Deductive Arguments and the Truth-Table Method
Intro to Philosophy, Spring 2012
Torrey Wang
Handout 2
I. Deductive reasoning
The following schematic arguments all exhibit forms of reasoning that are intended to be
deductive.1
Modus Tollens
1. A ⊃ B
2. ¬B
3. Therefore, ¬A
Barbara
1. Aab
2. Abc
3. Therefore, Aac
Affirming the Consequent
1. A ⊃ B
2. B
3. Therefore, A
But what is deductive reasoning? What sorts of argument qualify as a deduction?
First, notice that each of these arguments is composed of initial statements which are
expressly intended to logically imply its last statement, as suggested by the inclusion of the illative
“therefore” in each argument’s last statement. The statements doing the implying are these
arguments’ premises, which are denoted by ‘1’ and ‘2’ above. The statement being implied in each of
them is its conclusion, denoted by ‘3’. A deductive argument is hence any argument which seeks to
advance the logical truth of a conclusion by way of introducing premises which imply it. Deductive
reasoning, then, is any form of reasoning which involves giving such an argument.
In § II, we will look at the nature of the relationship of logical implication.
‘A’ and ‘B’ in Modus Tollens and Affirming the Consequent are understood as sentential variables which stand for any truthevaluable sentence in one’s language, while ‘⊃’ is the logical connective meaning “if…then…” and ‘¬’ is the sentential
operator meaning “not the case that…” ‘The ‘A’s in Barbara should be understood as the universal quantifier meaning
“all,” and serves to attribute whatever is named by the second of the lowercase letters to its right to whatever is named
by the first letter. (For example, “A(ll) b(oys) c(ry).”)
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II. Deductive validity
In the previous section, I characterized the notion of logical implication as a structural feature of
deductive arguments. In this section, I will characterize logical implication in terms of a deductive
argument’s susceptibility to logical evaluation.
An interesting feature of deductive arguments is that they do not require that we find out
whether their premises state actual truths in order for us to determine the truth of their conclusion.
Indeed, we can evaluate a deductive argument just by studying its form. This is because logical
implication is the necessary truth of a conclusion that results from assuming the truth of an
argument’s premises. How do we know that a set of premises logically imply some conclusion? That
is to say, how do we know that an argument is valid?
An argument is valid if and only if assuming its premises are true, its conclusion
necessarily follows (i.e., follows as a logical consequence).2
Validity
The necessity of a conclusion from premises whose truth is assumed is guaranteed by
accepted logical “rules of inference.” But how do you evaluate an argument according to a rule of
inference if you do not know of—or do not think you know of—any to boot? To assure you that
you do know of at least one, the most important one, no less, consider the following valid schematic
argument:
Modus Ponens
1. A ⊃ B
2. A
3. Therefore, B
What Modus Ponens says is simply that whenever the truth of a conditional is assumed, such as “if the
crocodile is a reptile, then it is cold-blooded,” then if the truth of its antecedent (“the crocodile is a
reptile”) is assumed also, its consequent (“the crocodile is cold-blooded”) necessarily follows as a logical
consequence.
Minimal reflection ought to convince you that implicit knowledge of the inference rule
captured by Modus Ponens—that from the truth of a conditional and its antecedent, one may infer its
consequent—is presupposed by your ability to construct and understand a deductive argument. If
you’re especially astute, you’ll notice that the rule of inference captured by Modus Ponens is what the
notion of validity essentially defines! Modus Ponens, the law of non-contradiction (that a statement
cannot both be true and false), and the law of excluded middle (that a statement cannot be neither
2 Notice that this means that if an argument contains no possible assignment of truth-values such that its premises are
true, it is by definition valid! More on this in § III.
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true nor false) give rise to all the rules of inference that not only logic studies, but that we appeal to
in ordinary reasoning.
The foregoing thus suggest a natural method for evaluating the validity of any deductive
argument: to figure out if a deductive argument is valid, one need only examine all the possible cases
in which its premises are true. Then, if every one of the conclusions corresponding to these cases is true
also, the argument is valid. A single case in which a false conclusion is matched to true premises
renders an argument invalid. Conversely, to figure out if a deductive argument is invalid, one
examines all the possible cases in which its conclusion is false. Then, if any of the premises
corresponding to these cases is true, the argument is invalid. If no true premises are matched to a
false conclusion in any case, the argument is valid.
In the next section, we look at a mechanical procedure for reliably determining the validity
or invalidity of a deductive argument whose premises and conclusion are expressible as unquantified
propositions—statements of the form “A and B,” “A or B,” “If A, then B,” or “not-A,” where ‘A’
and ‘B’ stand for individual statements. (So we won’t be talking about Aristotelian syllogism
anymore.)
