Logic-section 2 Flashcards
statement constants
what is an acceptable statement constant?
when letters stand for statements.
capital letters only.
statement variables
when can statement variables be used?
what is an acceptable statement variable?
statements already in SL can be and are symbolized as statement variables
a small uncapitalized letter
truth value
in logic a statement can either be T (true) or F (false)
what are the five operators of SL?
what are their symbols?
what does each operator mean?
give an example?
- negation, ~, not, ~x
- conjunction, &, and, x&y
- disjunction, v, or, xvy
- conditional, →, if…then, x→y
- biconditional,<—>, if and only if, x<—>y
what is the function of negation?
turning the truth value of a statement into its opposite
what is a truth table?
a table displaying all the possible truth values of a given statement under various conditions
and
what is the symbol for and?
what is the function of and in logic
&, the cunjuction operator
joins two statements together making a new statement
what is the process for analyzing an and-statement? Sum up the truth table of and-statements.
- break down the and-statement into two substatements and consider the truth value of each substatement
- if either substatement or both substatements are false the entire and-statement is false
or
what is the symbol and name for the or operator?
what is the function of the or operator?
V, disjuntion operator
or joins two statements together
how can an or-statement be analyzed? sum up the truth table for or-statements.
- consider the truth values of the substatements within an or-statement.
- if both substatements are false then and only then is the entire or-statement false
name and define two distinct meanings of or?
how does the disjunction operator handle these meanings in logic?
- inclusive or- means “this choice or that choice, or both”
- exclusive or- means “this choice or that choice, but not both”
or-statements are always inclusive in logic
if
what is the symbol and correct name for the if operator?
what is the function of if?
→, conditional operator
conditional operators represent if…then-statements
sum up the truth table for if…then statements?
- 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.
define an inverse
an inverse is created when both parts of an if…then statement are negated
define a contrapositive
reverse the order of the substatments and then negate these reordered substatements
always has the truth value of the original if…then-statement
define a converse
reverse the order of the substatements in the original if-statement
why do converse and inverse statements always have the same truth values
because they are contrapositives of each other
if and only if
biconditional operator, <–>
sum up the truth table for an if-and-only-if-statement?
- both parts of if-and-only-if statements are logically equivalent meaning they must have the same truth values
- both parts of the biconditional statement must have the same truth value in order for the whole statement to be considered true
arithmetic and SL have several important similarities, what are the similarities?
- binary operators turning two or more inputs into one single output
- unary operator turning one input into a single output
what are the binary operators of logic?
what is the unary operator of logic?
- &,V,→,<–>
- ~
what is the baisic idea behind substitution?
what is the implication for logic?
- in algebra letters can stand for numbers and similarly constants can stand for statements in logic
- 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
parenthesis and its related process of evaluating the statement
solve the innermost parenthesis first, then the process of evaluation will reduce the many values to one value
what is evaluation?
evaluation is done by replacing the constants of sl with their truth values and reducing the statement to one value
what is an interpretation?
a fixed set of truth values for all the sonstants in the statement
what is a sub-statement
any piece of the SL statement that can stand on its own a a complete statement
what is the scope of an operator
the smallest sub-statement that includes the operator in question
what needs to be known before an operator can be evaluated
the truth value of every other constant and operate within its scope
what is the main operator
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
what are the three rules of thumb when picking out the main operator
- pick the only operator outside of any parentheses
- when none are outside of parentheses: remove a set of parentheses
- if more than one operator is outside parentheses choose the one this is not the ~operator
what are eight forms of SL statements
Positive Forms
X&Y XvY X→Y X<->Y
Negative Forms
~(x&y) ~(x v y) ~(x→y) ~(x<->y)
what statements in SL can be represented by the 8 basic forms of SL
every SL statement can be represented by one of the 8 basic SL forms
what is brute force
mathematicians name referring to the method of exhausting all possible paths to a solution
time consuming; always effective
what is the process of setting up a truth table
- set up two sides of the table with all constants of the left and the statement on the right
- multiply 2 by itself equal to the number of constants in the statement to determine the number of rows
- 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
- divide each constant and its operator with a line in the right hand side of the table
what is the process of filling in the truth table
- copy the value of each constant in the proper column
- ~operators outside of parentheses negate all constants and operators within therefore the value of each constant must be known before evaluating these operators
- begin with the innermost set of parentheses
- repeat step 3 working outward until the table is complete
how do you read a truth table
- circle the column under the main operator
- the truth table reveals the value of the statement under every possible interpretation
- all answers are justified because the table lays out all the possibilities
what is a tautology
a statement that is always true regardless of the truth values
what is a contradiction
a statement that is always false regardless of the truth value of its constants
what is a contingent statement
a statement that is true under one intrpretation and false under another
what is semantic equivalence
when two statements have the same truth value under all interpretations
what makes a set of statements consistent
what makes a set of statements inconsistent
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
what makes an argument valid
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
what makes an argument invalid
having at least one interpretation in which all premises of true and the conclusion false
how can tautology and contradiction be linked together
they both can be turned into the other by adding a ~operator to the whole statement
how can inconsistency and contradiction be linked
statements from an inconsistent set can be linked with repeated use of the &operator forming a contradiction
how can semantic equivalence and tautology be linked
two semantically equivalent statements can be linked with the <->operator to form a tautology
how do you link validity with contradiction
take the statements from a valid argument and negate the conclusion
then link all of the satements with the &operator
what seperates quick tables from regular truth tables
quick tables start with the whole statement (truth value of the main operator) and finish with the value of the parts
when should quick tables be used
when problems have four or more constants or fit into the easy types category
outline the quick table process
- make a strategic assumption by declaring the satement true or false
- 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
what is a good assumption for proving tautologies
what is the desired outcome
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
assumptions and outcomes for contradictions
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
assumptions and outcomes for consistency and inconsistency
try to show all statements are consistent so assume all the statements are true
outcomes: interpretation found= consistent
no interpretation found= inconsistent
assumptions and outcomes for validity and invalidity
try to show validity therefore assume premises to be true and conclusion false
outcomes: interpretation found= invalid
no interpretation found= valid
what are the six easiest types of statements to work with in quick tables
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
four not so easy statements
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
