The Application of Deductive Logic
to: Logical Operators, Boolean Operators and Truth Tables, to Explicate the
Relationship Between Validity and “Non-logical” Reasoning
The results of an argument, based on logic and reasoning, highly
depend on the rules of logic and reasoning that are applied to the argument (
applied logic, 2010).
“Logical” and “non-logical” reasoning can be
applied to an argument (applied logic, 2010).
Therefore, the argument’s result may be based on “logical” or “non-logical”
reasoning (applied logic, 2010).
In
either case, the result of the argument may be referred to as “logically valid”
or true (Truth Table, n. d.).
The
reasoning involved in deciding the result of an argument, whether it be
“logical” or “non-logical,” is superfluous
insofar
as the nomenclature of the result is concerned; valid or invalid (applied
logic, 2010; Truth Table, n. d.).
Truth Tables may be based on both “logical”
or “non-logical” reasoning (applied logic, 2010). According to Wikipedia, the
definition of a
Truth Table is:
A Truth Table can be utilized to
determine whether or not a “propositional expression” is valid or true (Truth
Table, n. d.).
The reasoning, on which
the logicality or validity of a “propositional expression” is based, can be
either “logical” or “non-logical” (applied logic, 2010; Truth Table, n. d.).
Regardless of whether “logical” or “non-logical”
reasoning is applied to a “propositional expression,” the result may still be
determined to be “logically valid” or true (Truth Table, n. d.)
The determination of a “logically valid” or
true argument result, when applying “non-logical” reasoning to an argument,
contradicts “logical” reasoning and the rules of deductive logic (applied
logic, 2010).
The two logical value operations,
enumerated below, follow the “logical” reasoning rules which are must be
applied, in the case of deductive logic, for the analysis of the logicality of an
argument (applied logic, 2010).
Deductive
logic, and its concomitant “logical” reasoning rules, is the very antithesis of
the reasoning applied to a “non-logical” argument (applied logic, 2010;
Argument, n. d.).
The evaluation of the logicality
of logical value operations may be based on either “logical” or “non-logical”
arguments as well (Truth Table, n. d.).
The
following enumerated examples of logical value operations, present arguments
which are valid, because they are based on the “logical” reasoning rules of
deductive logic:
Logical Value Operations:
·
Logical conjunction: truth +truth= truth
·
Logical implication: truth +false= false (Truth
Table, n. d.)
Deductive logic states that only when both the premises are true can the
conclusion also be guaranteed to be true (applied logic, 2010).
In an argument in which one of the premises
is false, according to the rules of deductive logic, the conclusion must also be
false (applied logic, 2010).
According
to Wikipedia, what defines the validity of an argument is the logicality of the
argument’s form (Argument, n. d.).
The
veracity or “falsity” of an argument’s premises or conclusion do not, in of
themselves, determine the validity of an argument (Argument, n. d.).
In other words, “the validity of
an argument is not a guarantee of the truth of its conclusion,” and “a valid
argument may have false premises and a false conclusion” (Argument, n. d.).
Logical form, alone, is the reflection of a
valid argument (Argument, n. d.).
An
argument which lacks logical form represents an example of “non-logical” reasoning
and, therefore, the results of the argument are invalid (Argument, n. d.).
As
per
the rules of deductive logic, the result of arguments based on, “non-logical”
reasoning or “falsely based reasoning” are, invalid and therefore: false; not
true; not logically valid (applied logic, 2010; Invalid, 2010).
Some of the results of the following logical
value operations arguments are based on “non-logical” reasoning:
Logical Value Operations:
·
Logical disjunction: truth +truth =truth or
truth + false = truth
·
Logical equality: truth+ truth= true or false +
false=true
·
Exclusive disjunction: truth + false = true but
true + true =false
·
Logical NAND: truth + true= false
·
Logical NOR: false + false =true (Truth Table,
n.d.)
Despite the fact that some of the arguments enumerated above are based on
“non-logical” reasoning and are therefore invalid: “falsely based or reasoned,”
according to the rules of deductive logic, the conclusion may still be drawn
that the arguments are “logically valid” or true (Invalid, 2010; Truth Table,
n. d.).
When Boolean operators are
applied to Truth Tables, either “logical” or “non-logical” reasoning may be
applied to determine the result of the argument (Truth Table, n. d.).
In other words, when Boolean operators are
applied to Truth Tables, the resulting reasoning which occurs, to determine the
validity of the Truth Table, does not always follow the rules of deductive
logic (applied logic, 2010).
The
application of either “logical” or “non-logical” reasoning, to Boolean
operators: “And” and “Or,” and their interaction with Truth Tables, is evident
in the following examples posted by Professor Jorgensen to the class discussion
board:
AND
o Only if both bits are 1 (or True) will
the result bit be 1 so the "truth table" looks like this( first two
rows are operands, last (bottom) is the result of ANDing the operands.)
Boolean AND Truth Table
|
Operand
|
1
|
1
|
0
|
0
|
Operand
|
1
|
0
|
1
|
0
|
AND Result
|
1
|
0
|
0
|
0
|
Table 1.
OR
o If either of the operands is 1 then the result
is 1. If neither is 1 then the result is 0.
Boolean OR Truth Table
|
Operand
|
1
|
1
|
0
|
0
|
Operand
|
1
|
0
|
1
|
0
|
OR Result
|
1
|
1
|
1
|
0
|
Table 2. (Jorgensen,
2010)
Table 1., displays the
results of “ANDing” the operands: “1” and “0,” in an argument which applies the
Boolean operator of “AND” to the Truth Table (Jorgensen, 2010).
In,
Table
1., the argument
result of: “1,”
or true, only occurs when both operands are: “1”
or true (Jorgensen, 2010).
“Logical” reasoning, based on the rules of
deductive logic, is therefore applied to determine the results of the arguments
in
Table 1. (applied logic,
2010).
The result of the arguments
presented in
Table 1., because
they are based on the “logical” reasoning rules of deductive logic, are,
therefore, valid or true (applied logic, 2010).
However, in
Table 2., when the Boolean
operator of “Or” is applied to the operands: “1” and “O,” both “non-logical” and
“logical” reasoning are applied to the arguments (Truth Table, n. d.).
According to the application of the Boolean
operator “Or” to Truth
Table 2.,
if either of the operands is: “1,” or true, then the result is: “1” or true
(Jorgensen, 2010).
This means that when
both operands are: “1” or true then the argument’s result is: “1” or true—this
argument is, therefore, valid (applied logic, 2010).
However, it also means that if
one of the operands is: “0,” or false, the result of the argument is still
determined to be: “1” or true—this argument is, therefore, based on
“non-logical” reasoning (Jorgensen, 2010).
The resulting true and “logically valid” arguments, displayed in
Table 2., which reflect “non-logical”
reasoning, are in fact invalid, because they do not reflect the “logical” reasoning
rules of deductive logic (applied logic,
2010).
Deductive logic, and the “logical”
reasoning, on which its rules are based, by definition: determine the validity
of an argument, solely on the logicality of the argument’s form, and not on any
other factors (applied logic, 2010; Argument, n. d.).
Works
Cited
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