Propositional and Predicate Logic

• Logic is the study of reasoning, differentiating sound from unsound arguments

Propositional Logic

• Deals with propositions (statements that can be true or false) and their relationships
• Propositions are represented by variables (e.g., P, Q, R)
• Logical operators connect propositions to form compound propositions
• Truth tables define the meaning of logical operators by showing the truth value of a compound proposition for all possible truth values of its components
• Propositional logic is also known as sentential logic

Predicate Logic

• Extends propositional logic to include predicates, variables, and quantifiers
• Predicates are properties or relations that can be applied to objects
• Variables represent objects in a domain of discourse
• Quantifiers specify the quantity of objects that satisfy a predicate
• Universal quantification (∀) asserts that a predicate is true for all objects in the domain
• Existential quantification (∃) asserts that a predicate is true for at least one object in the domain
• Predicate logic is also known as first-order logic

Logical Fallacies

• Errors in reasoning that make an argument invalid or unsound
• Formal fallacies are errors in the structure of an argument
• Informal fallacies are errors in the content or context of an argument
• Common fallacies include:
  • Ad hominem: attacking the person instead of the argument
  • Appeal to authority: claiming something is true because an authority said so, without proper justification
  • Straw man: misrepresenting an opponent's argument to make it easier to attack
  • False dilemma: presenting only two options when more exist
  • Appeal to ignorance: claiming something is true because it hasn't been proven false, or vice versa
  • Hasty generalization: drawing a conclusion based on insufficient evidence
  • Post hoc ergo propter hoc: assuming that because one event followed another, the first event caused the second
  • Bandwagon fallacy: arguing that something is true because it is popular

Formal Proof Techniques

• Methods for demonstrating the validity of an argument using formal rules of inference
• Natural deduction involves deriving the conclusion from the premises using a set of inference rules
• Common inference rules include:
  • Modus ponens: If P, then Q. P is true. Therefore, Q is true
  • Modus tollens: If P, then Q. Q is false. Therefore, P is false
  • Hypothetical syllogism: If P, then Q. If Q, then R. Therefore, if P, then R
  • Disjunctive syllogism: P or Q. P is false. Therefore, Q is true
  • Addition: P. Therefore, P or Q
  • Simplification: P and Q. Therefore, P
  • Conjunction: P. Q. Therefore, P and Q
  • Resolution: (P or Q) and (¬Q or R). Therefore, P or R
• Axiomatic systems define a set of axioms and inference rules from which theorems can be derived

Logical Operators

• Symbols or words used to connect or modify propositions
• Common logical operators include:
  • Negation (¬): reverses the truth value of a proposition
  • Conjunction (∧): true if both propositions are true, otherwise false
  • Disjunction (∨): true if at least one proposition is true, otherwise false
  • Implication (→): true unless the first proposition is true and the second is false
  • Biconditional (↔): true if both propositions have the same truth value, otherwise false
• Operator precedence determines the order in which operators are applied in a compound proposition (¬ > ∧ > ∨ > → > ↔)