> Here is a summary of the first chapter on Sentential Logic from the lecture notes: > The first chapter introduces the basics of mathematical logic, focusing on the meaning of logical connectives like "and", "or", "implies", "if and only if", etc. Please define each of these and motivate them per their relevance in logic. > The chapter describes a formal language for these logical notions. It defines the symbols used, including negation (¬), implication (→), and sentential variables (Si for i ≥ 3). Expressions are defined as finite sequences of these symbols. Why do we need such a rigorous treatment of these? What does it avoid, or allow, or clarify? > Sentential formulas are defined recursively as meaningful expressions built up from the symbols according to certain rules. The principle of induction on sentential formulas is introduced. What are these rules? Expand on and explain induction. > The notion of a sentential assignment, which assigns truth values to sentential variables, is defined. This allows defining the meaning or truth value of any sentential formula under a given assignment. As best you can without writing out a table, please explain the concept of a truth assignment and what they do. > Tautologies are defined as formulas that are true under every assignment. Some common tautologies are listed. Give an example of a tautology that is less mathematical, something that can help relate it to something someone likely already knows. > A proof system for sentential logic is introduced, consisting of three axiom schemas and one inference rule (modus ponens). The notion of a proof of a formula φ from a set Γ of formulas is defined. Please state the axioms, and explain what an axiom is, and the role they play within a system of mathematics briefly. Several theorems are proved: - If Γ proves φ then φ is a tautology - The Deduction Theorem, which relates proofs of φ → ψ to proofs of ψ from φ - Tautologies are precisely the formulas provable from the empty set of formulas In summary, the chapter rigorously develops the syntax and semantics of propositional logic, introduces a proof system, and proves its soundness and completeness with respect to the semantic notion of tautology. This lays the logical foundations for the set theory developed in later chapters. # Expanded summary Here is an expanded summary addressing your points: ~~The logical connectives "and" (∧), "or" (∨), "implies" (→), and "if and only if" (↔) are~~ The logical connectives "and" (wedge), "or" (vee), "implies" (right arrow), and "if and only if" (double right arrow) are fundamental in logic. "And" represents a conjunction, where both parts must be true. "Or" represents a disjunction, where at least one part must be true. "Implies" represents a conditional, where if the first part (antecedent) is true, the second part (consequent) must be true. "If and only if" represents a biconditional, where both parts must have the same truth value. These allow expressing complex logical relationships between propositions. The rigorous treatment of the formal language avoids ambiguities and allows precise reasoning. It clarifies the exact meaning of formulas and enables analyzing their logical structure. This is necessary for defining concepts like tautologies and developing proof systems. Sentential formulas are built recursively: - Sentential variables are formulas - If φ is a formula, so is ¬φ - If φ and ψ are formulas, so are (φ → ψ) and (φ ∨ ψ) Induction on formulas allows proving properties for all formulas by proving the base cases (variables) and the inductive cases corresponding to the formula construction rules. A truth assignment gives a truth value (T/F) to each sentential variable. The truth values of complex formulas are then determined by the meanings of the connectives. E.g., φ ∧ ψ is true iff both φ and ψ are true; φ ∨ ψ is true iff at least one of φ or ψ is true, etc. So a truth assignment determines the truth value of every formula. A simple example of a tautology is "It is raining or it is not raining." This is always true regardless of whether it's actually raining - one of the two options must be the case. More generally, a formula of the form φ ∨ ¬φ is always a tautology. The axioms of the proof system are: - φ → (ψ → φ) - (φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)) - (¬φ → ¬ψ) → (ψ → φ) An axiom is a starting assumption taken to be true. In a proof system, axioms and inference rules allow deriving new true formulas. The soundness theorem says everything provable is true (a tautology), while the completeness theorem says everything true is provable. So the axioms and rules precisely capture logical truth. The Deduction Theorem is a metatheorem that relates proofs of an implication φ → ψ to proofs of ψ from the assumption φ. It reveals a deep connection between implication and proof. In summary, this chapter rigorously develops propositional logic, ensuring it rests on a solid, unambiguous foundation. The formalisms enable precise analysis of logical truth and proof, which are fundamental for mathematical reasoning and later developments in set theory. # Written out for text to speech Here are the rewritten pieces with symbols spelled out in English: The logical connectives "and" (wedge), "or" (vee), "implies" (right arrow), and "if and only if" (double right arrow) are If phi is a formula, so is not phi If phi and psi are formulas, so are (phi right arrow psi) and (phi vee psi) phi right arrow (psi right arrow phi) (phi right arrow (psi right arrow chi)) right arrow ((phi right arrow psi) right arrow (phi right arrow chi)) (not phi right arrow not psi) right arrow (psi right arrow phi) The Deduction Theorem is a metatheorem that relates proofs of an implication phi right arrow psi to proofs of psi from the assumption phi. It reveals a deep connection between implication and proof.