👁️‍🗨️Formal Logic I Unit 14 – Formal Logic: Applications Across Disciplines

Key Concepts and Terminology

• Formal logic studies the structure and validity of arguments using precise symbolic notation
• Propositions are declarative sentences that can be either true or false
• Logical connectives ($\wedge$, $\vee$, $\rightarrow$, $\leftrightarrow$, $\neg$) combine propositions to form compound statements
• Logical equivalence means two statements have the same truth value under all interpretations
• Tautologies are statements that are always true regardless of the truth values of their components
• Contradictions are statements that are always false regardless of the truth values of their components
• Validity refers to an argument where the conclusion necessarily follows from the premises
• Soundness refers to a valid argument with true premises

Foundations of Formal Logic

• Formal logic has roots in ancient Greek philosophy, particularly the works of Aristotle
• Propositional logic deals with the logical relationships between propositions without considering their internal structure
• First-order logic extends propositional logic by introducing quantifiers and predicates to analyze the internal structure of propositions
• Set theory provides a foundation for mathematical logic and is used to define logical concepts
• Boolean algebra is a branch of algebra that deals with the manipulation of logical expressions
• Formal systems consist of a set of axioms and inference rules used to derive theorems
• Consistency means a formal system does not lead to contradictions
• Completeness means a formal system can prove all true statements within its domain

Logical Operators and Symbols

• Conjunction ($\wedge$) represents "and" and is true when both propositions are true
• Disjunction ($\vee$) represents "or" and is true when at least one proposition is true
  • Inclusive disjunction allows for both propositions to be true
  • Exclusive disjunction (XOR) is true when exactly one proposition is true
• Implication ($\rightarrow$) represents "if...then" and is false only when the antecedent is true and the consequent is false
• Biconditional ($\leftrightarrow$) represents "if and only if" and is true when both propositions have the same truth value
• Negation ($\neg$) represents "not" and inverts the truth value of a proposition
• Universal quantifier ($\forall$) represents "for all" and is true when a predicate holds for every element in a domain
• Existential quantifier ($\exists$) represents "there exists" and is true when a predicate holds for at least one element in a domain

Argument Structure and Validity

• An argument consists of premises (assumed to be true) and a conclusion (claimed to follow from the premises)
• Modus ponens is a valid argument form: If $P \rightarrow Q$ and $P$ are true, then $Q$ must be true
• Modus tollens is a valid argument form: If $P \rightarrow Q$ and $\neg Q$ are true, then $\neg P$ must be true
• Hypothetical syllogism is a valid argument form: If $P \rightarrow Q$ and $Q \rightarrow R$ are true, then $P \rightarrow R$ must be true
• Disjunctive syllogism is a valid argument form: If $P \vee Q$ and $\neg P$ are true, then $Q$ must be true
• Constructing truth tables can determine the validity of an argument by checking all possible truth value assignments
  • An argument is valid if and only if there is no case where the premises are true and the conclusion is false

Proof Techniques and Methods

• Direct proof assumes the premises are true and uses logical steps to derive the conclusion
• Proof by contradiction assumes the negation of the conclusion and shows it leads to a contradiction with the premises
• Proof by cases divides the problem into exhaustive cases and proves the conclusion holds for each case
• Mathematical induction proves a statement holds for all natural numbers by proving a base case and an inductive step
• Proof by counterexample disproves a statement by finding an instance where it does not hold
• Proof by construction provides an explicit example or procedure to demonstrate the existence of an object or solution
• Proof by exhaustion verifies a statement by checking all possible cases (feasible for small finite sets)

Applications in Different Fields

• Computer science uses formal logic in algorithm design, programming languages, and artificial intelligence
  • Propositional logic is used in circuit design and boolean expressions
  • First-order logic is used in knowledge representation and automated theorem proving
• Mathematics relies on formal logic to define concepts, state theorems, and construct rigorous proofs
  • Set theory, number theory, and abstract algebra heavily utilize formal logic
• Philosophy employs formal logic to analyze arguments, paradoxes, and the foundations of reasoning
  • Modal logic extends formal logic to deal with concepts like necessity, possibility, and time
• Linguistics uses formal logic to study the structure and meaning of natural language
  • Predicate logic can represent the logical form of sentences and analyze entailment relations
• Law and politics use formal logic to construct and evaluate legal arguments and policies
  • Deontic logic formalizes concepts like obligation, permission, and prohibition

Common Fallacies and Pitfalls

• Affirming the consequent fallacy incorrectly concludes $P$ from $P \rightarrow Q$ and $Q$
• Denying the antecedent fallacy incorrectly concludes $\neg Q$ from $P \rightarrow Q$ and $\neg P$
• Begging the question (circular reasoning) fallacy assumes the conclusion in the premises
• Equivocation fallacy uses a term with multiple meanings in different parts of the argument
• False dilemma fallacy presents limited options as if they were exhaustive
• Ad hominem fallacy attacks the character of the arguer instead of addressing the argument
• Straw man fallacy misrepresents an opponent's argument to make it easier to refute
• Appeal to authority fallacy relies on the opinion of an authority figure without proper justification

Practice Problems and Examples

• Translate natural language arguments into formal logic notation and assess their validity
  • "If it rains, then the streets are wet. The streets are wet. Therefore, it rains." (invalid, affirming the consequent)
  • "If the car is out of gas, then it won't start. The car starts. Therefore, the car is not out of gas." (valid, modus tollens)
• Construct truth tables to determine the logical equivalence of statements
  • $P \rightarrow Q$ and $\neg P \vee Q$ are logically equivalent (have the same truth table)
• Prove theorems using various proof techniques
  • Prove by induction that the sum of the first $n$ odd numbers is $n^2$
• Identify and explain fallacies in real-world arguments
  • "Either you're with us, or you're against us." (false dilemma)
  • "My opponent's argument is wrong because they have a history of lying." (ad hominem)