Chapter 2: Propositional Logic: Syntax and Semantics
Summary
- Propositional logic is the logic of propositions and their inferential relations.
- A proposition is a statement, often expressed by a declarative sentence, which has a truth value.
- PL is the formal object language of propositional logic.
- The syntax of PL specifies its vocabulary and rules for making formulas.
- The vocabulary of PL includes uppercase letters, five operators— tilde (~), dot (•), vel (˅), horseshoe (⊃), triple-bar (≡)—and punctuation marks (), [], {}.
- Logical operators are tools for combining propositions or terms.
- Unary operators apply only to a single proposition. They never relate or connect two propositions. Negation, ~, is the only unary operator in PL.
- Binary operators relate or connect two propositions. All operators of PL, except negation, are binary operators.
- Negation, ~, is the logical operator used to translate ‘not’, ‘it is not the case that’, ‘it is false that’, and related terms. It is the only unary operator.
- Conjunction, •, is the logical operator used to translate ‘and’, ‘but’, and related terms. It is a binary operator. The formulas joined by a conjunction are called conjuncts.
- Disjunction, ˅, is the logical operator used to translate ‘or’, ‘unless’, and related terms. It is a binary operator. The formulas joined by a disjunction are called disjuncts.
- Material implication, ⊃, is the logical operator used to translate conditionals, ‘if . . . then . . . statements’, and related terms. It is a binary operator. The formula preceding the ⊃ is called the antecedent; the formula following the ⊃ is called the consequent. The order of the antecedent and consequent is significant; S ⊃ P is not logically equivalent to P ⊃ S.
- The biconditional, ≡, is the logical operator used for ‘if-and-only-if’, and related terms. It is a binary operator. The biconditional is a conjunction of a conditional with its converse; ‘A ≡ B’ is short for ‘(A ⊃ B) • (B ⊃ A)’.
- Formation rules specify how to combine the vocabulary of a language into well-formed formulas (wffs).
- A wff, or well-formed formula, is any logical symbol or string of symbols that are constructed properly. A wff is analogous to a grammatically correct sentence.
- PL has four formation rules. PL1: A single capital English letter is a wff. PL2: If α is a wff, so is ~α. PL3: If α and β are wffs, then so are: (α • β), (α ˅ β), (α ⊃ β), and (α ≡ β). PL4: These are the only ways to make wffs.
- An atomic formula of PL is any wff formed by a single use of PL1: a single capital English letter.
- A complex formula of PL is a wff formed in any way besides a single use of PL1.
- The main operator is the last operator added to a wff according to the formation rules.
- The semantics of PL specifies the rules for interpreting the symbols and formulas of the language.
- Bivalent logic is a two-valued logic. Every statement is interpreted as either true or false, and not both. The logic of PL is interpreted as bivalent.
- Compositionality is a semantic principle stating that the meaning of a complex sentence is determined by the meanings of its component parts. The language of PL is compositional.
- The truth value of a complex proposition is the truth value of its main operator.
- A truth table shows the truth value for a complex proposition given any truth values of its component propositions.
- The basic truth table is a way of representing the semantic rules governing each operator by showing the truth value of the operation, given any possible distribution of truth values of the component propositions.
- Negation, ~, is interpreted as true when the formula to which it applies is false; it is interpreted as false when the formula to which it applies is true.
- Conjunction, •, is interpreted as true only when both conjuncts are true; otherwise it is false.
- Disjunction, ˅, is interpreted as false only when both disjuncts are false; otherwise it is true.
- Material implication, ⊃, is interpreted as false only when the antecedent is true and the consequent is false; otherwise it is true.
- The biconditional, ≡, is interpreted as true when the component statements share the same truth value, (when they are both true or both false); otherwise it is false.
- A sufficient condition is something adequate or enough (though not necessarily required) for something else to obtain. In a material implication, the truth of the antecedent is the sufficient condition of the truth of the consequent.
- A necessary condition is something required (though not necessarily adequate or enough) for something else to obtain. In a material implication, the truth of the consequent is the necessary condition of the truth of the antecedent.
- A tautology is a proposition that is true in every row of its truth table. They are the logical truths of PL.
- Logical truths are propositions that are true on any interpretation.
- A contingency is a proposition that is true in some rows of its truth table and false in others.
- A contradiction is a proposition that is false in every row of its truth table.
- Two or more propositions are logically equivalent when they have the same truth values in every row of their truth tables.
- Two propositions are contradictory when they have opposite truth values in every row of their truth tables.
- Two or more propositions are consistent when they are true in at least one common row of their truth tables.
- Two propositions are inconsistent when there is no row of their truth tables in which both statements are true.
- A valuation is an assignment of truth values to simple component propositions.
- An invalid argument is one in which it is possible for true premises to yield a false conclusion.
- A valid argument has no row of its truth table in which the premises are true and the conclusion is false. In a valid argument, if the premises are true then the conclusion must be true.
- A counterexample to an argument is a valuation that makes the premises true and the conclusion false.
- If an argument has a counterexample, it is invalid. If an argument is invalid, it has at least one counterexample.
- A consistent valuation is an assignment of truth values to atomic propositions that makes a set of propositions all true. If it is not possible to make each statement true, then the set is inconsistent.