 | Sentential logic (or SL; also known as propositional logic) is one of two forms of classical formal logic. (The other form is quantifier logic, or QL, also known as predicate logic.) |
Logical arguments are communicated in language, but natural languages such as English tend to be imprecise. Words often have more than one meaning and sentences can be misinterpreted.
To help solve this problem, mathematicians and philosophers developed sentential logic, a language specifically designed to express logical arguments with precision and clarity. Because SL is a symbolic language, it has the added advantage of allowing for calculation according to precisely defined rules and formulas. Just as with mathematics, as long as you follow the rules correctly, the correct answer is guaranteed.
In the following sections, a few types of symbols are introduced that SL uses to achieve these goals.
Statement constants
If you ever spent a day in an algebra or pre-algebra class, you were probably exposed to that elusive little fellow known as x. Your teacher probably told you that x stood for a secret number and that it was your job to make x talk. He or she then showed you all sorts of sadistic ways of torturing poor little x until at last it broke down and revealed its true numerical identity. Oh, what fun that was.
Making letters stand for numbers is one thing mathematicians are really good at doing. So it isn't surprising that formal logic, which was developed by mathematicians, also uses letters as stand-ins. In the chapter introduction, I let on that logic uses statements rather than numbers, so it's logical to guess that in formal logic, letters stand for statements. For example:
Let K = Katy is feeding her fish.
Let F = The fish are joyfully flapping their little fins.
When a letter stands for a statement in English, the letter's called a statement constant. By convention, capital letters are used for constants.
 | For some reason, when it comes to constants, logicians tend to like the letters P and Q the most. Some folks say this is because P is the first letter in the word proposition, which means the same thing as statement, and Q just came along for the ride. My own personal theory is that after studying all that algebra in school, logicians were tired of using X and Y. |
Statement variables
When logicians got the idea to make letters stand for statements, they just ran with it. They realized that they could use a letter for any statement, even a statement in SL. When letters are used in this way, they're called statement variables.
| When a letter stands for a statement in SL, the letter's called a statement variable. By convention, small letters are used for variables. |
Truth value
Every statement in logic has a truth value that's either true or false. In formal logic, true is shortened to T and false to F.
For example, consider the truth values of these two statement constants:
Let N = The Nile is the longest river in Africa.
Let L = Leonardo DiCaprio is the king of the world.
As it happens, it's true that the Nile is the longest river in Africa, so the truth value of N is T. And it so happens that Leonardo DiCaprio is not the king of the world, so the truth value of L is F.
Boolean algebra, the precursor to formal logic, uses the value 1 to represent T and 0 to represent F. These two values are still used in computer logic.
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