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**4. Deductive Reasoning -- Good and Bad
**

4.1 Deductive Reasoning - Validity and Soundness

4.2 Deductive Logic -- Aristotle’s Square

4.3 Deductive Logic -- Aristotle’s Syllogistic Logic

4.4 Deductive Logic -- Sentence Connectives and Truth Tables

4.5 Deductive Logic -- Sentential Logic

4.6 Deductive Logic -- Summing Up

**4.1 Deductive Reasoning -- Validity and Soundness
**

Now that we know something about arguments, we will look at a very narrow variety of arguments that are notable for their seemingly “precise nature.”

We can represent this argument as follows:

Note that, as the argument states, if it is true that *all mammals are warm
blooded creatures* and it is likewise true that *all dogs are mammals*,
then **it must be true** that *all dogs are warm blooded*.
In some sense we might say that **“the truth” of the conclusion
is “contained in”” the truth” of the premises**.
Moreover, since it is in fact true (or so we believe) that “all mammals
are warm blooded” and “all dogs are mammals,” it must be in
fact true that “all dogs are warm blooded.”

It is important to see that there are two notions at work here: an “if--then”
or “hypothetical notion” and a straightforward “factual”
or “truth notion.” If it is true that “mammals are warm blooded”
and “dogs are mammals,” then it must be true that “dogs are
warm blooded.” This is the hypothetical claim: if one set of claims --
the premises -- are true, then so is the other -- the conclusion. But this “hypothetical
claim” does not make the straightforward claim that either premises or
conclusion is true. If we do make the “hypothetical claim,” however,
and in addition claim that the premises are in fact true, this gives us license
to claim that the conclusion is also in fact true.

In illustration of this distinction between “hypothetical” and “factual,”
consider the following argument and diagram.

EX. 4-1-c

All reptiles are warm blooded creatures.

All fish are reptiles.

-------------------

All fish are warm blooded creatures.

Here, as with our first argument, if it is true that “all reptiles are
warm blooded” and “all fish are reptiles,” then it must be
true that “all fish are warm blooded.” Again, there is a sense in
which all of the logical information expressed by the premises includes the
logical information expressed by the conclusion (the truth of the premises contains
the truth of the conclusion). The particular “if--then” structure
of the argument guarantees this. When a deductive argument has such a structure
we say that it is **valid**.

*A valid argument is an argument of such a form that if the premises
are true, the conclusion must be true.*

Both our “dog” and “fish” arguments are valid and indeed
they are instances of the very same valid argument form. Here it is:

Here ‘S’, ‘P’, and ‘M’ function as kinds
of “variables” that can take on different “values” in
particular arguments. In accord with a long tradition, we use ‘S’
to refer to the “subject term” of the conclusion, ‘P’
to refer to the “predicate term of the conclusion” and ‘M’
to refer to the “middle term” that appears in each of the premises.
We say more about this in section 4.2.

Returning to our “dog” (4.1-a-b) and “fish” (4.1-c-d)
arguments, clearly one of these arguments is “better” than the other
and this is because it not only has a “good if--then structure,”
but also it has premises that are “factual.” That is, the premises
of our “dog” argument are in fact true rather than false and thus
we say it is not only valid but also sound.

Sound arguments are valid arguments with all true premises. That is, a sound
argument (1) is a valid argument and (2) has all true premises

So we have outlined two important notions of what counts as a “good deductive
argument.” “Validity” has to do with the “form,”
or “if--then structure” of the argument. If the structure guarantees
that “if the premises are true, then the conclusion must be true,”
then the argument is valid. If in addition to having such a “valid structure”
an argument has all true premises, then the argument is sound. Thus, “soundness”
has to do with both “structure” and “content” and sound
arguments are deductive arguments of the “highest quality.”

But of course deductive arguments can fail to be valid and thus likewise fail
to be sound. Consider the following:

If, as the diagram depicts, it is true that both mammals and dogs are “warm
blooded,” this by itself does not allow us to infer that it must be true
that “all dogs are mammals.” We know that they are but the “structure
of the argument” does not guarantee this. This is an invalid argument
since it is not an argument of such a form that the truth of the premises guarantees
the truth of the conclusion. Note too that despite the fact that all of the
premises as well as the conclusion of this argument are in fact true, because
the argument is invalid, it is also unsound. Sound deductive arguments are ones
where true conclusions are conclusively inferable from true premises. If either
“guaranteed inferability” or “true premises” is lacking,
then the argument is unsound.

