There are four types of compound statements used in symbolic logic, namely:
1) conjunctive,
2) disjunctive,
3) conditional, and
4) biconditional
In these notes, I will focus only on conjunctive statements.
A conjunctive statement or conjunction is a compound statement connected by the word “and.” The component statements in a conjunction are called conjuncts. Let us consider this example:
Roses are red and jasmines are white.
Obviously, the above statement is a conjunction because it is connected by the word “and.” The first statement “Roses are red” is the first conjunct and the statement “Jasmines are white” is the second conjunct.
In my notes titled “Propositions and Symbols Used in Symbolic Logic” (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/), the symbol for “and” is • (dot). Now, if we let p stand for “Roses are red” and q for “Jasmines are white,” then the statement “Roses are red and jasmines are white” is symbolized as follows:
p • q
In some cases, a conjunctive statement does not use the word “and” as connective. Sometimes, the following words are used as connectives of a conjunctive statement:
But
However
Nevertheless
Even though
Whereas
Although
While
Still
Yet
Consider the following examples:
- Chocolate is delicious, but it is not a good food for people with diabetes.
- Lucas is playing, while Rob is studying.
- The teacher was already shouting, yet the students remain very noisy.
In cases where there are no words that signify a conjunction, a comma (,) or a semi-colon (;) may indicate that the statement is a conjunction. Consider the example below:
Although the human person is mortal, she can live long.
Symbolizing Conjunctive Statements
I have been symbolizing statements above and in my previous posts, but it is not until now that I will specifically talk about symbolizing statements.
Firstly, logicians usually put the variables or constants that will represent the statement right after the statement per se. Consider the examples below:
Chocolate is delicious, but it is not a good food for people with diabetes. (p, q)
Please note that the variables provided after the statement represent the component statements respectively. Thus, in the example above, the variable p represents the first component statement “Chocolate is delicious,” while q represents the second component statement “It is not a good food for people with diabetes.”
Secondly, when symbolizing statements, we need to put proper punctuations and negation if necessary. Thus, in the example above, the statement “Chocolate is delicious” is represented by p, while the statement “It is not a good food for people with diabetes” is represented by q. If we are not careful, we may symbolize the statement as follows: p • q. However, if we analyze the statement, we notice that the second component contains a negation sign “It is not the case.” Hence, the statement “Chocolate is delicious, but it is not a good food for people with diabetes” is symbolized as follows:
p • ~q
It is important to note that sometimes the word “and” is not truth-functional, that is, it does not connect two independent propositions. Thus, if this occurs, we should symbolize the proposition simply as a simple proposition. Consider the following example:
Bread and butter is a perfect combination.
Obviously, the “and” in the example above is not truth-functional because it does not connect two truth-functional propositions or sentences. This is because we cannot say that “Bread is a perfect combination” and “Butter is a perfect combination.” Hence, the proposition “Bread and butter is a perfect combination” is symbolized simply as:
p
However, if we have the example
“John and Mary are watching TV”
then we have to symbolize this as:
p • q
This is because the “and” here is truth-functional, that is, it connects two independent propositions or sentences. For sure, it is possible for us to say “John is watching TV” and “Mary is watching TV.” In other words, both John and Mary are watching TV.
Rules in Conjunction
- A conjunction is true if and only if both conjuncts are true.
- If at least one of the conjuncts is false, then the conjunction is false.
The truth table below illustrates this point.
The truth table above says:
1) If p is true and q is true, then p • q is true.
2) If p is true and q is false, then p • q is false.
3) If p is false and q is true, then p • q is false.
4) If p is false and q is false, then p • q is false.
Now, given the rule in conjunction, how do we determine the truth-value of the conjunctive statement p • ~q?
Let us suppose that the truth-value of p is true and q is false. So, if p is true and q false, then the statement p • ~q is true. To illustrate:
The illustration above says that p is true and q is false. Now, before we apply the rule in conjunction in the statement p • ~q, we need to simplify ~q first because the truth-value “false” is assigned to q and not to ~q. If we recall our discussion on the rule in negation, we learned that the negation of false is true. So, if q is false, then ~q is true. Thus, at the end of it all, p • ~q is true if p is true and q is false.