Introduction to Logic

• Logic is the formal systematic study of the principles of valid inference and correct reasoning
• It provides frameworks and tools to analyze and evaluate arguments, determine their validity, and construct sound and persuasive reasoning

Core Concepts in Logic

• Argument: A set of statements (premises) intended to support a conclusion
• Premise: A statement that is offered as evidence for the truth of the conclusion
• Conclusion: A statement that is claimed to follow from the premises
• Validity: A property of arguments where the conclusion necessarily follows from the premises
• Soundness: A property of arguments that are both valid and have true premises
• Inference: The process of drawing a conclusion from premises
• Fallacy: A flawed pattern of reasoning that leads to an invalid argument

Types of Logic

• Classical Logic: Includes propositional logic and predicate logic, based on principles such as the law of excluded middle and the law of non-contradiction

Propositional Logic

• Deals with propositions (statements that can be true or false) and logical connectives
• Proposition: A declarative statement that has a truth value (either true or false)
• Logical Connectives: Operators that combine or modify propositions
  • Conjunction (AND): Represented by ∧, true only if both propositions are true
  • Disjunction (OR): Represented by ∨, true if at least one proposition is true
  • Negation (NOT): Represented by ¬, reverses the truth value of a proposition
  • Implication (IF...THEN): Represented by →, false only if the first proposition is true and the second is false
  • Biconditional (IF AND ONLY IF): Represented by ↔, true if both propositions have the same truth value
• Truth Tables: Used to determine the truth value of compound propositions for all possible combinations of truth values of their constituent propositions

Predicate Logic

• Extends propositional logic by introducing predicates, quantifiers, and variables
• Predicate: A statement that involves variables and becomes a proposition when the variables are assigned specific values
• Variables: Symbols that represent objects or entities in a domain of discourse
• Quantifiers: Express the extent to which a predicate is true over a range of elements
  • Universal Quantifier (∀): "For all"
  • Existential Quantifier (∃): "There exists"
• Allows for more complex and nuanced expressions than propositional logic

Modal Logic

• Introduces modalities (e.g., necessity, possibility, belief, knowledge) to logical statements
• Modalities: Qualify the truth of a statement
  • Necessary: Represented by □, means "it is necessarily true that"
  • Possible: Represented by ◇, means "it is possibly true that"
• Used in philosophy, computer science, and linguistics to reason about knowledge, belief, time, and obligation

Deontic Logic

• Deals with moral and legal obligations and permissions
• Uses deontic operators to express concepts such as obligation, permission, and prohibition
  • Obligatory: It ought to be the case that
  • Permissible: It is permitted that
  • Forbidden: It is forbidden that
• Used in ethics, law, and artificial intelligence to formalize reasoning about moral and legal norms

Temporal Logic

• Used to reason about time and the order of events
• Introduces temporal operators to express when events occur relative to each other
  • Always: Always in the future
  • Eventually: Eventually in the future
  • Until: Occurs until occurs
• Used in computer science to verify the correctness of concurrent and reactive systems

Fuzzy Logic

• Deals with reasoning that is approximate rather than fixed and exact
• Introduces the concept of degrees of truth, allowing statements to be partially true
• Membership Functions: Assign a degree of membership (between 0 and 1) to elements in a fuzzy set
• Used in control systems, artificial intelligence, and decision-making processes

Informal Fallacies

• Errors in reasoning that are not formal logical errors but are still invalid or misleading
• Fallacies of Relevance: Premises are not relevant to the conclusion
  • Ad Hominem: Attacking the person making the argument rather than the argument itself
  • Appeal to Authority: Claiming something is true because an authority says so, without sufficient justification
  • Appeal to Emotion: Manipulating emotions to persuade rather than using logical reasoning
  • Straw Man: Misrepresenting an opponent's argument to make it easier to attack
• Fallacies of Ambiguity: Arise from the ambiguous use of language
  • Equivocation: Using a word or phrase in different senses within the same argument
  • Amphiboly: Ambiguity arising from grammatical structure
• Fallacies of Presumption: Involve unwarranted assumptions
  • Begging the Question: Assuming the conclusion in the premises
  • False Dilemma: Presenting only two options when more exist

Applications of Logic

• Mathematics: Used to formalize mathematical theories and prove theorems
• Computer Science: Used in programming languages, artificial intelligence, and formal verification
• Philosophy: Used to analyze arguments, explore metaphysical questions, and develop ethical theories
• Linguistics: Used to study the structure of language and meaning
• Law: Used to analyze legal arguments and construct legal reasoning systems
• Artificial Intelligence: Used in expert systems, knowledge representation, and automated reasoning

Proof Techniques

• Direct Proof: Directly showing that the conclusion follows from the premises
• Proof by Contradiction: Assuming the negation of the conclusion and deriving a contradiction
• Proof by Contraposition: Proving the contrapositive of the statement
• Proof by Induction: Proving a statement for a base case and then showing that if it holds for an arbitrary case, it also holds for the next case

Metalogic

• The study of the properties of logical systems themselves
• Includes topics such as completeness, soundness, decidability, and compactness
• Completeness: A logical system is complete if every valid formula can be proven within the system
• Soundness: A logical system is sound if every provable formula is valid
• Decidability: Whether there exists an effective method for determining whether a formula is valid
• Compactness: If every finite subset of a set of formulas has a model, then the entire set has a model

Axiomatic Systems

• Logical systems built from a set of axioms and inference rules
• Axioms: Basic statements that are assumed to be true without proof
• Inference Rules: Rules that allow the derivation of new statements from existing ones
• Used to formalize mathematical and logical theories

An overview of logic, the study of valid inference and correct reasoning. Key concepts include arguments, premises, conclusions, validity, and soundness. Explores classical logic, propositional logic, and predicate logic.