.
4. Conjunctive syllogism uses a conjunctive premise stating two choices cannot be true together.
It provides examples and
Based from the book : "Logic Made Simple for Filipinos" by Florentino Timbreza here is the summary made into powerpoint of Lesson 12: The Categorical Syllogism.
It Includes:
Introduction to categorical syllogism
General Axioms of the Syllogism
Eight Syllogistic Rules
Figures and Moods of the Categorical Syllogism
Examples in these slides are our own, there were no examples derived from the book.
The document discusses syllogistic figures and principles of categorical syllogism. It explains:
1) Four syllogistic figures defined by the location of the middle term in the premises.
2) Five principles of categorical syllogism including reciprocal identity, reciprocal non-identity, dictum de omni, dictum de nullo, and contradiction.
3) Basic rules of categorical syllogism requiring three terms, limitations on negatives, and conclusions matching premises.
4) For a syllogism to be valid it must be correct in form and have true content.
1. Propositional logic (L1) has a simple syntax consisting of sentence letters like P and Q, and logical connectives like negation (¬), conjunction (∧), and disjunction (∨).
2. The semantics of L1 specifies the truth conditions of sentences based on the truth values (True/False) assigned to basic sentences under an interpretation.
3. The meanings of the logical connectives are given by their truth tables, which determine the truth value of complex sentences based on the truth values of their components. For example, a conjunction is only true if both conjuncts are true.
ON CATEGORICAL SYLLOGISM
LEGEND
U - universal / P - particular
A - affirmative / N - negative
3 PROPOSITIONS
- MAJOR Premise (first premise)
- MINOR Premise (second premise)
- Conclusion
Rule in constructing the argument:
UUU / UPP / PUP
AAA / ANA / NAN
ON THE TERMS (3 terms)
MAJOR term
- the Predicate of the Conclusion and is found in the Major premise (subject or predicate)
MINOR term
- the Subject of the Conclusion and is found in the Minor premise (subject or predicate)
MIDDLE term
- found in the major and minor premises but not in the conclusion
ON FIGURE (CATEGORICAL SYLLOGISM)
FIGURE-refers to the arrangement of MIDDLE term in the two premises in the syllogism
CONT.
ON MOODS (CATEGORICAL SYLLOGISM)
MOODS refers to the pattern of the types of proposition used in the syllogism.
This document discusses propositional logic and covers topics like propositions, common logical operators like negation and conjunction, proving the equivalence of logical formulas, constructing logical formulas based on truth tables, and simplifying logical formulas using laws like De Morgan's laws and distribution laws. Examples are provided for each topic to illustrate key concepts in propositional logic.
Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكروDr. Khaled Bakro
Discrete Mathematics chapter 2 covers propositional logic. A proposition is a statement that is either true or false. Propositional logic uses propositional variables and logical operators like negation, conjunction, disjunction, implication and biconditional. Truth tables are used to determine the truth value of compound propositions formed using these operators. Logical equivalences between compound propositions can be shown using truth tables or by applying equivalence rules.
This document discusses syllogisms in ordinary language. It begins by outlining objectives related to identifying ways arguments can deviate from standard form, reducing the number of terms in a syllogism, and translating categorical propositions. It then covers reducing terms to three, translating propositions into standard form, using parameters for uniform translation, identifying enthymemes and sorites, and disjunctive and hypothetical syllogisms. It concludes with discussing types of dilemmas and methods for responding to them.
The Foundations: Logic and Proofs: Propositional Logic, Applications of Propositional Logic, Propositional Equivalence, Predicates and Quantifiers, Nested Quantifiers, Rules of Inference, Introduction to Proofs, Proof Methods and Strategy.
The document discusses propositional logic including:
- Propositional logic uses propositions that can be either true or false and logical connectives to connect propositions.
- It introduces syntax of propositional logic including atomic and compound propositions.
- Logical connectives like negation, conjunction, disjunction, implication, and biconditional are explained along with their truth tables and significance.