**4.2 Deductive Logic -- Aristotle’s
Square
**

Over 2000 years ago the Greek philosopher Aristotle wrote a series of works, known collectively as the Organon, in which he laid the foundations for all of what we currently understand as reasoning or logic. In particular, Aristotle’s account of deductive logic continues to be employed today. What follows in this section and the next is a brief exposition of some of the useful highlights that have grown from Aristotle’s account.

We begin with what has come to be called “Aristotle’s Square of Opposition.” The “Square,” depicted below, illustrates some of the “truth-relations” that Aristotle found to hold among various kinds of statements. The four primary kinds of statements he considers were later given the “nicknames” of ‘A’, ‘E’, ‘I’, and ‘O’ and as a group have come to be called categorical statement-kinds.

Each of these statement-kinds points to a relation between two particular kinds
or classes of things, an “S” or subject-kind and a “P”
or predicate kind. Moreover, each statement kind is said to have a quantity
and a quality. Quantity is said to be either universal or particular. A and
E statement-kinds are said to be universal in quantity in that they say something
about “all members of a kind or class.” I and O statement-kinds
are particular in quantity since they aim to say something merely about “some
(at least one) member of a kind or class.” A and I statement-kinds are
affirmative in quality since they say something “positive” while
E and O statement-kinds are negative in quality since they “deny”
something. (Indeed, the nicknames ‘A’, ‘I’, ‘E’,
and ‘O’ were taken from the first two vowels of each of the two
Latin terms ‘affirmo’ and ‘nego’.) Thus we can identify
each statement-kind in terms of its quantity and quality as follows: A = universal
affirmative; E = universal negative; I = particular affirmative; O = particular
negative.

Here is Aristotle’s Square of Opposition:

Working with only the four statement-kinds A, E, I, and O, Aristotle identifies
some important “truth-relations” existing between the statement-kinds.
If it is true that All S is P (All salamanders are pink), then it must be false
that No S is P (No salamanders are pink). However, if it is false that All S
is P, then we cannot determine the truth of No S is P. (That is, if we have
some salamanders slithering around on the table and it’s false that “all
of them are pink,” we can’t tell from this alone whether none of
them are pink or whether some are and some aren’t.) Thus, while statements
of kind A and E can’t be true together, they can be false together. This
is what we call the contrary relation. “Contraries” can’t
both be true but they can both be false.

Statements of kinds A and O as well as statements of kinds E and I are contradictories.
They must have “opposite truth values.” Thus if it is true that
“some salamanders on the table are pink,” then it must be false
that “none of them are pink” and if there are salamanders slithering
on the table and it is false that “some of them are pink,” then
it must be true that “none of them are pink.” Analogous claims hold
true for statements of kinds A and O.

The relationship of implication holds between Aristotelian statements of kind
A and corresponding statements of kind I and implication also holds between
corresponding E and I statements. If there are salamanders on the table and
it is true that “all of them are pink,” then it must be true that
“some (at least one) of them are pink.” Similarly, if a survey of
the salamanders reveals that “none of them are pink,” that is, E
is true, then it must likewise be true that O, “some of them are not pink.”
Implication, however, is a “one-way relation.” While the truth of
A statements imply the truth of corresponding I statements, the truth of I statements
does not imply the truth of A statements. That is, the fact that some salamanders
on the table are pink does not imply that all of them are. Likewise for O and
E statements.

Interestingly, however, we learn from Aristotle’s Square that if I statements
are false, then corresponding A statements must also be false. Similarly for
O and E statements. We can prove this using the Square as follows:

1. If I is false, then E must be true -- Why? They are contradictories.

2. But if E is true, then A must be false -- Why? E and A are contraries.

--------------------------------------------------------------------------------

3. If I is false, then A is false.

Finally, we point to the sub-contrary relation that holds between statements
of kind I and O. Here Aristotle identifies the fact that if there are salamanders
on our table, then while it is easy to see how it can be true that both some
are pink (I), and some are not (O), it is impossible that both of these statements
are false. Thus, while statements of kinds I and O can be true together, they
cannot be false together. This is the sub-contrary relation.

It is worth noting some additional truth-relations between the four statement
kinds that do not appear on the Square. But, in order to identify them, we must
first understand the distinction between a kind or class and its complement.
Straightforwardly, we say that the class S of salamanders includes all of the
salamanders that exist. The complement of S is the class non-S or S (we say
“S-bar”). S is the class that includes everything in the World that
is not a salamander. (This includes you, me, the Atlantic Ocean, the door knob
on your bedroom … everything that is not a salamander.)

