Chapter 2 Review Guide

How are statements about only-if statements translated into if-then form?

"p only if q" is the same (can be translated into) "if not q then not p"

What does the notation a ≤ x < b mean?

(a v x) ^ b

What are negations for the following forms of statements? -p∧q -p∨q -p→q

- ~p ∨ ~q - ~p ^ ~q - p ^ ~q

What is a tautology, and what is a contradiction?

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 contradiction is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables.

What does it mean for an argument to be sound?

An argument is called sound if and only if it is valid *and* all its premises are true.

How do you identify the logical form of an argument?

By identifying the premises and conclusions of the argument/statement

How do you test to see whether two statement forms are logically equivalent?

Construct a truth table and determine if the desired columns are the same.

What are converse error and inverse error?

Converse error: p →q q ∴ p Inverse error: p →q ∼p ∴∼q

What are the converse and inverse of a statement of the form "If p then q"?

Converse: "If q then p" Inverse: "If ~p then ~q"

How do you annotate a truth table to explain how it shows that two statement forms are or are not logically equivalent?

Draw arrows underneath the table linking the two together to show they are equivalent.

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 the contrapositive of a statement of the form "If p then q"?

If ~q then ~p

What does it mean to say that something is true only if something else is true?

Known as the biconditional of two statements, denoted by ↔

How do you test to see whether a given form of argument is valid?

Make a truth table and see if both columns are equivalent

How do you construct a truth table for a general compound statement?

Make columns headed p, q, r, p∧q, ∼r, and (p∧q) ∨∼ r. Enter the eight logically possible combinations of truth values for p, q, and r in the three left-most columns. Then fill in the truth values for p∧q and for ∼r. Complete the table by considering the truth values for (p∧q) and for ∼r and the definition of an or statement. Since an or statement is false only when both components are false, the only rows in which the entry is F are the third, fifth, and seventh rows because those are the only rows in which the expressions p∧q and ∼r are both false. The entry for all the other rows is T.

What are modus ponens and modus tollens?

Modus ponens: If p then q. p therefore q. Modus tollens: If p then q. ~q. therefore ~p.

Can an invalid argument have a true conclusion?

No, in order for a conclusion to be true its argument must be valid.

If a conditional statement is true, can its converse also be true?

No, its converse is not logically equivalent to the conditional statement.

Given a conditional statement and its contrapositive, converse, and inverse, which of these are logically equivalent and which are not?

Not: conditional and inverse; conditional and converse; conditional and contrapositive Are: converse and inverse; contrapositive and converse

What is the conjunction of statements p and q?

The conjunction of p and q is "p and q" denoted p ^ q.

How do you annotate a truth table to explain how it shows that an argument is or is not valid?

The corresponding columns of the truth table should be equivalent

What is the disjunction of p and q?

The disjunction of p and q is "p or q" denoted p ∨ q.

What are De Morgan's laws?

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 does it mean for a form of argument to be valid?

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 for two statement forms 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.

Can you express converses, inverses, and contrapositives of conditional statements in ordinary English?

Yes

Can a valid argument have a false conclusion?

Yes, a valid argument can have a false conclusion (unsound)

What is a statement?

a sentence that is either true or false

What is a conditional statement?

an if-then statement

How do you use valid forms of argument to solve puzzles such as those of Raymond Smullyan about knights and knaves?

idk

What is a biconditional statement?

if and only if

Given a conditional statement, what is its hypothesis (antecedent)? Conclusion (consequent)?

in a conditional statement "if p then q" p is the hypothesis and q is the conclusion

How are statements about necessary and sufficient conditions translated into if-then form?

necessary: r is a necessary condition for s means (translated to) "if s then r" sufficient: r is a necessary and sufficient condition for s means (translated to) "r if, and only if, s"

If p and q are statements, how do you symbolize "p but q" and "neither p nor q"?

p but q: p ^ q neither p nor q: ~p and ~q

What does it mean to say that something is a necessary condition for something else?

r is a necessary condition for s means: "if not r then not s"

What does it mean to say that something is a sufficient condition for something else?

r is a sufficient condition for s means: "if r then s"

What are the truth table definitions for ~p, p ^ q, p ∨ q, p -> q, and p <-> q?

too much to type here lol

Which of modus ponens, modus tollens, converse error, and inverse error are valid and which are invalid?

valid: modus ponens and modus tollens invalid: converse error and inverse error

What is exclusive or?

when or is used in its exclusive sense, the statement " p or q" means "p or q but not both" or "p or q and not both p and q," which translates into symbols as (p∨q) ∧∼ (p∧q). This is sometimes abbreviated p⊕q or p XOR q.

What is the order of operations for the logical operators?

~ negations first ∧, ∨and and or second →, ↔ third

What is the double negative property?

∼(∼p) ≡ p

How is Theorem 2.1.1 used to show that two statement forms are logically equivalent?

∼(∼p∧q)∧(p∨q) ≡ (∼(∼p)∨∼ q)∧(p∨q) by De Morgan's laws ≡ (p∨∼ q)∧(p∨q) by the double negative law ≡ p∨(∼q ∧q) by the distributive law ≡ p∨(q∧∼ q) by the commutative law for∧ ≡ p∨c by the negation law ≡ p by the identity law.