Chapter 2: The Logic of Compound Statements

What is a *contradication*?

A *contradication* is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables. A statement whose form is a contradication is a *contradictory statement*.

What is a *tautology*?

A *tautology* is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables. A statement whose form is a tautology is a *tautological statement*.

If it is not possible to show hypothesis true and the conclusion false at the same time is called what?

A conditional statement that is true by virtue of the fact that its hypothesis is false is often called *vacuously true* or *true by default*.

What is a *syllogism* and its components?

An argument form consisting of two premises and a conclusion is called a *syllogism*. The first and second premises are called the *major premise* and *minor premise*, respectively.

What is *∼p*?

Given a statement p, the sentence "∼p" is read "not p" or "It is not the case that p" and is called the *negation of p*.

What is *p ∧ q*?

Given another statement q, the sentence "p ∧ q" is read "p and q" and is called the* conjunction of p and q*.

What is the *biconditional of p and q*?

Given statement variables p and q, the*biconditional of p and q* is "p if, and only if, q" and is denoted p ↔ q. It is true if both p and q have the same truth values and is false if p and q have opposite truth values. The words if and only if are sometimes abbreviated iff.

What are the variables of a statement called?

If p and q are statement variables, the *conditional* of q by p is "If p then q" or "p implies q" and is denoted p → q. It is false when p is true and q is false; otherwise it is true. We call p the *hypothesis* (or *antecedent*) of the conditional and q the *conclusion* (or *consequent*).

What is the definition of *conjunction*?

If p and q are statement variables, the *conjunction* of p and q is "p and q," denoted p ∧ q. It is true when, and only when, both p and q are true. If either p or q is false, or if both are false, p ∧ q is false.

What is the definition of *disjunction *?

If p and q are statement variables, the *disjunction *of p and q is "p or q," denoted p ∨ q. It is true when either p is true, or q is true, or both p and q are true; it is false only when both p and q are false.

What is a *biconditional*?

It p and q are statements, p *only if* q means "if not q then not p," or, equivalently, "if p then q."

What is the *converse* of a conditional?

Suppose a conditional statement of the form "If p then q" is given. The *converse*is "If q then p." Symbolically, The *converse*of p → q is q → p. A conditional statement and its converse are not logically equivalent. The converse and the inverse of a conditional statement are logically equivalent to each other.

What is the *inverse* of a conditional?

Suppose a conditional statement of the form "If p then q" is given. The *inverse*is "If ∼p then ∼q." Symbolically, The *inverse*of p → q is ∼p → ∼q. A conditional statement and its inverse are not logically equivalent. The converse and the inverse of a conditional statement are logically equivalent to each other.

What is the *contrapositive* of a conditional?

The *contrapositive* of a conditional statement of the form "If p then q" is If ∼q then ∼p. Symbolically, The contrapositive of p → q is ∼q → ∼p.

What is happens in *De Morgan's Law?

The negation of an and statement is logically equivalent to the or statement in which each component is negated. The negation of an or statement is logically equivalent to the and statement in which each component is negated.

what is *p ∨ q*?

The sentence "p ∨ q" is read "p or q" and is called the *disjunction of p and q*.

In logic, which terms are undefined?

The words sentence, true, and false are the initial undefined terms.

How is *valid* used for arguments?

To say that an argument form is *valid* means that no matter what particular statements are substituted for the statement variables in its premises, if the resulting premises are all true, then the conclusion is also true. To say that an argument is *valid* means that its form is valid.

What does it mean to be *logically equivalent*?

Two statement forms are called *logically equivalent* if, and only if, they have identical truth values for each possible substitution of statements for their statement variables. The logical equivalence of statement forms P and Q is denoted by writing P ≡ Q. Two statements are called *logically equivalent* if, and only if, they have logically equivalent forms when identical component statement variables are used to replace identical component statements.

p but q

means p and q

neither p nor q

means ∼p and ∼q

What is *modus ponens*?

*Modus ponens* is a form of syllogism in logic. It has the following form: If p then q. p ∴ q Latin meaning "method of affirming" (the conclusion is an affirmation)

What is *modus tollens*?

*modus tollens* has the following form: If p then q. ∼q ∴ ∼p Latin meaning "method of denying" (the conclusion is a denial).

Theorem 2.1.1 Logical Equivalences

1. Commutative laws: p ∧ q ≡ q ∧ p 2. Associative laws: (p ∧ q) ∧ r ≡ p ∧ (q ∧ r) 3. Distributive laws: p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) 4. Identity laws: p ∧ t ≡ p p ∨ c ≡ p 5. Negation laws: p ∨ ∼p ≡ t p ∧ ∼p ≡ c 6. Double negative law: ∼(∼p) ≡ p 7. Idempotent laws: p ∧ p ≡ p p ∨ p ≡ p 8. Universal bound laws: p ∨ t ≡ t p ∧ c ≡ c 9. De Morgan's laws: ∼(p ∧ q) ≡ ∼p ∨ 10. Absorption laws: p ∨ (p ∧ q) ≡ p 11. Negations of t and c: ∼t ≡ c ∼c ≡ t

Order of Operations for Logical Operators

1. ∼ Evaluate negations first. 2. ∧, ∨ Evaluate ∧ and ∨ second. When both are present, parentheses may be needed. 3. →,↔ Evaluate → and ↔ third. When both are present, parentheses may be needed.

What is *fallacy*? Describe three types.

A *fallacy* is an error in reasoning that results in an invalid argument. Three common fallacies are using *ambiguous premises*, and treating them as if they were unambiguous, *circular reasoning* (assuming what is to be proved without having derived it from the premises), and *jumping to a conclusion* (without adequate grounds).

What is *rule of inference*?

A *rule of inference* is a form of argument that is valid. Thus modus ponens and modus tollens are both rules of inference. The following are additional examples of rules of inference: Generalization Specialization Elimination Transitivity Proof by Division into Cases

What is a *statement form*?

A *statement form* (or *propositional form*) is an expression made up of statement variables (such as p, q, and r) and logical connectives (such as ∼, ∧, and ∨) that becomes a statement when actual statements are substituted for the component statement variables.

What is a *statement*?

A *statement* (or *proposition*) is a sentence that is true or false but not both.

What is an *argument* and its componets?

An *argument* is a sequence of statements, and an *argument form* is a sequence of statement forms. All statements in an argument and all statement forms in an argument form, except for the final one, are called *premises* (or *assumptions* or *hypotheses*). The final statement or statement form is called the *conclusion*. The symbol ∴ , which is read "therefore," is normally placed just before the conclusion.

What does *sound* and *unsound* mean?

An argument is called *sound* if, and only if, it is valid and all its premises are true. An argument that is not sound is called *unsound*.

What is the definition of *negation*?

If p is a statement variable, the *negation* of p is "not p" or "It is not the case that p" and is denoted ∼p. It has opposite truth value from p: if p is true, ∼p is false; if p is false, ∼p is true.

What are *necessary* and *sufficient* conditions?

If r and s are statements: r is a *sufficient*condition for s means "if r then s." r is a *necessary*condition for s means "if not r then not s." r is a necessary condition for s also means "if s then r." r is a necessary and sufficient condition for s means "r if, and only if, s."

What is the *Contradiction Rule*?

If you can show that the supposition that statement p is false leads logically to a contradiction, then you can conclude that p is true.

What is a *truth table*?

The *truth table* for a given statement form displays the truth values that correspond to all possible combinations of truth values for its component statement variables.