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Logic > Logic-section 2 > Flashcards

Logic-section 2 Flashcards

1
Q

statement constants

what is an acceptable statement constant?

A

when letters stand for statements.

capital letters only.

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2
Q

statement variables

when can statement variables be used?

what is an acceptable statement variable?

A

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

a small uncapitalized letter

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3
Q

truth value

A

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

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4
Q

what are the five operators of SL?

what are their symbols?

what does each operator mean?

give an example?

A
  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
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5
Q

what is the function of negation?

A

turning the truth value of a statement into its opposite

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6
Q

what is a truth table?

A

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

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7
Q

and

what is the symbol for and?

what is the function of and in logic

A

&, the cunjuction operator

joins two statements together making a new statement

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8
Q

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

A
  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
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9
Q

or

what is the symbol and name for the or operator?

what is the function of the or operator?

A

V, disjuntion operator

or joins two statements together

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10
Q

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

A
  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
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11
Q

name and define two distinct meanings of or?

how does the disjunction operator handle these meanings in logic?

A
  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

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12
Q

if

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

what is the function of if?

A

→, conditional operator

conditional operators represent if…then-statements

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13
Q

sum up the truth table for if…then statements?

A
  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.
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14
Q

define an inverse

A

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

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15
Q

define a contrapositive

A

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

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

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16
Q

define a converse

A

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

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17
Q

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

A

because they are contrapositives of each other

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18
Q

if and only if

A

biconditional operator, <–>

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19
Q

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

A
  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
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20
Q

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

A
  1. binary operators turning two or more inputs into one single output
  2. unary operator turning one input into a single output
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21
Q

what are the binary operators of logic?

what is the unary operator of logic?

A
  1. &,V,→,<–>
  2. ~
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22
Q

what is the baisic idea behind substitution?

what is the implication for logic?

A
  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
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23
Q

parenthesis and its related process of evaluating the statement

A

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

24
Q

what is evaluation?

A

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

25
Q

what is an interpretation?

A

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

26
Q

what is a sub-statement

A

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

27
Q

what is the scope of an operator

A

the smallest sub-statement that includes the operator in question

28
Q

what needs to be known before an operator can be evaluated

A

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

29
Q

what is the main operator

A

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

30
Q

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

A
  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
31
Q

what are eight forms of SL statements

A

Positive Forms

X&Y XvY X→Y X<->Y

Negative Forms

~(x&y) ~(x v y) ~(x→y) ~(x<->y)

32
Q

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

A

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

33
Q

what is brute force

A

mathematicians name referring to the method of exhausting all possible paths to a solution

time consuming; always effective

34
Q

what is the process of setting up a truth table

A
  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
35
Q

what is the process of filling in the truth table

A
  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
36
Q

how do you read a truth table

A
  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
37
Q

what is a tautology

A

a statement that is always true regardless of the truth values

38
Q

what is a contradiction

A

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

39
Q

what is a contingent statement

A

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

40
Q

what is semantic equivalence

A

when two statements have the same truth value under all interpretations

41
Q

what makes a set of statements consistent

what makes a set of statements inconsistent

A

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

42
Q

what makes an argument valid

A

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

43
Q

what makes an argument invalid

A

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

44
Q

how can tautology and contradiction be linked together

A

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

45
Q

how can inconsistency and contradiction be linked

A

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

46
Q

how can semantic equivalence and tautology be linked

A

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

47
Q

how do you link validity with contradiction

A

take the statements from a valid argument and negate the conclusion

then link all of the satements with the &operator

48
Q

what seperates quick tables from regular truth tables

A

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

49
Q

when should quick tables be used

A

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

50
Q

outline the quick table process

A
  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
51
Q

what is a good assumption for proving tautologies

what is the desired outcome

A

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

52
Q

assumptions and outcomes for contradictions

A

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

53
Q

assumptions and outcomes for consistency and inconsistency

A

try to show all statements are consistent so assume all the statements are true

outcomes: interpretation found= consistent

no interpretation found= inconsistent

54
Q

assumptions and outcomes for validity and invalidity

A

try to show validity therefore assume premises to be true and conclusion false

outcomes: interpretation found= invalid

no interpretation found= valid

55
Q

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

A

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

56
Q

four not so easy statements

A

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

57
Q
A

Logic (3 decks)

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