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 daytoday decisionmaking. 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 rightarrow 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 reallife 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 fullblown
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.
Symbol 

Font 

Description 
˅ 

Times New Roman 

resembles lowercase "v" 
→ 

Times New Roman 

rightpointing arrow 
↔ 

Times New Roman 

leftandrightpointing arrow 
\ 

Symbol 

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.