An Overview of Arguments
Definition An argument consists of a set of statements called premises along with another
statement called the conclusion. In a sense, arguments are just extensions of conditional
statements with multiple components in the premise.
Example The following is a simple (and famous) argument: “Aristotle was human and all humans
are mortal. Therefore, Aristotle was mortal.” Here the premise consists of the conjunction “Aristotle
was human and all humans are mortal,” while the conclusion is the statement “Aristotle was mortal”.
Criterion for Validity
An argument is valid if the conclusion is TRUE whenever all the
premises in the argument are assumed to be TRUE. Otherwise, the argument is invalid.
Checking for Validity: Analyzing an Argument
There are various ways to analyze an
argument: using truth tables, using standard forms, using Euler diagrams, and even using circuits.
In any of these cases, always start by writing the argument in its symbolic form:
P1
P2
(Premise 1)
(Premise 2)
(Premise n)
Pn
_______________
Conclusion
Q
Example In the argument given above, if we let h be the statement “Aristotle was human” and let
m be the statement “Aristotle was human,” then we have the following symbolic form for this
argument:
(Premise 1)
h
(Premise 2)
hm
_____________________
(Conclusion)
m
Analyzing an Argument with Truth Tables To check for validity using truth tables, construct a
truth table that shows all the possible truth values for each of the premises in the argument as
well as all truth values for the conclusion (the final column in the table). Applying the criterion
for validity, you must then check that for every row in the table where all premises are TRUE,
you also have a TRUE conclusion. If this is not the case, then the argument is invalid.
Analyzing an Argument with Circuits The criterion for validity also implies that an argument
is valid if and only if the compound statement  P1  P2   Pn   Q is a tautology. This is the
symbolic form used when checking for the validity of an argument using a circuit. In that case,
you must show that the light bulb on the right end of the circuit is always ON in all cases.
Analyzing an Argument Through Standard Forms Perhaps the best (and simplest) way to
check whether an argument is valid or not is by using standard forms (when applicable). Below
are tables with some of the most commonly used standard forms for arguments. These are
widely used in courts and debates.
VALID FORMS
MODUS PONENS
pq
p
_____
q
MODUS TOLLENS
pq
~q
_____
~ p
LAW OF SYLLOGISM
(TRANSITIVITY)
pq
qr
_____
 pr
DISJUNCTIVE
SYLLOGISM
pq
~p
_____
q
pq
q
_____
p
FALLACY OF THE
INVERSE
pq
~p
_____
~q
INVALID FORMS
FALLACY OF THE
CONVERSE
Analyzing an Argument with Euler Diagrams
Euler diagrams are only applicable to
argument that involve quantifiers. For a detailed discussion on those types of arguments, read
Section 3.6.