Propositional Logic

• Logic is the study of reasoning and argumentation
• Used to determine good and bad arguments

Propositional Logic

• Studies statements and their logical relationships
• A proposition is a declarative sentence that can be either true or false, but not both
• Propositional variables represent propositions (e.g., P, Q, R)
• Logical connectives combine propositions to form compound propositions

Common Logical Connectives

• Negation (¬): "not P" - true if P is false, and false if P is true
• Conjunction (∧): "P and Q" - true if both P and Q are true; otherwise, false
• Disjunction (∨): "P or Q" - true if either P or Q (or both) are true; false only if both are false
• Implication (→): "if P, then Q" or "P implies Q" - false only if P is true and Q is false; otherwise, true; P is the hypothesis/antecedent, Q is the conclusion/consequent
• Biconditional (↔): "P if and only if Q" or "P is equivalent to Q" - true if P and Q have the same truth value (both true or both false); false otherwise

Truth Tables

• Used to define the meaning of logical connectives by specifying the truth value of a compound proposition for all possible truth values of its constituent propositions
• Negation:
  • P | ¬P
  • T | F
  • F | T
• Conjunction:
  • P | Q | P ∧ Q
  • T | T | T
  • T | F | F
  • F | T | F
  • F | F | F
• Disjunction:
  • P | Q | P ∨ Q
  • T | T | T
  • T | F | T
  • F | T | T
  • F | F | F
• Implication:
  • P | Q | P → Q
  • T | T | T
  • T | F | F
  • F | T | T
  • F | F | T
• Biconditional:
  • P | Q | P ↔ Q
  • T | T | T
  • T | F | F
  • F | T | F
  • F | F | T

Tautology, Contradiction, and Contingency

• Tautology: A compound proposition that is always true, regardless of the truth values of its variables
• Contradiction: A compound proposition that is always false, regardless of the truth values of its variables
• Contingency: A compound proposition that is neither a tautology nor a contradiction

Logical Equivalence

• Two propositions are logically equivalent if they have the same truth value in all possible cases
• Can be shown using truth tables or logical equivalences (e.g., De Morgan's Laws)
• De Morgan's Laws:
  • ¬(P ∧ Q) ≡ ¬P ∨ ¬Q
  • ¬(P ∨ Q) ≡ ¬P ∧ ¬Q

Predicate Logic

• Extends propositional logic to include predicates, variables, quantifiers, and objects
• Predicate: A statement that contains variables and may be true or false depending on the values of those variables, it defines a property or relationship
• Variable: A symbol representing an object or entity
• Quantifier: Expresses the extent to which a predicate is true over a range of elements

Types of Quantifiers

• Universal Quantifier (∀): "for all" - ∀x P(x) means P(x) is true for every x in the domain
• Existential Quantifier (∃): "there exists" - ∃x P(x) means P(x) is true for at least one x in the domain

Negating Quantified Statements

• ¬(∀x P(x)) ≡ ∃x ¬P(x) (Not all x have property P is equivalent to: There exists an x that does not have property P)
• ¬(∃x P(x)) ≡ ∀x ¬P(x) (It is not the case that there exists an x with property P is equivalent to: All x do not have property P)

Logical Fallacies

• Errors in reasoning that invalidate an argument
• Formal Fallacy: A defect in the structure of an argument which makes the argument invalid
• Informal Fallacy: A defect in the content of an argument which makes the argument weak

Common Logical Fallacies

• Ad Hominem: Attacking the person making the argument instead of the argument itself
• Appeal to Authority: Arguing that a statement is true because an authority figure said it, without providing other evidence
• Appeal to Ignorance: Arguing that a statement is true because it has not been proven false, or vice versa
• Bandwagon Fallacy: Arguing that something is true because it is popular
• Straw Man: Misrepresenting an opponent's argument to make it easier to attack
• False Dilemma: Presenting only two options when more exist
• Hasty Generalization: Drawing a conclusion based on insufficient evidence
• Post Hoc Ergo Propter Hoc: Assuming that because one event preceded another, the first event caused the second

Deductive Reasoning

• Starts with general statements (premises) and reaches a specific conclusion
• If the premises are true and the argument is valid, the conclusion must be true
• Example:
  • Premise 1: All men are mortal.
  • Premise 2: Socrates is a man.
  • Conclusion: Therefore, Socrates is mortal.

Inductive Reasoning

• Starts with specific observations and arrives at a general conclusion
• The conclusion is likely but not guaranteed to be true, even if the premises are true
• Example:
  • Observation: Every swan I have seen is white.
  • Conclusion: Therefore, all swans are white. (This is false, as black swans exist)
• Inductive reasoning is used to form hypotheses and theories

Venn Diagrams

• Diagram that utilizes overlapping circles to represent sets and their relationships
• The universal set is typically represented by a rectangle enclosing all sets under consideration.
• Each circle represents a set, and the overlapping areas represent the intersection of sets

Shading in Venn Diagrams

• Shading is used to represent set operations visually
• A ∪ B (A union B): Shade everything in set A and set B
• A ∩ B (A intersection B): Shade the region where set A and set B overlap
• A' (Complement of A): Shade everything outside of set A within the universal set
• A - B (A minus B): Shade the region of A that does not overlap with B
• Shading can be used to determine set identities and simplify set expressions