- Other concepts discussed include precedence of connectives, logical equivalence, properties of operators, and limitations of propositional logic.
- Examples are provided to illustrate propositional logic concepts like truth tables, logical equivalence, and translating English statements to symbolic form.
The document discusses logic and propositions. It begins by defining a proposition as a statement that is either true or false. It then provides examples of propositions and non-propositions. The document also discusses arguments and their validity. An argument is valid if the premises guarantee the conclusion. It discusses logical operators like conjunction, disjunction, negation and implication. Truth tables are used to determine the truth values of compound propositions formed using logical operators. Laws of algebra are also discussed for propositional logic.
This book is written by LOIBANGUTI, BM, it is just an online copy provided for free. No part of this book mya be republished. but can be used and stored as a softcopy book, can be shared accordingly.
This document provides an overview of a course on discrete mathematics. The course objectives are to develop mathematical reasoning skills, learn how to analyze counting problems, and learn how to specify and analyze algorithms. Topics covered include formal logic, proofs, sets and combinatorics, algorithms and complexity, induction, counting, discrete probability, relations, graphs and trees, Boolean algebra, and finite-state and Turing machines. The primary textbook is Discrete Mathematics and Its Applications by Rosen. Chapter 1 covers propositional logic, including logical connectives like negation, conjunction, disjunction, implication, equivalence, and truth tables. Propositional equivalences like De Morgan's laws are also discussed. Bit operations on strings of Boolean values are introduced.
This document provides an overview of general philosophy of science and introduces some key concepts in logic that are relevant to philosophy of science. It discusses how general philosophy of science seeks to understand science across many disciplines rather than focusing on a single field. It also introduces deductive logic and different types of arguments, focusing on the distinction between valid and sound arguments. The document examines how logic has been used as a tool in philosophy of science but may have limitations, as scientific theories are not always logical deductions from evidence alone.
The document discusses key concepts in logic including propositions, truth tables, logical connectives like conjunction and disjunction, quantifiers, and valid arguments. Some key points:
- A proposition is a statement that is either true or false.
- Truth tables define the truth values of logical connectives and conditionals.
- Quantifiers like "all" and "some" are used to make generalized statements about sets.
- Venn diagrams can represent relationships between sets graphically.
- An argument is valid if the premises necessarily make the conclusion true.
This document provides information about logic and mathematical logic. It begins by asking the reader to answer 10 logic questions. It then defines key logical terms and concepts like propositions, truth tables, logical connectives, and logical operators like negation, conjunction, and disjunction. It provides examples and explanations of how to determine the truth value of simple and compound propositions using truth tables and logical operators. The document aims to explain the foundations and basic components of propositional logic.
The document discusses deductive and inductive arguments. It provides examples of valid and invalid deductive arguments using categorical propositions and conditional premises. It also discusses inductive arguments, noting that inductive conclusions generalize from specific premises rather than necessarily following from them. The document then compares deductive and inductive arguments and discusses their uses in everyday life and mathematics. It concludes by introducing some common rules of inference for deductive arguments.
This document provides an overview of propositions and logic. It discusses:
- Propositions as statements that can be classified as true or false.
- Simple and compound propositions, where compound propositions combine two or more simple propositions.
- Logical operations on propositions including negation, conjunction, disjunction, conditional, and biconditional statements.
- Truth tables as a way to systematically determine the truth value of compound propositions based on the truth values of their components.
Artificial Intelligence (AI) | Prepositional logic (PL)and first order predic...Ashish Duggal
The following are the topics in this presentation Prepositional Logic (PL) and First-order Predicate Logic (FOPL) is used for knowledge representation in artificial intelligence (AI).
There are also sub-topics in this presentation like logical connective, atomic sentence, complex sentence, and quantifiers.
This PPT is very helpful for Computer science and Computer Engineer
(B.C.A., M.C.A., B.TECH. , M.TECH.)
This document provides information about symbolic logic, including:
1. It defines propositions as statements that are either true or false, and assigns them the truth values of TRUE or FALSE.
