Logical Operators and Abbreviated Truth Tabels

 Logical Operators and Abbreviated Truth Tables

    Logical operators are the mathematical symbols of philosophy. There are six operators used in statement logic. How they are used is very straightforward. 

    Each logical operator has its own definition and its own requirements for what makes it true. Understanding them in an argument is key to creating and using truth tables. With a proper understanding, you can unlock a whole world of knowledge that you already live in. Bringing life to your language and the language of those around you. 

What Logical Operators Mean

    As we discussed in a previous post, the use of statement logic requires the use of six different logical operators. We'll start by providing the definition of each operator. These definitions are a little different from a normal definition thought. These definitions are defining when a statement is true. 

    We'll start by defining a conditional statement (an if/then).

    Now, for what the table means. The middle yellow section tells us the truth value of the conditional statement based on the combined truth value of the two statements sandwiching the operator. The two sandwiching statements have names two. The one that comes first is called the antecedent and the one that comes after is called the consequent.

    So, for an if/then statement, the operator is true three out of four times. The only time it is false is when the antecedent is true but the consequent is false. We now have the definition of what a conditional statement true. 

    Let's do a bi-conditional now. 

    Here we can see that a bi-conditional operator is false two out of four times. It is only true when the truth values of the antecedent and consequent are the same. We'll do the next couple rather quickly as by now I assume you understand what is going on here.

    This is what a negation looks like. 

    We see that a negation is true anytime its consequent statement is false. Remember that a negation flips the truth value of the statement it is attached to. 

    Now we'll do a conjunction (and) statement. 

    A conjunction is only true if both of its statements are true at the same time. As a side note, when applying logic to English, the word 'but' functions as a conjunction. So if you were to say "I like vanilla but not chocolate" that would translate to "I like vanilla and not chocolate" which would go into statement logic like "V . ~C."

    The last operator that needs a logical definition is the either/or statement. Also known as a disjunct.

    The disjunct is only false if both statements are false. 

    The reason this is the last operator that needs a logical definition is because the last operator, the conclusion, is not really dependent on the truth values of any statements. What determines the truth value of the conclusion is the same as whatever determines the truth value of the statements that make up the conclusion. 

    In other words, the conclusion doesn't need its own definition because its definition is going to be the same definition of whatever operators are included in it. 

Applying These Definitions to Abbreviated Tables

    I'm not going to walk through the steps of creating and constructing a truth table. If you are interested in a step-by-step process you can check out my truth table page. I'm just going to hop right into it. This is an argument that I created for the purpose of teaching. 

    

    I like to build my truth table in a spreadsheet. I find it helps to keep me organized. Notice that I have boxed off all of the main logical operators in a solid line, with any secondary operators in a smaller, dotted line. 

    You may have also noticed that there are six different statements separated from the argument by a thick solid line. Six statements is a lot of truth values to wrangle. To do it in a regular table would require 64 different combinations of just the truth values. That doesn't even get to the argument. 

    That's why we use something called an abbreviated table for arguments that have a lot of individual statements. Abbreviated truth tables work because the definition of a valid argument is what it is. (Remember that the definition is that there is never a false conclusion with all true premises.)

    Because of that definition, all we have to prove is that there is one combination where there is a false conclusion with all true premises. 

    So where we need to start is by creating a combination that gives us a false conclusion.

    Now that we have our false conclusion, we need to start coming up with different combinations that will give us true conclusions. I like to work backward. 

    This is why those definitions above were so important. They allow us to identify complex arguments and determine if they are valid or not. I'm going to do the rest of the table in one go. 

    Here we can see that this argument is invalid. It has at least one combination of truth values that gives us a false conclusion with all true premises. Much faster than filling out 64 individual combinations. 

Conclusion

    Truth tables are easy ways of determining whether or not an argument is valid. There are a few other ways. One of them is a method of proofs. Proofs are much more complicated than truth tables, but are generally the preferred method of determining validity. I will address them in a later post. 

    As always, please ask questions and share your thoughts :)