Philosophy Chapter 7

Translate the logical operators: - . v -> <->

- = not (negation) . = and (conjunction) v = or (disjunction) -> = if-then (conditional) <-> = if and only if (biconditional)

What are the four logically significant relationships statements might stand in to each other? - - - -

-Equivalence -Contradictoriness -Consistency -Inconsistency

Abbreviate this statement: Roses are not blue.

-R

What are the three logically significant categories that a single statement may belong? - - -

-Tautology -Contradiction -Contingency

Two statements are _________ ________ if and only if they agree in truth value on every assignment of truth values to their atomic components.

are logically equivalent

The "<->" sign (called the double-arrow) is used to symbolize ______________.

biconditionals

The following are stylistic variants of ______________: Mary is a teenager if and only if she is from 13 to 19 years of age. Mary is a teenager just in case she is from 13 to 19 years of age. Mary's being a teenager is a necessary and sufficient condition for Mary's being from 13 to 19 years of age.

biconditionals

We can symbolize atomic statements within compound statements using _____________ letters.

capital letters.

The "->" sign (called the arrow) is used to symbolize ________________.

conditionals.

A __________ is true if both __________ are true; otherwise it is false.

conjunction; conjunct

The statements composing a conjunction re called ________.

conjuncts.

A statement is __________ if and only if it is true on some assignments of truth values to its atomic components and false on others.

contingent

A statement is a _________ if and only if it is false on every assignment of truth values to its atomic components.

contradiction

A _____________ is false if both its ___________ are false; otherwise, it is true.

disjunction; disjuncts Ex: "Either Abraham Lincoln or Andrew Jackson was born in 1809" (or both were). This is true because Lincoln was born in 1809

Symbolize the following compounds and name the type of compound: It is false that Christopher is a Buddhist or a Hindu. It is not true that if Joshua finishes his dissertation this year, he is guaranteed a tenure-track job. It is not the case that both the Orioles will win and the Mariners will win.

negation of a disjunction : -(B v H) negation of a conditional : -(F -> T) negation of a conjunction: -(O . M)

Fill in the truth tables: p/-p ----- ?/? ?/? p q / p . q ------------ ? ? / ? ? ? / ? ? ? / ? ? ? / ? p q / p v q ------------ ? ? / ? ? ? / ? ? ? / ? ? ? / ? p q / p -> q ------------ ? ? / ? ? ? / ? ? ? / ? ? ? / ? p q / p <-> q ------------ ? ? / ? ? ? / ? ? ? / ? ? ? / ?

p/-p ----- T/F F/T p q / p . q ------------ T T / T T F / F F T / F F F / F p q / p v q ------------ T T / T T F / T F T / T F F / F p q / p -> q ------------ T T / T T F / F F T / T F F / T p q / p <-> q ------------ T T / T T F / F F T / F F F / T

When we assign each atomic statement a distinct capital letter, we provide what we will call a ______ __ ____________.

scheme of abbreviation

A _________ ___________ is a condition that guarantees that a statement is true (or that a phenomenon will occur).

sufficient condition

A statement is a ____________ if and only if it is true on every assignment of truth values to its atomic component.

tautology

What type of compound do these logical operators stand for: tilde: dot: vee: arrow: double-arrow:

tilde = negation dot = conjunction vee = disjunction arrow = conditional double-arrow = biconditional

Translate these logical operators: tilde: dot: vee: arrow: double-arrow:

tilde = not dot = and vee = or arrow = if-then double-arrow = if and only if

The "-" symbol, called ________, is used to translate the English word "___" and its stylistics variants. This type of command is called a ________.

tilde; not; negation

A compound statement is _____-__________ if its truth value is completely determined by the truth value of the atomic statements that compose it.

truth-functional

T/F, in a truth table, a statement is a contradiction if it is false on every row.

True

T/F, in a truth table, a statement is a tautology if it is true on every row.

True

T/F, in a truth table, a statement is contingent if it is true on some rows and false on other rows.

True

T/F, in a truth table, two statements are logically equivalent if they have the same truth value on each row.

True

T/F, nothing counts as a WFF unless it can be demonstrated to be one by applications of the symbolic expression conditions.

True

T/F, symbolizing an argument enables us to apply certain powerful techniques to determine validity.

True

T/F, the definition of logical inconsistency allows for two statements to both be false.

True

T/F, the essential insight behind abbreviated truth tables is this: If there is an assignment of truth values to one row of a truth table, making all the premises true while the conclusion false, then the argument form in question is invalid.

True

T/F, the tautologies of statement logic belong to a class of statements that are true simply by virtue of their form.

True

T/F, the truth value of certain compound statements is a function of the truth value of the atomic statements that make them up.

True

T/F, the word "unless" can be translated by mean of the arrow as well as the vee.

True

T/F, we should assume that statements of the form Either A or B are inclusive disjunctions.

True

T/F, we should treat biconditionals as conjunctions of material conditionals.

