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CHAPTER TWO: SENTENCE LOGIC

• §2.1: The Fundamentals

  • The basic unit is a declarative sentence (truth-value-bearing). Questions or imperatives do not have truth-values.

  • A sentence’s truth value can depend on the occasion of utterance; some sentences have context-dependent truth-values; for present purposes, truth values are two-valued and context-independence is assumed where convenient.

  • English-only distinctions such as 'statement' vs 'sentence' are often ignored in practice; for this book, sentences and statements are interchangeable.

  • Some sentences (e.g., “It is false that …”) are truth-functional operators (negation, conjunction, disjunction, conditional, biconditional) acting on truth-values.

• §2.2: Truth Functions

  • A truth function determines the truth value of a complex sentence from the truth values of its components.

  • Truth-functional operators include negation, conjunction, disjunction, the conditional (horseshoe), and the biconditional (triplebar).

  • Examples of truth-function operators in English: conjunction, disjunction, conditional statements, and negation words like ‘It is false that …’.

  • Some phrases (e.g., ‘It is obvious that …’) are not truth-functional operators; such phrases cannot be treated with the truth-functional machinery of sentence logic.

• §2.3: Sentence Letters and Symbols

  • Use uppercase Roman letters as sentence letters to stand for simple sentences (e.g., A, B, C, …).

  • There are 26 letters; subscripts can extend the set, but readers are advised to avoid unnecessary subscripts for readability.

  • Five symbols are introduced to form sentences from others:

  • Negation: ~ or, in typographic convention, ¬; rough equivalents: 'It is false that A'.

  • Conjunction: ^ or sometimes ∧; rough equivalent: 'A and B'.

  • Disjunction: v or ∨; rough equivalent: 'A or B' (inclusive sense unless context dictates exclusive).

  • Horseshoe (conditional): > or →; rough equivalent: 'If A, then B'.

  • Triplebar (biconditional): = or ↔; rough equivalent: 'A if and only if B'.

  • Important reading note: There are no brackets around ~A; tilde negation applies to the formula immediately to its right.

• §2.4: Metalanguage

  • Object language is the symbolic language; metalanguage is English (with some symbols) used to discuss the object language.

  • The book’s object language will not be used to talk about itself; metatheory is acknowledged but not deeply explored in this volume.

  • Distinguishing object language vs metalanguage helps to avoid semantic pitfalls when discussing rules and proofs.

  • The English language often contains expressions that do not function as truth-functional operators; these are noted to clarify the scope of sentence logic.

  • Notion of truth tables is the primary tool for defining operators; the main operator of a WFF is the last operator added in its construction.

• §2.5: Well-Formed Formulas (WFFs)

  • WFFs are the valid combinations of symbols per the rules of the language.

  • An atomic formula is a single sentence letter (e.g., A, B).

  • If A is a WFF, then ~A is also a WFF.

  • Complex formulas like (A ^ B) are WFFs only if the main operator and the subformulas are well-formed; extraneous brackets must not be used unnecessarily because they can create ambiguity (but brackets can be dropped when unambiguous).

  • The main operator of a formula is the last operator added when building the formula; it determines how subformulas are interpreted.

  • Examples and constraints on WFFs include: (A ^ B) is a WFF; (A v B) is a WFF; (A v B) is ambiguous without brackets; proper use of brackets is necessary to remove ambiguity (e.g., A v (B v C) vs (A v B) v C).

• §2.6–§2.13: TRANSLATION AND EXERCISES (Overview)

  • Translation practice: translating English sentences and arguments into the symbolic language, with attention to structure and the proper placement of sentence letters, operators, and parentheses.

  • Emphasis on breaking complex sentences into simpler components, replacing pronouns with explicit nouns, and ensuring translations are truth-functionally equivalent to the intended meaning.

  • Examples span negation, conjunction, disjunction, the horseshoe, and the triplebar; also, translating complicated sentences and disjunctive/conditional structures with correct bracketing.

• §2.14–§2.15: DEFINING THE OPERATORS; TRUTH TABLES

  • Operators are defined by truth tables, specifying all possible truth-value combinations for subformulas.

  • Tilde (~) defines negation; conjunction (^) defines ‘and’; disjunction (v) defines ‘or’ (inclusive by default); horseshoe (>) defines ‘if … then’; triplebar (=) defines ‘if and only if’.

  • The truth tables illustrate how each operator yields a truth-value for all input combinations. The memory aid: horseshoe is truth-functional, but many English conditional sentences are not purely truth-functional; nonetheless, it is used to translate English conditionals.

• §2.16–§2.18: TRUTH-TABLES; TYPES OF FORMULAS; TRUTH-FUNCTIONAL EQUIVALENCES

  • Truth-functionally true: a formula that is true across all possible truth-value assignments to its atomic constituents.

