10 Apr 2010
03 Oct 2013

Symbolic Logic

Alphabet Soup and Symbol Salad

NOTE: Most students probably will not find techniques of symbolic logic to be of much practical use in everyday life.  However, familiarity with concepts introduced and practiced in this section should serve as preparation for discussions in later sections, particularly Categorical Logic, which should be of practical value to everyone.

Symbolic logic is rational thinking in its purest and simplest form, an abstraction of normal reasoning.  It is essentially the study of the general forms of logical reasoning without the content of specific cases.  As such, it isn't of much use to most people in day-to-day decision-making.  However, it's a valuable tool for understanding the underpinnings of logic, and also for discussing the mechanical details of conventional forms of reasoning, such as categorical logic.  Thus, we won't study symbolic logic in great depth, but only sufficiently to pick up whatever will be helpful in understanding other concepts.

Symbolic logic is in a way similar to algebra in mathematics.  Algebra uses letters of the alphabet to represent actual numerical values, and arithmetic symbols to represent the operations to be performed on these values.  Recall the simple geometric formula for the area of a rectangle: A = l ´ w, where A represents area, l represents length, and w represents width.  To find the area of any rectangle (for example, a floor to be tiled or carpeted), we simply plug in the figures for the actual length and width—say, 7 meters long by 6 meters wide—then perform the multiplication operation indicated by the ´ symbol, and we get an area of 7 ´ 6, or 42 square meters.  The algebraic formula shows us how to figure the area of any rectangle, no matter what its particular dimensions might be.  With the universal relationship of a rectangle's area to its length and width encapsulated in an easily remembered algebraic formula, we don't have to stop and think the whole relationship through from scratch each time we need to find the area of a different rectangle.  And we do much the same thing for the areas of other geometric figures, like triangles and circles, using a standard mathematical formula for each.

In symbolic logic, we do something similar, using letters to represent statements, and symbols to represent logical operations.  Some techniques of symbolic logic are quite complicated—not the sort of thing most people would choose to study, even if they were paid to do so.  Our study of symbolic logic will be confined to what's called statement logic or proposition logic, which is relatively simple and straightforward, and thus helpful in grasping the principles of organized reasoning.

For example, let's use the letter w to represent the statement "the weather is chilly," g to represent "I go to the gym," and a right-arrow symbol* "" to represent an IF...THEN relationship between the two statements.  In this way, we can represent the complex idea, "If the weather is chilly, then I go to the gym," as "w → g."  We can likewise represent any other two similarly related statements exactly the same way: "the weather is warm" and "I go swimming," or "they are women now" and "they were once girls" can be put into structurally identical IF...THEN relationships and expressed symbolically as "w → g."

Now, what advantage there might be in doing this might not be immediately apparent.  But consider that the same logical rules apply to all IF...THEN relationships, and then we glimpse the advantage of using this sort of shorthand to represent the relationships of all such patterns of thinking in general, and what inferences we can and cannot logically draw from them.  This relieves us of having to figure out the implications of each example individually—or having to rewrite an entire line of thought longhand whenever we want to try revising something.  Not only that, but real-life logical relationships can get quite complicated, often involving several logically connected relationships.  Evaluating the overall implications of such relationships as represented by a compact group of letters and symbols is certainly easier than wrestling with pages of full-blown statements.

Next: Logical Operations

*NOTE: This and following sections employ standard logic symbols, which are installed on most PC's manufactured since 2003.  If you're using an older computer, you might need to update your existing fonts or install new ones in order to view the symbols properly.








Times New Roman


resembles lower-case "v"


Times New Roman


right-pointing arrow


Times New Roman


left-and-right-pointing arrow





three dots arranged in a triangle

If, on each line of the preceding chart, the character you see in the Symbol column on the left corresponds  to what's in the Description column on the right, your computer's installed fonts are good to go.