2. It gives examples of propositions and non-examples, and asks the reader to determine if certain statements are propositions.
3. It introduces propositional variables, logical connectives like "and" and "or", and truth tables to evaluate compound propositions formed from variables and connectives.
Propositional logic is a good vehicle to introduce basic properties of logicpendragon6626
Propositional logic uses symbols and logical connectives to evaluate the validity of compound statements based on the validity of atomic statements. Natural deduction and resolution are deductive systems that use inference rules to prove statements. Natural deduction is sound and complete, while resolution is also complete. Propositional resolution can check validity by constructing a refutation tree, and linear resolution with Horn clauses is efficient for this task like the logic programming language Prolog.
The document discusses the square of opposition, a diagram used in classical logic to represent relationships between types of propositions. It outlines the four basic proposition forms - A (universal affirmative), E (universal negative), I (particular affirmative), O (particular negative) - and the logical relationships between them. Specifically, A and O are contradictory, as are E and I. A and E are contrary, I and O are subcontrary, and A propositions are subaltern to I propositions. However, modern logic rejects the assumption that all categories have members, so the square of opposition is updated to show only contradictory relationships.
1. The document discusses key concepts in propositional logic including negation, conjunction, disjunction, and exclusive disjunction.
2. Negation, conjunction, and disjunction are explained using examples and truth tables. Negation switches the truth value, conjunction requires both to be true, and disjunction requires at least one to be true.
3. Exclusive disjunction is introduced as giving true if one or the other is true but not both, as represented in its truth table.
With vocabulary
1. The Statements, Open Sentences, and Trurth Values
2. Negation
3. Compound Statement
4. Equivalence, Tautology, Contradiction, and Contingency
5. Converse, Invers, and Contraposition
6. Making Conclusion
This document summarizes Chapter 1 of the textbook "Discrete Mathematics" by R. Johnsonbaugh. It covers the topics of logic, proofs, and propositional logic. Key points include:
- Logic is the study of correct reasoning and is used in mathematics and computer science.
- A proposition is a statement that can be determined as true or false. Connectives like AND, OR, and NOT can combine propositions.
- Truth tables define the truth values of compound propositions formed from connectives.
- Quantifiers like "for all" and "there exists" are used to make universal and existential statements.
- A proof is a logical argument establishing the truth of a theorem using definitions, ax
This document discusses conditional statements and their equivalent forms. It covers the following key points:
- Conditional statements can be expressed in equivalent forms, such as "if p then q" or "q only if p".
- Related statements to a conditional include the converse ("if q then p"), inverse ("if not p then not q") and contrapositive ("if not q then not p").
- The converse and contrapositive of a conditional statement are logically equivalent to the original statement. The inverse is not logically equivalent.
- Examples are provided to demonstrate writing conditionals in different forms and identifying the converse, inverse and contrapositive of conditional statements.
This document discusses mathematical foundations of computer science. It covers topics such as statements (propositions), logic operators (NOT, AND, OR), compound statements, logical equivalence, conditional statements, and arguments. Specifically, it defines statements as sentences that are either true or false. It introduces logic operators and how they can be used to construct new statements from old ones. It also discusses logical forms such as truth tables and De Morgan's laws. Finally, it discusses conditional statements, logical equivalence, and valid arguments.
The document defines key concepts in propositional logic including propositions, truth tables, logical connectives like negation, conjunction, disjunction, implication, biconditional, and logical equivalences. It provides the definitions and truth tables for logical connectives and discusses examples like DeMorgan's laws and implications. Boolean variables and operations that correspond to logical connectives are also introduced.
This is the summary of Church Going. This is one of the poem of Philip Larkin. Philip Larkin is one of the most prominent poet of English Literature and Language.
