➕Logic and Formal Reasoning Unit 3 – Propositional Logic: Arguments & Proofs

Key Concepts in Propositional Logic

• Propositional logic focuses on the logical relationships between propositions, which are declarative sentences that can be either true or false
• Connectives (logical operators) such as negation ($\neg$), conjunction ($\wedge$), disjunction ($\vee$), implication ($\rightarrow$), and equivalence ($\leftrightarrow$) combine propositions to form compound statements
• Truth values (true or false) are assigned to propositions and determine the truth value of compound statements based on the truth tables of the connectives
• Tautologies are propositions that are always true regardless of the truth values of their constituent propositions, while contradictions are always false
• Logical equivalence between two propositions means they have the same truth value for all possible truth value assignments of their constituent propositions
• Propositional logic provides a foundation for analyzing and evaluating arguments based on their logical structure and the truth values of their premises and conclusions
• Propositional logic has limitations, such as the inability to represent relationships between objects or quantify over them, which are addressed by more advanced logical systems like first-order logic

Building Blocks: Propositions and Connectives

• Propositions are the fundamental units of propositional logic and represent declarative sentences that can be either true or false
• Atomic propositions are the simplest form of propositions and cannot be broken down into smaller propositions (e.g., "The sky is blue")
• Compound propositions are formed by combining atomic propositions using logical connectives
• Negation ($\neg$) is a unary connective that reverses the truth value of a proposition (e.g., $\neg p$ is true when $p$ is false)
• Conjunction ($\wedge$) is a binary connective that represents the logical "and" and is true only when both propositions are true (e.g., $p \wedge q$)
• Disjunction ($\vee$) is a binary connective that represents the logical "or" and is true when at least one of the propositions is true (e.g., $p \vee q$)
  • Inclusive disjunction allows for both propositions to be true, while exclusive disjunction (XOR) requires exactly one proposition to be true
• Implication ($\rightarrow$) is a binary connective that represents a conditional statement, where the truth of the first proposition (antecedent) implies the truth of the second proposition (consequent) (e.g., $p \rightarrow q$)
• Equivalence ($\leftrightarrow$) is a binary connective that represents a biconditional statement, where two propositions have the same truth value (e.g., $p \leftrightarrow q$)

Argument Structure and Validity

• An argument in propositional logic consists of premises (propositions assumed to be true) and a conclusion (a proposition that follows from the premises)
• The structure of an argument determines its validity, which is the property of the conclusion being true whenever the premises are true
• Modus ponens is a valid argument form where if $p$ is true and $p \rightarrow q$ is true, then $q$ must be true
• Modus tollens is a valid argument form where if $p \rightarrow q$ is true and $q$ is false, then $p$ must be false
• Hypothetical syllogism is a valid argument form where if $p \rightarrow q$ and $q \rightarrow r$ are true, then $p \rightarrow r$ must be true
• Disjunctive syllogism is a valid argument form where if $p \vee q$ is true and $\neg p$ is true, then $q$ must be true
• Constructing valid arguments requires using the rules of inference and ensuring that the structure of the argument is correct
• Invalid arguments may have true premises and a false conclusion, or they may have a structure that does not guarantee the truth of the conclusion given the truth of the premises

Truth Tables and Logical Equivalence

• Truth tables are a method for determining the truth value of a compound proposition based on the truth values of its constituent propositions
• Each row in a truth table represents a possible combination of truth values for the constituent propositions, and the final column represents the truth value of the compound proposition for that combination
• Truth tables can be used to determine the logical equivalence of two propositions by comparing their truth values for all possible combinations of truth values of their constituent propositions
• De Morgan's laws state that $\neg(p \wedge q)$ is logically equivalent to $\neg p \vee \neg q$, and $\neg(p \vee q)$ is logically equivalent to $\neg p \wedge \neg q$
• The law of double negation states that $\neg(\neg p)$ is logically equivalent to $p$
• The law of the excluded middle states that $p \vee \neg p$ is a tautology
• The law of contradiction states that $p \wedge \neg p$ is a contradiction
• Logical equivalence is an important concept in propositional logic, as it allows for the simplification and manipulation of complex propositions while preserving their truth values

