Formal Logic Terms and Diagrams Defined

Every Formal Logic relationship features at least two separate variables linked in a relationship. The
variables represent groups or ideas. For instance, in the example on the previous page, “A” represented
“authors” and “H” represented “happy.” These variables—and LH and F—were linked in relationships
that were represented by the diagrammatic elements of “some,” “---> ,” and “<---> .” Below, we
discuss the components of Formal Logic diagrams in more detail.
1. Choosing Symbols to Represent Each Variable
Choosing symbols to represent each group or idea is easy: simply choose the letter or letters that, to you,
best represent the element. For most people, the best symbols are the first letter of each word or words. For
example, using “A” to represent “authors” makes it easy to remember “authors” when you are examining
your diagram. The exact letters you choose to represent each group are not critical; what is important is
that you use those same letters to represent the group throughout your diagram and inferences. This is
especially important as some terms are negated. For example, if you represent “happy” with “H” as you
begin your diagram, and later you are presented with a seemingly new element, “unhappy,” do not create a
new variable, “UH.” Instead, simple negate “happy” and use “ H .”
2. Conditional Reasoning Terms and Diagrams
Many of the relationship indicators used within Formal Logic problems are terms you are familiar with
from Conditional Reasoning. Conditional indicators such as “if” and “only” yield exactly the same
diagrams that you used in Sufficient and Necessary problems. Let us briefly review those terms and their
resultant diagrams:

A. The Single Arrow (---> )
Introduced by sufficient and necessary words such as: if...then, when, all, every, and only,
where both elements are positive or both elements are negative.
Example Statement: All X’s are Y’s (X and Y both positive)
Diagram: X---> Y
Example Statement: If you are not T, then you are not V (T and V both negative)
Diagram: T---> V
You can, of course take the contrapositive of this diagram and force both terms to
be positive:
V---> T
B. The Double Arrow (<---> )
Introduced by “if and only if” or by situations where the author implies that the arrow goes
“both ways,” such as by adding “vice versa” after a conditional statement.
Example Statement: X if and only if Y
Diagram: X <---> Y
Example Statement: All W’s are Z’s, and all Z’s are W’s
Diagram: W <---> Z
Double-arrow statements allow for only two possible outcomes: the two variables occur
together, or the neither of the two variables occur.
C. The Double-Not Arrow (<-I--> )
Introduced by conditional statements where exactly one of the terms is negative, or by
statements using words such as “no” and “none” that imply the two variables cannot “go
together.”
Example Statement: No X’s are Y’s
Diagram: X <-I--> Y
Example Statement: If you are a T, then you are not a V
Diagram: T <-I--> V
Remember, the statement above produces a diagram, T V , which can
more properly be diagrammed as T <-I--> V.
3. New Terms and Diagrams
A. Relationships involving Some
The word “some” can be defined as at least one, possibly all. Take a careful look at that
definition—it is different than what most people expect because some includes the
possibility of all. Thus, if I say, “Some of my friends graduated last week,” using the
definition above it could in fact be true that all of my friends graduated last week.
When diagramming statements involving some, simply place the word some between the
two elements:
Example Statement: Some X’s are Y’s
Diagram: X some Y
When “some” appears as “some are not” (as in “Some X’s are not Y’s”), the interpretation
changes due to the not. “Some are not” can be defined as at least one is not, possibly all are
not. Thus, if I say, “Some of my friends are not present,” it could be true that none of my
friends are present.
One of the most popular ways to introduce the idea that some are not is to use the phrase
not all, which is functionally equivalent to some are not.
When diagramming statements involving some are not, simply place the word some
between the two elements and negate the second element:
Example Statement: Some W’s are not Z’s
Diagram: W some Z
Example Statement: Not all T’s are V’s
Diagram: T some V
The LSAT introduces the concept of some in a variety ways, including the following relationship
indicators:
some
at least some
at least one
a few
a number
several
part of
a portion
B. Relationships involving Most
The word “most” can be defined as a majority, possibly all. Again, take a careful look at
that definition—it is different than what most people expect because most includes the
possibility of all. Thus, if I say, “Most of my friends graduated last week,” using the
definition above it could in fact be true that all of my friends graduated last week.
When diagramming statements involving most, simply place the word most between the
two elements and place an arrow under the most pointing at the second element:
Example Statement: Most X’s are Y’s
Diagram: X most Y
When “most” appears as “most are not,” the interpretation changes due to the not. “Most
are not” can be defined as at majority are not, possibly all are not. Thus, if I say, “Most of
my friends are not present,” it could be true that none of my friends are present.
When diagramming statements involving most are not, simply place the word most
between the two elements, place an arrow under the most pointing at the second element,
and negate the second element:
Example Statement: Most W’s are not Z’s
Diagram: W most Z
The LSAT introduces the concept of most in a variety ways, including the following relationship
indicators:
most
a majority
more than half
almost all
usually
typically
C. Final Note
Some students ask if there is a contrapositive for some and most statements. The answer is No.
Only the arrow statements like all have contrapositives; some and most do not because they do not
necessarily encompass an entire group.