Section 1.1: Individual Constants (3)
- Every individual constant must name an (actually existing) object.
- No individual constant can name more than one object.
- An object can have more than one name, or no name at all.
Section 1.2: Predicate Symbols (2)
- Every predicate symbol comes with a single, fixed “arity,” a number that tells you how many names it needs to form an atomic sentence.
- Every predicate is interpreted by a determinate property or relation of the same arity as the predicate.
Section 1.3: Atomic Sentences (3)
- Atomic sentences are formed by putting a predicate of arity n in front of n names (enclosed in parentheses and separated by commas).
- Atomic sentences are built from the identity predicate, =, using infix notation: the arguments are placed on either side of the predicate.
- The order of the names is crucial in forming atomic sentences.
Section 2.1: Valid & Sound Arguments (4)
- An argument is a series of statements in which one, called the conclusion, is meant to be a consequence of the others, called the premises.
- An argument is valid if the conclusion must be true in any circumstances in which the premises are true.
- We say that the conclusion of a logically valid argument is a logical consequence of its premises.
- An argument is sound if it is valid and the premises are all true.
Section 2.2: Methods of Proof (2)
- A proof of a statement S from premises P1, … , Pn is a step-by-step demonstration which shows that S must be true in any circumstances in which the premises P1, … , Pn are all true.
- Informal and formal proofs differ in style, not in rigor.
Section 2.2: Methods of Proof - Principles of the Identity Relation (4)
- = Elim: If b = c, then whatever holds of b holds of c. This is also known as the indiscernibility of identicals.
- = Intro: Sentences of the form b = b are always true (in FOL). This is also known as the reflexivity of identity.
- Symmetry of Identity: If b = c, then c = b.
- Transitivity of Identity: If a = b and b = c, then a = c
Section 2.5: Demonstrating Nonconsequence (2)
- To demonstrate the invalidity of an argument with premises P1, … , Pn and conclusion Q, find a counterexample: a possible circumstance that makes P1, … , Pn all true but Q false.
- Such a counterexample shows that Q is not a consequence of P1, … , Pn.
Section 3.1: Negation Symbol (3)
- If P is a sentence of FOL, then so is ¬P.
- The sentence ¬P is true if and only if P is not true.
- A sentence that is ether atomic or the negation of an atomic sentence is called a literal.
Section 3.2: Conjunction Symbol (2)
- If P and Q are sentences of FOL, then so is P ∧ Q.
- The sentence P ∧ Q is true if and only if both P and Q are true.
Section 3.3: Disjunction Symbol (2)
- If P and Q are sentences of FOL, then so is P ∨ Q.
- The sentence P ∨ Q is true if and only if P is true or Q is true (or both are true).
Section 3.6: Equivalent Ways of Saying Things - DeMorgan’s Laws (3)
- Double negation: ¬¬P ⇔ P
- DeMorgan: ¬(P ∧ Q) ⇔ (¬P ∨ ¬Q)
- DeMorgan: ¬(P ∨ Q) ⇔ (¬P ∧ ¬Q)
Section 4.1: Tautologies & Logical Truths - Let S be a sentence of FOL built up from atomic sentences by means of truth-functional connectives alone. A truth table for S shows how the truth of S depends on the truth of its atomic parts. (4)
- S is a tautology if and only if every row of the truth table assigns TRUE to S.
- If S is a tautology, then S is a logical truth (that is, logically necessary).
- Some logical truths are not tautologies.
- S is TT-possible if and only if at least one row of the truth table assigns TRUE to S.
Section 4.2: Logical & Tautological Equivalence - Let S and S’ be sentences of FOL built up from atomic sentences by means of truth-functional connectives alone. To test for tautological equivalence, we construct a joint truth table for the two sentences. (3)
- S and S’ are tautologically equivalent if and only if every row of the joint truth table assigns the same values to S and S’.
- If S and S’ are tautologically equivalent, then they are logically equivalent.
- Some logically equivalent sentences are not tautologically equivalent.
Section 4.3: Logical & Tautological Consequence - Let P1, … , Pn and Q be sentences of FOL built up from atomic sentences by means of truth functional connectives alone. Construct a joint truth table for all of theses sentences. (3)
- Q is a tautological consequence of P1, … , Pn if and only if every row that assigns TRUE to each of P1, … , Pn also assigns TRUE to Q.
- If Q is a tautological consequence of P1, … , Pn, then Q is also a logical consequence of P1, … , Pn.
- Some logical consequences are not tautological consequences.
