Rough Draft Independent Research: Boolean Truth Tables And Logical Deductive Reasoning

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 mathematical  table used in logic—specifically in connection with Boolean algebra, boolean [sic]functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables (Enderton, 2001) (Truth Table, n. d.)

 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

Argument. (n. d.). Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Argument

applied logic. (2010)  In Encyclopædia Britannica Online. Retrieved from http://www.britannica.com/EBchecked/topic/30698/applied-logic

Invalid. (2010) Webster’s Dictionary. Retrieved from http://dictionary.reference.com/browse/invalid?o=100074

Invalid. (2010) Webster’s Thesaurus. Retrieved from http://thesaurus.com/browse/invalid?o=100074&__utma=1.1108970625.1272913250.1274862810.1274867741.14&__utmb=1.3.9.1274867752917&__utmc=1&__utmx=-&__utmz=1.1274867741.14.10.utmcsr=yahoo|utmccn=(organic)|utmcmd=organic|utmctr=Websters&__utmv=-&__utmk=88169371

Jorgensen, P. (2010, May 19) Boolean explained a bit more [discussion board post]. Retrieved from Online class Discussion Board: https://campus.fsu.edu/webapps/portal/frameset.jsp?tab_id=_2_1&url=%2fwebapps%2fblackboard%2fexecute%2flauncher%3ftype%3dCourse%26id%3d_6308801_1%26url%3d

Truth Table. (n. d.). Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Truth_table