True

T/F, you must at least provide a column of truth values under every logical operator and under every atomic statement that stands alone as a premise or the conclusion.

True

T/F, with the truth tables we are looking for a row, and hence a possible situation, in which the premises are all true but the conclusion is false.

True (and if we can find such a row (or situation), then the argument form is invalid.

T/F, one false conjunct renders an entire conjunction false.

True, Ex: "St. Augustine and Abraham Lincoln were both born in 354" is false, for although St. Augustine was born in 354, Lincoln was not.

T/F, a statement can be a compound even if it has only one statement as a component.

True, Ex: It's not the case that Ben Jonson wrote Hamlet. This statement is a compound statement with the phrase "It's not the case that" and B as the atomic statement.

T/F, negations are truth-functional compounds

True, Ex: the statement "Bertrand Russell was born in 1872" is true; so it's negation, "Bertrand Russell was not born in 1872," is false

T/F, if the premises of an argument when made into a conjunction, form a contingent statement, then if the conclusion of the argument is a contradiction, the argument is invalid.

True, because the conclusion will be false in every row, while the premises will be true in at least one row.

T/F, we cannot introduce a dot, vee, arrow, or double-arrow without parentheses around the resulting expression.

True.

T/F, we may alternate brackets and parenthesis because multiple parenthesis can become confusing.

True.

T/F, we use parenthesis to form a punctuation.

True.

Using the scheme of abbreviation, rewrite this compound statement: It is not the case that Ben Jonson wrote Hamlet.

It is not the case that B.

____________ ___________ are used to symbolize English expressions.

Logical operators.

Remembering Truth Conditions: __________: Always the opposite __________: Always false except when both _________ are true __________: Always true except when both ______ are false __________: Always true except when the antecedent is true and the consequent is false __________: Always true except when its two constituent statements have different truth values.

Negation Conjunction; conjuncts Disjunction: disjuncts Material Conditional Material Biconditional

The ________ _________ captures that part of the meaning of the English conditional that is essential for the validity of the basic argument forms of statement logic.

material conditional

The truth table of a __________ ___________ affirms the invalidity of the fallacy of denying the antecedent.

material conditional

A _________ __________ is a condition that, if lacking, guarantees that a statement is false ( or that a phenomenon will not occur).

necessary condition

Label the main and minor operators in each: -(B v H) -(F -> T) -(O . M)

The tilde "-" is the main operator because it governs (B v H), the minor operator is the "v" because it governs B and H The tilde "-" is the main operator because it governs (F -> T), the minor operator is the "->" because it governs F and T The tilde "-" is the main operator because it governs (O . M), the minor operator is the "." because it governs O and M

How will you know if the premises of an argument are inconsistent?

There will be no row in the truth table in which all of the premises are true.

What is the main operator of these statements: E . -F (G v H) . K (L -> M) . (N v O)

These are all conjunctions and the main operator is the dot.

A __________ __________ is a conjunction of two material conditionals; it is true when its constituent statements have the same truth value and false when they differ in truth value.

A material biconditional

A ____________ ____________ is a conditional that is false only when its antecedent is true and its consequent is false; otherwise, it is true.

A material conditional

A ___________ _________ is a lowercase letter that serves as a placeholder for any statement - for example p, q, r, s.

A statement variable

A ______-________ ___________ (WFF) is a grammatically correct symbolic expression.

A well-formed formula

An _____________ statement is one that does not have any other statement as a component. A _______________ statement is one that has at least one atomic statement as a component.

An atomic statement A compound statement

An ______________ of statement logic is any sequence of symbols in this vocabulary (A-Z and -,v,.,->.<->).

An expression of statement logic

Is this an atomic statement or a compound statement? Roses are red.

Atomic statement

Using the scheme of abbreviation, rewrite this compound statement: China has a large population, and Luxembourg has a small population.

C and L.

Is this an atomic statement or a compound statement? Either Palermo is the capital of Sicily, or Messina is the capital of Sicily.

Compound statement

Using the scheme of abbreviation, rewrite this compound statement: The Democrats win if and only if the Republicans quarrel.

D if and only if R.

Using the scheme of abbreviation, rewrite this compound statement: Either Palermo is the capital of Sicily, or Messina is the capital of Sicily.

Either P or M.

T/F, a limitation of the truth table method is that it becomes unwieldy as arguments become longer.

True

T/F, a negation has the opposite truth value of the statement negated.

True

T/F, a triple dot is used to mark a conclusion.

True

T/F, all arguments having contradiction among their premises are unsound because contradictions are always false.

True

T/F, an argument that has a contradiction among its premises is a valid argument

True

T/F, biconditionals are, in effect, conjunctions of conditionals.

True

T/F, every argument whose conclusion is a tautology is valid - regardless of the content of the premises.

True

T/F, any argument that has a tautology as its premise but a contingent statement as its conclusion is valid.

False, any argument that has a tautology as its premise but a contingent statement as its conclusion is INVALID

T/F, any argument with logically consistent premises will be valid yet unsound.

