Propositions and Symbols Used in Propositional Logic

Just as in traditional or Aristotelian logic, our main goal in propositional logic (or symbolic logic) is to determine the validity of arguments. But because arguments are composed of propositions, and because we need to symbolize the argument first before we can determine its validity using a specific rule, we need therefore to discuss the types of proposition and symbols used in symbolic logic.

Please note that symbolic logic uses only declarative statements or propositions because any other types of proposition are not truth-functional, that is, they cannot be either true or false.  For example, the interrogative proposition “What is your name?” is not truth-functional because we cannot assign any truth-value to it, that is, it cannot be either true or false.

In similar manner, the exclamatory proposition “What an exciting journey!” cannot be used in symbolic logic because, again, we cannot assign a truth-value to it. Hence, again, we can only employ declarative propositions in symbolic logic because they are the only types of proposition that can either be true or false. Think, for example, of the proposition “Donald Trump is a racist president.” Depending on the context, we may say “Yes, it is true that Donald Trump is a racist president,” or we may say “It is false that Donald Trump is a racist president.”

There are two types of declarative proposition used in symbolic logic, namely, simple and compound proposition.

On the one hand, a simple proposition is one that is composed of only one proposition. For example, “Donald Trump is the president of the United States.” As we can see, this proposition has only one component.

On the other hand, a compound proposition is composed of two or more propositions, such as:

1) Jack is singing, while Jill is dancing.

2) If the road is wet, then either it rains today or the fire truck spills water on the road.

As you notice, the first example is made up of two propositions, namely:

Jack is singing.
Jill is dancing.

The second example, on the other hand, is composed of three propositions, namely:

The road is
It rains today.
The fire truck spills water on the road.

Now, logicians usually use the lower case of the English alphabet p through z to symbolize propositions. They are called variables. The upper-case A through Z are called constants. For example, if we let p stand for the proposition “Jack is singing,” then it is symbolized as p. Thus, instead of saying “Jack is singing,” we just say p.

The symbol •(dot), which is read as “and,” is used to symbolize the connective of a conjunctive proposition. As I will discuss in the succeeding posts, a conjunctive proposition is connected by the word “and.” Let’s take, for example, the proposition “Jack is singing and Jill is dancing.” If we let p stand for “Jack is singing,” and q for “Jill is dancing,” then the proposition “Jack is singing and Jill is dancing” is symbolized as follows:

p • q

The symbol v (wedge), which is read as “Either…or” or just “or” is used to symbolize the connective of a disjunctive proposition. As I will discuss in the succeeding posts, disjunctive propositions are connected by the words “Either…or” or simply “or.” If we let p stand for “Jack is singing” and q for “Jill is dancing,” then the proposition “Either Jack is singing or Jill is dancing” is symbolized as follows:

p v q

Please note that the proposition above is an inclusive disjunction. There is another way to symbolize an exclusive disjunction. But I will discuss this other type of disjunctive proposition when I go to the four types of compound propositions.

The symbol ⊃ (horse shoe), which is read as “If…then” or just “then” is used to symbolize the connective of a conditional proposition. As I will discuss in the succeeding posts, conditional propositions are connected by the words “If…then” or just “then.” 

Now, if we let p stand for “Jack is singing” and q for “Jill is dancing,” then the proposition “If Jack is singing, then Jill is dancing” is symbolized as follows:

p ⊃ q

The symbol ≡ (triple bar), which is read as “If and only if,” is used to symbolize the connective of a biconditional proposition. As I will discuss in the succeeding posts, biconditional propositions are connected by the words “If and only if.” If we let p stand for “Jack is singing” and q for “Jill is dancing,” then the proposition “Jack is singing if and only if Jill is dancing” is symbolized as follows:

p ≡ q

The symbol  /∴  (forward slash and triple dots) is read as “therefore.” This is symbol is used to separate the premises and the conclusion in an argument. For example, if the premises in the argument are 1) p ⊃ q,   2) p and the conclusion is q, then the argument is symbolized as follows:

p ⊃ q
p /∴ q

Lastly, the symbol  ~  (tilde), which is read as “not,” is used to negate a proposition. As I will show later, any proposition can be negated. Thus, the proposition “Jack is not singing” is symbolized as follows:

~ p

Below is the summary of some of the basic symbols used in symbolic logic.