  • Truth-functionally false: true in no assignment (always false).

  • Truth-functionally indeterminate: true in some assignments, false in others.

  • Equivalences: two formulas are truth-functionally equivalent if the biconditional between them is truth-functionally true; for example, A ∧ B is equivalent to B ∧ A; and many equivalences can be shown by a single truth table.

• §2.19–§2.24: ARGUMENTS; TRANSLATING ARGUMENTS; EXERCISES

  • An argument is a set of premisses and a conclusion; an argument is truth-functionally valid if there is no row in the corresponding truth table where all the premises are true and the conclusion is false.

  • Solutions often involve translating the argument, placing premisses and conclusion on a single truth table, and checking for a counterexample row.

  • Emphasis on constructing fresh translations rather than relying on memorized solutions.

• §2.25: SUMMARY OF TOPICS

  • Quick reference to terminology and topics covered in Chapter 2, including truth values, truth-functions, WFFs, metalanguage vs object language, and the five primary operators.

CHAPTER THREE: DERIVATIONS IN SENTENCE LOGIC

• §3.1: DERIVATION BASICS

  • A derivation is a numbered sequence of formulas with a vertical scope line indicating the extent of the derivation.

  • Occurrences are justified by four kinds of lines: premiss, assumption, previously derived formula, or rule of inference.

  • Derivations parallel truth-table checks: a valid derivation exists iff the conclusion follows from the premises under the rules; derivations can be extended with more structure (scope lines).

• §3.2: CONJUNCTION INTRODUCTION

  • If you have A and B, you can infer A ∧ B on the same scope line.

  • The two conjuncts must be on the same scope line as the new conjunction; the main operator is ∧; line references show prerequisites.

  • Examples illustrate that the main operator of the new formula must be the latest operator introduced.

• §3.3: CONJUNCTION ELIMINATION

  • From A ∧ B, you may derive either A or B on the same scope line.

• §3.4: NEGATION ELIMINATION

  • From ¬¬A, you may derive A. Two consecutive tildes can be removed one at a time; rule is limited to a formula whose main operator is a tilde immediately followed by another formula.

• §3.5: DISJUNCTION INTRODUCTION

  • From A, you can derive A ∨ B; from B, you can derive A ∨ B; the main operator must be the disjunction of the resulting formula.

• §3.6: HORSESHOE ELIMINATION

  • From A and A → B, you may derive B.

• §3.7: TRIPLEBAR ELIMINATION

  • From A = B and B = C, you may infer A = C; similarly, from A = B and A = C you may infer B = C, etc.

• §3.8–§3.7. etc.: EXERCISES; CONSTRUCTING DERIVATIONS; ASSUMPTIONS; REITERATION

  • Emphasis on careful use of scope lines, correct application of rules to main operators, and avoiding illegitimate reiterations.

• §3.9–§3.13: CONSTRUCTING DERIVATIONS; HORSESHOE INTRODUCTION; TRIPLEBAR INTRODUCTION

  • Horseshoe Introduction (I): to derive A → B, assume A on a new scope line and derive B on that scope line; end line and use I to introduce the horseshoe as the main operator.

  • Triplebar Introduction (I): to derive A = B, derive A from B at one scope and B from A at another scope; use =I to infer A = B.

• §3.15: NEGATION INTRODUCTION

  • Indirectly derive ¬A by assuming A, deriving a contradiction, and applying ¬I.

• §3.16: DISJUNCTION ELIMINATION

  • From A ∨ B, and subderivations of C from A and from B, infer C. Requires two subderivations on two branches.

• §3.17–§3.23: EXERCISES; ADDITIONAL STRATEGIES; INDIVIDUAL DERIVATION PROBLEMS; CATEGORICAL DERIVATIONS

  • Extensive worked examples illustrate multiple layers of subderivations, the need for correct barrier management, and the interplay between introduction/elimination rules.

• §3.24: SUMMARY OF TOPICS

  • Quick recap of derivation rules, strategies, and the relationship between derivations and truth tables.

CHAPTER FOUR: PREDICATE LOGIC

• §4.1: INDIVIDUALS AND PREDICATES

  • Predicate logic generalizes sentence logic by incorporating individuals and predicates.

  • Constants: lowercase letters a, b, c for individuals; standard use of these constants in translations.

  • Predicates: uppercase letters followed by a lower-case (e.g., P(a)) denote properties or relations; predicates differ from sentence letters since their arguments make them open sentences when combined with constants.

  • Example translations show how subjects and predicates combine (e.g., Peter is tired: Tp; Greek is hard: Gx). Mixing sentence logic with predicate logic is possible but not common; predicate logic is usually used throughout.

• §4.2: ONE-PLACE AND MULTI-PLACE PREDICATES

  • One-place predicates: F x; multi-place predicates: R x y (two-place), etc.

  • For relationships (e.g.,