False, any argument with logically inconsistent premises will be valid yet unsound.

T/F, if a biconditional statement is a tautology, then its two constituent statements are logically equivalent.

True

T/F, if an argument has a tautology as its conclusion, then there is no way to assign truth values so that the conclusion is false.

True

T/F, placing parenthesis around a compound then putting a tilde on the outside of the parenthesis makes it clear that the atomic statement is being negated, not the compound.

False, placing parenthesis around a compound then a tiled outside the parenthesis makes it clear that the COMPOUND is being negated not just the atomic statement.

T/F, the dot can be used to convey relationships: Mike and Kirsten are married. William and Peter are twins.

False, the dot does not convey relationships.

T/F, the dot can convey temporal order.

False, the dot does not convey temporal order.

T/F, if at least one premise is a contradiction (or if the premises are inconsistent), then there is no way to assign values so that the premises are all true, and a complete truth table may be needed to establish this.

True

T/F, if you find a row where the premises are true but the conclusion is false then the argument form is invalid.

True

T/F, the dot can correctly translate every use of the English word "and."

False, the dot does not correctly translate every use of the English word "and." Ex: Stuart climbed Mount Baker and looked inside the sulfur cone. Stuart looked inside the sulfur cone and climbed Mount Baker. The first sentence indicates temporal order and the dot does not convey temporal order.

Using the scheme of abbreviation, rewrite this compound statement: If Sheboygan is in Wisconsin, then Sheboygan is in the U.S.A.

If S, then U.

___________ ___________ is when there is three statements that cannot all be true.

Inconsistent triad

Order the summary of truth-table method ? Look for a row where all the premises are true and the conclusion is false. Assuming you've done everything correctly up to this point, if there is one, the argument is invalid; if not, it's valid. ?. Assign truth values mechanically -? The number of rows for atomic statements that you need is 2n where n is the number of atomic statements ? In the case of complex compound statements, work out the truth values of simpler compounds first, then work your way "outward" to the main logical operator. -? Start assigning truth values to atomic statements in columns by first assigning truth values to the far right statement: alternate Ts and Fs in the column beneath it. The next column to the left: alternate pairs of Ts and Fs. the next column to the left: alternate quadruples of Ts and Fs. the next column to the left: alternate groups of eight, and so on (doubling). -? Place the capital letters of atomic statements in sequence from left to right in the order that they appear in our symbolization. ? Identify the main logical operator of each premise and the conclusion

1. Assign truth values mechanically - Place the capital letters of atomic statements in sequence from left to right in the order that they appear in our symbolization. -The number of rows for atomic statements that you need is 2n where n is the number of atomic statements -Start assigning truth values to atomic statements in columns by first assigning truth values to the far right statement: alternate Ts and Fs in the column beneath it. The next column to the left: alternate pairs of Ts and Fs. the next column to the left: alternate quadruples of Ts and Fs. the next column to the left: alternate groups of eight, and so on (doubling). 2. Identify the main logical operator of each premise and the conclusion 3. In the case of complex compound statements, work out the truth values of simpler compounds first, then work your way "outward" to the main logical operator. 4. Look for a row where all the premises are true and the conclusion is false. Assuming you've done everything correctly up to this point, if there is one, the argument is invalid; if not, it's valid.

A symbolic expression is a WFF under the following conditions: 1. 2. 3. 4. 5. 6.

1. Capital letters (which stand for atomic statements) are WFFs. 2. If p is a WFF, then so is -p. 3. If p and q are WFFs, then so is (p . q). 4. If p and q are WFFs, then so os (p v q). 5. if p and q are WFFs, then so is (p -> q). 6. If p and q are WFFs, then so is (p <-> q).

What are the two abbreviated truth table principles? 1. 2.

1. If there is any assignment of values in which the premises are all true and the conclusion is false, then the argument is invalid. 2. If more than one assignment of truth values will make the conclusion false, then consider each such assignment; if each assignment that makes the conclusion false makes at least one premise false, then the argument is valid.

Two (or more) statements are __________ ________ if and only if they are both (all) true on some assignment of truth values to their atomic components.

logically consistent

In a truth table, two statements are _________ ___________ if they have a different truth value on every row.

logically contradictory

Two statements are ___________ __________ if and only if they disagree in truth value on every assignment of truth values to their atomic components.

logically contradictory

In a truth table, two (or more) statements are _________ _________ if they have the same truth value on at least one row.

logically cosistent

Two statements are ________ _______ if they have the same truth value on each row.

logically equivalent

In a truth table, two statements are _________ __________ if there is no row on which they are both (all) true.

logically inconsistent

Two (or more) statements are _________ ____________ if and only if they are never both (all) true on any assignment of truth values to their atomic components.

logically inconsistent

The __________ logical operator in a compound statement is the one that governs the largest component or components of a compound statement. A _________ logical operator governs smaller componets.

main logical operator; minor logical operator.

A ________ ___________ is true when its constituent statements have the same truth value and false when they differ in truth value.

material biconditional