Negations, Conjunctions, Disjunctions, and De Morgan's Laws

Basic Notation

In symbolic logic, De Morgan's Laws are powerful tools that can be used to transform an argument into a new, potentially more enlightening form. We can make new conclusions based off what may be considered old knowledge we have at hand. But like all rules, we have to understand how to apply it. We start off with two statements that are somehow related to each other, commonly symbolized as p and q. We can link them together in many ways, but for the purpose of this hub we only need be concerned with conjunctions and disjunctions as our main instruments of logical conquest.

Negation

A ~ (tilde) in front of a letter means that the statement is false and negates the truth value present. So if statement p is "The sky is blue," ~p reads as, "The sky is not blue" or "It is not the case that the sky is blue." We can paraphrase any sentence into a negation with "it is not the case that" with the positive form of the sentence. We refer to the tilde as a unary connective because it is only connected to a single sentence. As we will see below, conjunctions and disjunctions work on multiple sentences and are thus known as binary connectives (36-7).

p
q
p ^ q
T
T
T
T
F
F
F
T
F
F
F
F

Conjunction

A conjunction is symbolized as

p ^ q

with the ^ representing "and" while p and q are the conjuncts of the conjunction (Bergmann 30). Some logic books may also use the symbol "&," known as an ampersand (30). So when is a conjunction true? The only time a conjunction can be true is when both p and q are true, for the "and" makes the conjunction dependent on the truth value of both the statements. If either or both of the statements are false, then the conjunction is false also. A way to visualize this is through a truth table. The table on the right represents the truth conditions for a conjunction based off of it's constituents, with the statements we are examining in the headings and the value of the statement, either true (T) or false (F), falling underneath it. Every single possible combination has been explored in the table, so study it carefully. It is important to remember that all possible combinations of true and false are explored so that a truth table does not mislead you. Also be careful when choosing to represent a sentence as a conjunction. See if you can paraphrase it as an "and" type of sentence (31).

p
q
p v q
T
T
T
T
F
T
F
T
T
F
F
F

Disjunction

A disjunction, on the other hand, is symbolized as

p v q

with the v, or wedge, representing "or" and p and q being the disjuncts of the disjunction (33). In this case, we require only one of the statements to be true if we want the disjunction to be true, but both statement can be true as well and still yield a disjunction that is true. Since we need one "or" the other, we can have just a single truth value to get a true disjunction. The truth table on the right demonstrates this.

When deciding to use a disjunction, see if you can paraphrase the sentence into an "either...or" structure. If not, then a disjunction may not be the right choice. Also be careful to make sure that both sentences are full sentences, not inter-dependent on one another. Finally, take note of what we call the exclusive sense of "or." This is when both choices cannot be correct at the same time. If you can either go to the library at 7 or you can go to the baseball game at 7, you cannot pick both as true at once. For our purposes, we deal with the inclusive sense of "or," when you can have both choices as true simultaneously (33-5).

p
q
~(p ^ q)
~p v ~q
T
T
F
F
T
F
T
T
F
T
T
T
F
F
T
T

De Morgan's Law #1: Negation of a Conjunction

While each law does not have a number-order to it, the first one I will discuss is called "negation of a conjunction." That is,

~(p ^ q)

This means that if we constructed a truth table with p, q, and ~(p ^ q) then all the values we had for the conjunction will be the opposite truth value that we established before. The only false case would be when p and q are both true. So how can we transform this negated conjunction into a form that we can understand better?

The key is to think when the negated conjunction would be true. If either p OR q were false then the negated conjunction would be true. That "OR" is the key here. We can write out our negated conjunction as the following disjunction

~p v ~q

The truth table on the right further demonstrates the equivalent nature of the two. Thus,

~(p ^ q) = ~p v ~q

p
q
~(p v q)
~p ^ ~q
T
T
F
F
T
F
F
F
F
T
F
F
F
F
T
T

De Morgan's Law #2: Negation of a Disjunction

The "second" of the laws is called the "negation of the disjunction." That is, we are dealing with

~(p v q)

Based off the disjunction table, when we negate the disjunction, we will only have one true case: when both p AND q are false. In all other instances, the negation of the disjunction is false. Once again, take note of the truth condition, which requires an "and." The truth condition we arrived at can be symbolized as a conjunction of two negated values:

~p ^ ~q

The truth table on the right again demonstrates how these two statements are equivalent. Thus

~(p v q) = ~p ^ ~q

Source

Works Cited

Bergmann, Merrie, James Moor, and Jack Nelson. The Logic Book. New York: McGraw-Hill Higher Education, 2003. Print. 30, 31, 33-7.

© 2012 Leonard Kelley

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Comments 2 comments

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calculus-geometry 3 years ago from Germany

This is nicely laid out and easy to follow, thanks!


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1701TheOriginal 3 years ago Author

Thank you, glad you enjoyed it!

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