Rules of Inference

• Rules of inference are principles that allow for the derivation of new propositions (conclusions) from existing propositions (premises) in a logically valid way
• Modus ponens (affirming the antecedent) states that if $p$ and $p \rightarrow q$ are true, then $q$ must be true
• Modus tollens (denying the consequent) states that if $p \rightarrow q$ is true and $q$ is false, then $p$ must be false
• Hypothetical syllogism (chain rule) states that if $p \rightarrow q$ and $q \rightarrow r$ are true, then $p \rightarrow r$ must be true
• Disjunctive syllogism (process of elimination) states that if $p \vee q$ is true and $\neg p$ is true, then $q$ must be true
• Conjunction introduction states that if $p$ is true and $q$ is true, then $p \wedge q$ must be true
• Conjunction elimination states that if $p \wedge q$ is true, then both $p$ and $q$ must be true
• Disjunction introduction states that if $p$ is true, then $p \vee q$ must be true for any proposition $q$
• Applying rules of inference correctly is crucial for constructing valid arguments and proofs in propositional logic

Proof Techniques and Strategies

• Proofs in propositional logic demonstrate the validity of an argument by showing that the conclusion follows logically from the premises using rules of inference
• Direct proof starts with the premises and applies rules of inference to derive the conclusion step by step
  • Each step in a direct proof must be justified by citing the rule of inference or logical equivalence used
• Indirect proof (proof by contradiction) assumes the negation of the conclusion and shows that this leads to a contradiction with the premises, thus proving the original conclusion by reductio ad absurdum
• Proof by cases (case analysis) involves breaking down a proposition into distinct cases and proving each case separately
• Proof by induction is not applicable in propositional logic, as it deals with finite, unordered propositions rather than infinite sequences or recursively defined structures
• Proof by equivalence involves transforming a proposition into a logically equivalent form that is easier to prove
• Proof by contrapositive involves proving $\neg q \rightarrow \neg p$ instead of $p \rightarrow q$, as they are logically equivalent
• Choosing the appropriate proof technique for a given problem is a skill that develops with practice and exposure to various types of propositions and arguments

Common Fallacies and Pitfalls

• Fallacies are flawed arguments that may appear convincing but are logically invalid
• Affirming the consequent fallacy incorrectly concludes $p$ from $p \rightarrow q$ and $q$, ignoring the possibility of other conditions that could imply $q$
• Denying the antecedent fallacy incorrectly concludes $\neg q$ from $p \rightarrow q$ and $\neg p$, ignoring the possibility of other conditions that could imply $q$
• Begging the question (circular reasoning) fallacy occurs when the premise and the conclusion are essentially the same, providing no new information
• Equivocation fallacy occurs when a term is used with different meanings in the premises and the conclusion, leading to an invalid argument
• False dilemma (false dichotomy) fallacy presents two options as if they were the only possibilities, ignoring other alternatives
• Slippery slope fallacy argues that a small step will inevitably lead to a chain of events culminating in a significant (usually negative) outcome without sufficient justification
• Avoiding fallacies requires carefully analyzing the structure and content of arguments, ensuring that the premises provide genuine support for the conclusion and that the reasoning is valid

Applications in Real-World Reasoning

• Propositional logic provides a foundation for analyzing and evaluating arguments in various domains, such as philosophy, law, science, and everyday reasoning
• In philosophical arguments, propositional logic can be used to assess the validity of moral, epistemological, or metaphysical claims
• Legal reasoning often relies on propositional logic to determine the consistency and validity of arguments based on evidence, precedents, and legal principles
• Scientific reasoning employs propositional logic to formulate and test hypotheses, derive predictions, and draw conclusions from experimental data
• In everyday life, propositional logic can help individuals make sound decisions by evaluating the logical consistency of arguments and claims encountered in media, advertisements, and personal interactions
• Propositional logic also forms the basis for more advanced logical systems, such as first-order logic and modal logic, which are used in fields like computer science, artificial intelligence, and linguistics
• Understanding propositional logic can improve critical thinking skills and the ability to construct and evaluate arguments in various contexts
• While propositional logic has its limitations, such as the inability to represent relations between objects or quantify over them, it remains a powerful tool for analyzing and reasoning about the world around us