This presentation discusses Gross National Product (GNP) and its utilization in Pakistan. GNP is defined as the total money value of all final goods and services produced by the residents of a country in one year. It is a flow concept that measures final output produced by a country's citizens. The components that make up GNP expenditures are consumption, investment, government expenditures, and net exports. A formula is provided that calculates GNP as the sum of these components plus net factor income from abroad. Data is then presented on Pakistan's GNP in 2018 which is broken down by sectors such as textiles, food, agriculture, transportation, services, and industrial sectors.
This document discusses the different forms and structures of verb tenses in English, including:
- There are three main tenses: present, past, and future. Each has indefinite, continuous, perfect, and perfect continuous forms.
- Examples are provided of sentence structures using subject+verb combinations for each tense, including affirmative, negative, interrogative, and WH- question forms.
- Structures covered include simple present and past, present and past continuous, present and past perfect, and future tenses.
The presentation topic was about prepositions. It was presented by Malik Muhammad Ali, a student with the roll number BSENL-17-25. The document provided basic information about the topic and presenter of an upcoming presentation.
this is the full history of philosophy both western and eastern philosophy in detailed and year wise tabular view. this is year by year history of philosophy and it is more precise one.
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Argument Forms
1. Bahauddin Zakariya University Multan – Bahadur Sub Campus, Layyah
Course: Philosophy
Presented to: Mr. Riaz Khan Dasti
Presented by: Malik Muhammad Ali
Roll no. BSENL-17-25
Topic: Argument Forms
2. Argument form:
The statement of the propositional calculus are propositions, they can
be combined to form logical arguments, complete with one or more
premises and a single conclusion that may follow validity from them.
For example A É B (D.B) É ~E (𝐴Ú𝐸) É (DºB)
𝐴
𝐵
𝐷.𝐵
~E
𝐴Ú𝐸
𝐷ºB
All the three arguments share a same structure. The main É is the first
premises of each statement, the second statement is the antecedent of
that statement; and the conclusion is the consequent. We can exhibit it
more clearly as p É q:
𝑝
𝑞
Many arguments are the substitution of a same argument from. We can
see from above example. If we change B . D then B . D will be the first
premises of second argument.
Testing Validity:
Recognizing individual arguments as substitution-instances of more
general argument forms are an important skill because, as we’ve
already seen, the validity of any argument depends solely upon its
logical form. It is also the form of argument when all the premises are
right but the conclusion is false. The same truth table is used for this to
determine the argument form. If all the possible combination of truth
table is true and it verifies the conclusion then it will be valid and it will
3. be a valid form of argument. If all the structure of premises is true but
the conclusion is false then it is invalid and argument form is invalid.
Modus Ponens:
Consider an example when we construct a truth-table that lits each of
the four combinations of truth-table. Where p É q
𝑝
𝑞
1st
Premise 2nd
Premise Conclusion
p q p É q p Q
T T T T T
T F F T F
F T T F T
F F T F F
In mathematics (calculus) it is also called implication or conditional. In
this the conclusion will only false when hypothesis is true and
conclusion is false.
Let’s have another table
1st premise 2nd Premise Conclusion
p q p É q P q
T T T T T
T F F F T
F T T T F
F F T F F
In this we have the both premises are true in third argument but
conclusion is false, it means sometimes our premises are true but
conclusion is false, the inference is invalid.
4. E.g. 1. If the moon is made of blue cheese, then pig fly.
2. The moon is made of blue cheese.
3. Therefore, pigs fly. This forms argument as follow as p → q
Modus Tollens:
It is the form of valid inference is Modus Tollen (denoted by M.T).
As p É q will be as
~q
~p
1st
premise 2nd
premise Conclusion
p q p É q ~q ~p
T T T F F
T F F T F
F T T F T
F F T T T
This statement shows that the argument is true when both they are
false but conclusion is true. In Calculus ( ~ ) means converse (opposite)
As p É q
~p
~q
1st
premise 2nd
premise Conclusion
p q p É q ~p ~q
T T T F F
T F F F T
F T T T F
F F T T T
In this, in third argument, both of the premises are true while the
conclusion is false. This shows the fallacy of denying the antecedent.
