Logic-section 2

Question 1

statement constants

what is an acceptable statement constant?

Answer 1

when letters stand for statements.

capital letters only.

Question 2

statement variables

when can statement variables be used?

what is an acceptable statement variable?

Answer 2

statements already in SL can be and are symbolized as statement variables

a small uncapitalized letter

Question 3

truth value

Answer 3

in logic a statement can either be T (true) or F (false)

Question 4

what are the five operators of SL?

what are their symbols?

what does each operator mean?

give an example?

Answer 4

1. negation, ~, not, ~x
2. conjunction, &, and, x&y
3. disjunction, v, or, xvy
4. conditional, →, if…then, x→y
5. biconditional,<—>, if and only if, x<—>y

Question 5

what is the function of negation?

Answer 5

turning the truth value of a statement into its opposite

Question 6

what is a truth table?

Answer 6

a table displaying all the possible truth values of a given statement under various conditions

Question 7

and

what is the symbol for and?

what is the function of and in logic

Answer 7

&, the cunjuction operator

joins two statements together making a new statement

Question 8

what is the process for analyzing an and-statement? Sum up the truth table of and-statements.

Answer 8

1. break down the and-statement into two substatements and consider the truth value of each substatement
2. if either substatement or both substatements are false the entire and-statement is false

Question 9

or

what is the symbol and name for the or operator?

what is the function of the or operator?

Answer 9

V, disjuntion operator

or joins two statements together

Question 10

how can an or-statement be analyzed? sum up the truth table for or-statements.

Answer 10

1. consider the truth values of the substatements within an or-statement.
2. if both substatements are false then and only then is the entire or-statement false

Question 11

name and define two distinct meanings of or?

how does the disjunction operator handle these meanings in logic?

Answer 11

1. inclusive or- means “this choice or that choice, or both”
2. exclusive or- means “this choice or that choice, but not both”

or-statements are always inclusive in logic

Question 12

if

what is the symbol and correct name for the if operator?

what is the function of if?

Answer 12

→, conditional operator

conditional operators represent if…then-statements

Question 13

sum up the truth table for if…then statements?

Answer 13

1. an if-statement is false only when the first part of it is true and the second part is false, all other patterns mean the statement is true.

Question 14

define an inverse

Answer 14

an inverse is created when both parts of an if…then statement are negated

Question 15

define a contrapositive

Answer 15

reverse the order of the substatments and then negate these reordered substatements

always has the truth value of the original if…then-statement

Question 16

define a converse

Answer 16

reverse the order of the substatements in the original if-statement

Question 17

why do converse and inverse statements always have the same truth values

Answer 17

because they are contrapositives of each other

Question 18

if and only if

Answer 18

biconditional operator, <–>

Question 19

sum up the truth table for an if-and-only-if-statement?

Answer 19

1. both parts of if-and-only-if statements are logically equivalent meaning they must have the same truth values
2. both parts of the biconditional statement must have the same truth value in order for the whole statement to be considered true

Question 20

arithmetic and SL have several important similarities, what are the similarities?

Answer 20

1. binary operators turning two or more inputs into one single output
2. unary operator turning one input into a single output

Question 21

what are the binary operators of logic?

what is the unary operator of logic?

Answer 21

1. &,V,→,<–>
2. ~

Question 22

what is the baisic idea behind substitution?

what is the implication for logic?

Answer 22

1. in algebra letters can stand for numbers and similarly constants can stand for statements in logic
2. once constants are put into place the correct truth values of the constants can be substituted in then given operation performed resulting in the truth value of the statement

Question 23

parenthesis and its related process of evaluating the statement

Answer 23

solve the innermost parenthesis first, then the process of evaluation will reduce the many values to one value

Question 24

what is evaluation?

Answer 24

evaluation is done by replacing the constants of sl with their truth values and reducing the statement to one value

Question 25

what is an interpretation?

Answer 25

a fixed set of truth values for all the sonstants in the statement

Question 26

what is a sub-statement

Answer 26

any piece of the SL statement that can stand on its own a a complete statement

Question 27

what is the scope of an operator

Answer 27

the smallest sub-statement that includes the operator in question

Question 28

what needs to be known before an operator can be evaluated

Answer 28

the truth value of every other constant and operate within its scope

Question 29

what is the main operator

Answer 29

it is the operator whose scope includes the entire SL statement and therefore cannot be operated on until last its value is the value of the whole statement

Question 30

what are the three rules of thumb when picking out the main operator

Answer 30

1. pick the only operator outside of any parentheses 2. when none are outside of parentheses: remove a set of parentheses 3. if more than one operator is outside parentheses choose the one this is not the ~operator

