Chapter 02 Summary

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.