5. There is also another possibility that:
1. If the moon is made of blue cheese, then pig fly.
2. Pigs don’t fly.
3. Therefore, moon is not made of cheese.
In this argument will be valid because by seeing 1st
and 2nd
follows that
moon is not made up of cheese so argument is valid.
Hypothetical Syllogism:
1st
premise 2nd
premise Conclusion
P q r p É q q É r p É r
T T T T T T
T T F T F F
T F T F T T
T F F F T F
F T T T T T
F T F T F T
F F T T T T
F F F T T T
A larger truth table is required to demonstrate the validity of the
argument form called hypothetical syllogism. In this we take more than
one variable. p É q
p Ér
𝑞É 𝑟
But in this table truth table also expresses the validity of the
argument.
6. Disjunctive Syllogism:
p Ú q
~p
q
Then the table will be:
1st
premise 2nd
premise Conclusion
P q p Ú q ~p q
T T T F T
T F T F F
F T F T T
F F F T F
In Calculus Ú stands for union. If one of the argument p or q is true then
it will be true in 1st premise or p Ú q. but in this conclusion will bne true
in which sense when all the premises will be true.
For invalid inference, we have p Ú q
𝑝
~q
1st
premise 2nd
premise Conclusion
P q p Ú q P ~q
T T T T F
T F T T T
F T T F F
F F F F T
1st
row show that it is a possibility that conclusion will be false when
both the premise are true.
Argument Form:
In logic, the argument form or test form of an argument results from
replacing the different words, or sentences that make up the argument
7. With letters, along the line of algebra; the letters represent logical
variables.
Here is an example:
A
All humans are mortal. Socrates is human. Therefore, Socrates is
mortal. By re-writing the arguments.
B
All humans are mortal.
Socrates is human.
Therefore, Socrates is mortal.
Replace them with letters.
C
All S are P.
A is S.
Therefore a is P.
In this we have used S for human or humans, P for mortal and a for
Socrates. In this C is the original form of argument in A. we must have
to consider one thing that form of argument is what makes an
argument valid or cogent.
Logic Keys:
“Not” is symbolized as ¬
8. “If-then” is symbolized as →
“and” is symbolized as ˄
“or” is symbolized as ˅
“if and only if” is symbolized as ↔
The major task of logic is to deduct the logical consequences of a set of
sentences. In this order we have to identify the logic in logical
sentences.
If the premises are intended to provide conclusive support for the
conclusion, the argument is deductive one. If the premises are
intended to support the conclusion only to a lesser degree, the
argument is inductive one.
Immediate Inference:
The simplest possible arguments that can be constructed from
categorical propositions are those with one premise and, one
conclusion.
A: All A’s are B’s. All B’s are A’s
E: No A’s are B’s. No B’s are A’s.
I: Some A’s are not B’s. Some B’s are not A’s.
O: Some A’s are not B’s. Some B’s are not A’s.
i
No snakes are birds. Some pets are cats.
No birds are snakes.
9. Some pets are cats.
By immediate inferences we analyze the complex form of argument.
Categorical Syllogism:
The next more complex form of argument is one with two categorical
propositions as premises and one categorical proposition as conclusion.
In this first premise occurs and then the subject term of the conclusion
occurs in second premise, the argument is called a categorical
syllogism. There are basically three letters are used (A,E,I,O) is called
mod of syllogism. The possible moods are AAA, AIO, EIO & so on. In
these moods we have three terms S, M and P (subject, middle, and
predicate) arranged are called figure of syllogism.
Figure 1 Figure 2 Figure 3 Figure 4
M – P P – M M – P P – M
S – M S – M M – S M – S
S – P S – P S – P S – P
All P’s are M’s = All cantaloupes are fruits.
All M’s are S’s = All fruits are seed-bearers.
But syllogism must have to prove the validity of argument.
All P’s are M’s = No scientist is children.
All S’s are M’s = some infants are children.
But Some S’s are not P’s = some infants are not.