Question 31

what are eight forms of SL statements

Answer 31

_Positive Forms_ X&Y XvY X→Y X\<-\>Y _Negative Forms_ ~(x&y) ~(x v y) ~(x→y) ~(x\<-\>y)

Question 32

what statements in SL can be represented by the 8 basic forms of SL

Answer 32

every SL statement can be represented by one of the 8 basic SL forms

Question 33

what is brute force

Answer 33

mathematicians name referring to the method of exhausting all possible paths to a solution time consuming; always effective

Question 34

what is the process of setting up a truth table

Answer 34

1. set up two sides of the table with all constants of the left and the statement on the right 2. multiply 2 by itself equal to the number of constants in the statement to determine the number of rows 3. set up the left hand constant column so that every possible truth value is accounted for: alternating TF for the first constant and doubling for each subsequent column ex. tftf ttff ttttffff, move from right to left 4. divide each constant and its operator with a line in the right hand side of the table

Question 35

what is the process of filling in the truth table

Answer 35

1. copy the value of each constant in the proper column 2. ~operators outside of parentheses negate all constants and operators within therefore the value of each constant must be known before evaluating these operators 3. begin with the innermost set of parentheses 4. repeat step 3 working outward until the table is complete

Question 36

how do you read a truth table

Answer 36

1. circle the column under the main operator 2. the truth table reveals the value of the statement under every possible interpretation 3. all answers are justified because the table lays out all the possibilities

Question 37

what is a tautology

Answer 37

a statement that is always true regardless of the truth values

Question 38

what is a contradiction

Answer 38

a statement that is always false regardless of the truth value of its constants

Question 39

what is a contingent statement

Answer 39

a statement that is true under one intrpretation and false under another

Question 40

what is semantic equivalence

Answer 40

when two statements have the same truth value under all interpretations

Question 41

what makes a set of statements consistent what makes a set of statements inconsistent

Answer 41

having at least one interpretation making all of the statements in the set true when no interpretation exists that makes all the statements of a set true

Question 42

what makes an argument valid

Answer 42

when all premises are T and the conclusion is necessarily T the argument is valid valid arguments have no interpretations in which all premises are true and the conclusion is false

Question 43

what makes an argument invalid

Answer 43

having at least one interpretation in which all premises of true and the conclusion false

Question 44

how can tautology and contradiction be linked together

Answer 44

they both can be turned into the other by adding a ~operator to the whole statement

Question 45

how can inconsistency and contradiction be linked

Answer 45

statements from an inconsistent set can be linked with repeated use of the &operator forming a contradiction

Question 46

how can semantic equivalence and tautology be linked

Answer 46

two semantically equivalent statements can be linked with the \<-\>operator to form a tautology

Question 47

how do you link validity with contradiction

Answer 47

take the statements from a valid argument and negate the conclusion then link all of the satements with the &operator

Question 48

what seperates quick tables from regular truth tables

Answer 48

quick tables start with the whole statement (truth value of the main operator) and finish with the value of the parts

Question 49

when should quick tables be used

Answer 49

when problems have four or more constants or fit into the easy types category

Question 50

outline the quick table process

Answer 50

1. make a strategic assumption by declaring the satement true or false 2. each statement assumed to be true has two possible outcomes a. an interpretation under your assumption will be found b. disprove the assumption by showing such interpretation is not possible

Question 51

what is a good assumption for proving tautologies what is the desired outcome

Answer 51

assume the statement is false to try and show the statement is a contradiction outcome:if an interpretation is found under this assumption the statement is not a tautology

Question 52

assumptions and outcomes for contradictions

Answer 52

prove the two statements are semantically inquivalent by connecting them with the \<-\>operator and assume the statement is false outcomes: if interpretation is found the statements are semantically inequivalent therefore contradictory if interpretation is not found then not contradictory

Question 53

assumptions and outcomes for consistency and inconsistency

Answer 53

try to show all statements are consistent so assume all the statements are true outcomes: interpretation found= consistent no interpretation found= inconsistent

Question 54

assumptions and outcomes for validity and invalidity

Answer 54

try to show validity therefore assume premises to be true and conclusion false outcomes: interpretation found= invalid no interpretation found= valid

Question 55

what are the six easiest types of statements to work with in quick tables

Answer 55

x&y (T) or ~(x&y)(F) where x is T and y is T XvY(F) or ~(XvY)(T) where x is F and y is F x→y(F) or ~(x→y)(T) where x is T and Y is F

Question 56

four not so easy statements

Answer 56

x\<-\>Y(T) or ~(x\<-\>Y)(F) where x is T and y is true, or where x is F and y is F x\<-\>y (F) or ~(x\<-\>y)(T) where x is T and y is F, or where x is F and y is T