III. The Truth-Table method
The method just described for evaluating an argument for validity or invalidity has probably
seemed to you to be unintuitive and perhaps unhelpfully so. Let us then see a concrete application
of the principles involved in carrying out such a method by employing the time-honored Truth-Table
procedure for evaluating validity. For the purposes of this introductory handout, I shall introduce
just a few of the most familiar rules of inference, which should be immediately obvious to you.
(Though note the last two, which are presupposed, and hence not technically rules of inference.)
Inference Rule
Description
Simplification
Given (A and B), infer A.
Disjunctive Syllogism
Given (A or B) and not-A, infer B.
Double Negation Elimination
Given not-not-A, infer A.
Conditional3
Given (If A, then B), infer (not-A or B)
Non-Contradiction
not-(A and not-A)
Excluded Middle
(A or not-A)
3 I find that it is helpful to think of the truth-conditions of a true conditional as something of an analogue to the truthconditions of a valid argument. This is my paltry redemption of footnote 2 of this handout.
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The following is a step-by-step explanation of how to construct a truth-table and to conduct
a logical analysis with it, using an argument which is an instance of Affirming the Consequent.
Affirming the Consequent*
1. If (A) the Bible is true, then (B) the man named Jesus existed.
2. (B) The man named Jesus existed.
3. (A) Therefore, the Bible is true.
Step 1
Determine how many unique statements there are in the argument you wish to analyze,
and assign to each an uppercase letter for convenient reference in the future.
A quick check shows that Affirming the Consequent* consists of two distinct statements, namely, (A)
The Bible is true, and (B) The man named Jesus existed. With this information, we can figure out
how many rows to include in our truth-table according to the following rule in Step 2.
Step 2
Create n columns and 2n rows for n unique statements.
It is easy to see the rationale behind this rule. By the rule of Excluded Middle, each unique
statement is either true or false. Since there are 2 unique statements in our argument, there are 4
unique ways of assigning truth-values to them that we must consider. But, perhaps more pressingly,
you might instead be wondering, what are the rows in a truth-table supposed to tell us about an
argument? Roughly, each row represents a unique way there is of assigning truth-values to an
argument’s premises. This in turn tells us how many total ways there are of an argument’s going right
or wrong. That is to say, we are interested in assigning truth-values to the premises of a row because
each row is supposed to model one way in which the world might turn out being. Each row hence
describes in our language a possible case, or way, of an argument’s being true or false according to
how the world actually is or might turn out. Simply put, a truth-table uncovers all the possible ways of
being for our world that there are with respect to the premises of any given argument.
According to Step 2, then, we create 2 columns and 4 rows for Affirming the Consequent*. The
relevant columns appear under “Statements,” while I’ve gone ahead and filled in the truth-values for
each row underneath it.
Step 3
Systematically fill in truth-values for the n unique statements.4
Assigning truth-values to premises for a truth-table can seem a daunting task if one is working with many unique
statements, but there is a simple formula one can follow to mitigate the headache to follow. Suppose an argument you’re
evaluating contains 3 statements. You can ensure you cover all the unique possible ways of assigning truth-values to its
premises in each row by first assigning ‘T’ to the first 4 rows of the first statement, ‘F’ to the last 4 rows of the same, ‘T’
to the first 2 rows of the second statement, ‘F’ to the next 2 of the same, and then ‘T’ to the next 2 of the same, etc., and
lastly, by alternating assigning ‘T’ and ‘F’ to each row of the third statement.
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Hence:
Statements
(A) The
Bible is true
Premise 1
(B) The
If the Bible is true,
man named then the man named
Jesus existed Jesus existed.
1
T
T
2
T
F
3
F
T
4
F
F
Premise 2
Conclusion
The man named
Jesus existed.
The Bible is true.
And, if you’ve been attentive, you probably noticed that I’ve fast-forwarded one additional step:
Step 4
Create a column for each premise and conclusion.
We’ve now finished the preliminary portion of constructing our truth-table, and are just
about ready to begin evaluating our argument. Keeping in mind our definition of validity, recall that
what we really want to do is to examine just those cases in which each premise of an argument is
true. Thus, we shall concern ourselves only with those possible ways of an argument going right or
wrong that involve only true premises.
Step 4.5
Step 5
Fill in truth-values for each premise of each row.
Identify just those rows, i.e., possible assignments of truth-values, whose premises are
all true, and consider just those rows for logical evaluation.
The table now looks like the following.
Statements
(A) The
Bible is true
Premise 1
(B) The
If the Bible is true,
man named then the man named
Jesus existed Jesus existed.
