logic test 3

Summary of the Abbreviated Truth Table Method

1. After placing the argument in a truth table, determine whether there are multiple ways in which the conclusion can be false. 2. If there is just one way, place an F under the (main operator of the) conclusion and a T under the (main operator of) each premise. a. To show invalidity, uniformly assign Ts and Fs to all of the components of the conclusion and the premises; write Ts and Fs under the atomic statements on the left of the table. b. To show validity, uniformly assign Ts and Fs to all of the components of the conclusion and the premises; write a backslash under (the main operator of) the premise you were led to say was both true and false. Do NOT write Ts and Fs under the atomic statements on the left. 3. If there is more than one way for the conclusion to be false, place on F under the (main operator of the) conclusion for each way it can be false, thereby creating as many rows as there are ways for the conclusion to be false. On each row, place a T under the main operator of each premise. a. To show invalidity, follow instruction 1b for at least on row. b. to show validity, follow instruction 2b for EVERY row.

Summary of Truth-Table Method

1. Assign values mechanically a. place the capital letters of atomic statements in sequence from left to right in the order that they appear in our symbolization b. the number of rows for atomic statements that you need is 2^n, where n is the number of atomic statements. c. 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 orf 8 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.

conjunction

always false except when both conjuncts are true

negation

always the opposite

disjunction

always true except when both disjuncts are false

material biconditional PART 2

always true except when its two constituent statements have different truth values

material conditional PART 2

always true except when the antecedent is true and the consequent is false

truth-functional

compound statement that has a truth value that is completely determined by the truth value of the atomic statements that compose it

material conditional

conditional that is false only when its antecedent is true and its consequent is false. otherwise, it is true.

material biconditional

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

minor logical operator

governs smaller components

well-formed formula

grammatically correct symbolic expression

tautology

if an only if it is true on every assignment of truth values to its atomic components

logically equivalent

if an only if they agree in truth value on every assignment of truth values to their atomic components

contradiction

if and only if it is false on every assignment of truth values to its atomic components

contingent

if and only if it is true on some assignments of truth value to its atomic components and false on others

logically consistent

if and only if they are both (all) true on some assignment of truth values to their atomic components

logically inconsistent

if and only if they are never both all true on any assignment of truth values to their atomic components

logically contradictory

if and only if they disagree in truth value on every assignment of truth values to their atomic components

main logical operator

in a compound statement, is the one that governs the largest component or components of a compound statement

statement variable

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

atomic statement

one that does not have any other statement as a component

compound statement

one that has at least one atomic statement as a component