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Premise 2
Conclusion
(B) The man named
Jesus existed.
(A) The Bible is true.
1
T
T
T
T
2
T
F
3
F
T
T
T
4
F
F
T
How were the truth-values for the 4 rows under Premises 1 and 2 obtained? With respect to
Premise 1, as I suggested earlier in stating our abbreviated inference rules, it is a rule of logic that a
conditional is true just in case either its antecedent is false or its consequent is true. (This is the
Conditional inference rule that I stated earlier.) By importing the truth-values already assigned to
statements (A) and (B) during Step 3 to each row into our evaluation of their corresponding
Premise 1, then, I was able to determine the truth-values for all 4 rows of Premise 1 according to the
Conditional rule. With respect to Premise 2, I simply copied their truth-values over from the truthvalues that we already assigned to their corresponding statement (B)’s during Step 4.5.
In Step 5, we were told to ignore all rows in which there is even a single false premise.
Following that directive, we get the following truncated truth-table (deleting irrelevant rows for
maximum perspicuity):
Statements
(A) The
Bible is true
Premise 1
(B) The
If the Bible is true,
man named then the man named
Jesus existed Jesus existed.
Premise 2
Conclusion
(B) The man named
Jesus existed.
(A) The Bible is true.
1
T
T
T
T
3
F
T
T
T
We’ve now arrived at our final step. All we have left to do now is to fill in the remaining
boxes under “Conclusion” using information borrowed from their corresponding statement (A)’s—
remembering, again, to ignore rows in which one or more premises are false. Once we’ve done this,
we simply see if any of the rows which contain only true premises is matched to a false conclusion. If
there is such a row, we will have proved that Affirming the Consequent*, and indeed, any argument
which shares its form, is invalid.
Step 6
Fill in truth-values for each conclusion of each remaining row. If there are no rows
with false conclusions, the argument is valid. Otherwise, it is invalid.
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Our final truth-table should now look like this:
Statements
(A) The
Bible is true
Premise 1
(B) The
If the Bible is true,
man named then the man named
Jesus existed Jesus existed.
Premise 2
Conclusion
(B) The man named
Jesus existed.
(A) The Bible is true.
1
T
T
T
T
T
3
F
T
T
T
F
Since the shaded row, row 3, describes one possible way to assign truth-values to Affirming
the Consequent* such that its conclusion is false even though its premises are true, we have indeed
proven that this argument is invalid.
IV. Checking your work
Suppose you have just finished evaluating a deductive argument by way of the Truth-Table
method described above. How do you know, without embroiling yourself in further tedious truthtabulating (i.e., drawing up another truth-table), that your truth-table is accurate? Such a concern
must necessarily disquiet those of us who don’t have much faith already, as it is, in our ability to
track multiple things at the same time, let alone keep track of so many columns and rows, statement
letters, and rules of inference, in showing that an argument is valid or invalid.
The most obvious response is the best one, I think. One can check whether the result
delivered by the Truth-Table method is accurate by envisaging, or imagining, scenarios which can be
modeled by the invalidity-causing row in her original truth-table. If she can’t do this without
violating one or more of the premises for that row, then she must have messed up somewhere in
assigning truth-values to premises and conclusion. Such a mistake is very common, for the reason I
just cited. And, if that was the only row in which an assignment of truth-values was such as to
render the argument invalid, it may turn out that one’s initial verdict of invalidity is off the mark. In
the case of Affirming the Consequent*, we check out work by first supposing, in accordance with the
case described by row 3, it possible that Jesus existed (Premise 2), that the conditional that if the
Bible is true, then Jesus existed (Premise 1) is true also, but that the Bible is true is false. Then, we
ask ourselves: is this actually possible? Indeed, it is. Suppose the person the Bible identifies as Jesus
had been a rather ordinary person, and was only made the subject of a fictional work, which we now
call the Bible, that was created by corrupt leaders in the Roman Empire hoping to use it for political
gain. In such a case, premises 1 and 2 would be true, and the conclusion false. Affirming the
Consequent* is indeed invalid.
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This method thus provides an easy way of checking for accuracy in one’s use of the TruthTable method without forcing one to need to draw another truth-table. This can save quite a bit of
time. On the other hand, checking in this manner may yet turn out to be an extremely involved task,
especially if one’s premises are highly convoluted, and involve lots of, say, embedded conditionals or
disjunctions or negations of disjunctions of conditionals, etc. In such a scenario, one would like to
have a proof method instead which encodes the information yielded by the Truth-Table method in a
more streamlined fashion. But that is a topic for those of you who wish to take a course dedicated to
symbolic logic, not Intro to Philosophy!
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