Given this understanding of a class and its complement, we can perform certain
operations on categorical statements (A, E, I and O) and look to see what truth-relations
the “changed statements” bear to the originals. We begin with the
operation obversion. To obtain the obverse of a given categorical statement,
we:

1. Change the quality of the given statement to its opposite (affirmative to
negative or negative to affirmative);

2. Substitute for the term in the 2nd position (“P” or “predicate
position”) its complement.

So, for example:

A quick inspection shows that each categorical statement is logically equivalent
to its obverse. That is, if the statement is true, then so is its obverse and
if the statement is false its obverse is false as well. Thus, logically equivalent
statements have the same truth value and performing the operation of obversion
on a categorical statement (A, E, I, or O) yields a logically equivalent statement.

A second important operation is conversion. To obtain the converse of a given
categorical statement, we merely switch the terms that are in the 1st or subject
position with those in the 2nd or predicate position. Thus:

Are any of these categorical statements logically equivalent to their respective
converse statements? Clearly it does not follow that if it is true that all
salamanders are pink, then it is also true that all pink things are salamanders!
It is easy for us to imagine a world where not only all of the salamanders are
pink but there are many other pink things (roses, flamingoes …) as well.
We can quickly see that conversion fails to yield logically equivalent statements
when performed on A and O statements. (Think for yourself about the conversion
of the above O statement and why it need not have the same truth value as its
converse.)

But conversion does yield logically equivalent statements when performed on
E and I statements. From a logical standpoint it makes no difference whether
we say that “No salamanders are pink” or whether we say that “No
pink things are salamanders.” Similarly, “Some salamanders are pink”
is logically equivalent to “Some pink things are salamanders.” So
we say “conversion works for E and I” but “conversion fails
for A and O.”

Finally, one more useful operation that we can perform on categorical statements,
contraposition. To obtain the contrapositive of a given categorical statement,
we:

1. Switch the terms in the subject and predicate positions;

2. Substitute for each its complement.

While this is not as easy to figure out as the converse and obverse relations
above, it turns out that the contraposition operation when performed on A and
O statements does in fact yield logically equivalent expressions. It fails,
however, when performed on E and I statements.

To sum up, Aristotle’s Square and his further investigations of truth-relations
that hold between various categorical statements provide important starting
points for constructing systems of deductive reasoning that allow us to test
the validity of certain kinds of deductive arguments. However, it is important
to see that in this section we have already been discussing various forms of
valid deductive arguments. When we say, for example, that an A statement is
logically equivalent to its contrapositive, part of what we are claiming is
that the following is a valid argument:

All S is P

----------------------

All non-P is non-S

But there is much more to be said about valid reasoning based on Aristotle’s
Square and to a brief account of what has come to be called Aristotelian Syllogistic
Logic, we now turn.

**4.3 Deductive Reasoning -- Aristotle’s Syllogistic
Logic
**

Consider a simple argument:

Based on Aristotle’s work, this is an example of what today we call a
Standard Form Categorical Syllogism. This needs some explaining.

First, the above argument is a syllogism. This means simply that it is an argument
with two premises and a conclusion. Second, it is a categorical syllogism. As
such it is a syllogism that employs only categorical statements -- A, E, I,
and O kinds of statements from Aristotle’s Square. Finally, it has standard
form. That is, it is a categorical syllogism that:

Surprisingly, perhaps, it turns out that this definition allows us to identify
exactly 256 varieties of standard form categorical syllogism many of which continue
to be among the most commonly employed arguments today.

We identify these 256 syllogistic forms in terms of what we call mood and figure.
First, a SFCS (standard-form categorical syllogism) has a mood that is nothing
more than a listing from top to bottom of the kinds of categorical statements
that compose the argument. In our example above, each premise as well as the
conclusion of the argument is an A-type categorical statement. Thus the mood
of this argument is AAA. But in addition to mood, SFCSs can also differ as regards
the arrangements of the terms in the argument. There are four possible arrangements
or figures.

Figure 1

M P

S M

----

S P

Figure 2

P M

S M

----

S P

Figure 3

M P

M S

----

S P

Figure 4

P M

M S

----

S P

If we look at our example 4.3-a we see that it is figure 1. Thus we say that
the above argument is an AAA-1 standard form categorical syllogism. Each SFCS
has four possible kinds of categorical statement (A, E, I, or O) that can occur
in each premise or conclusion and in addition it has a figure. Thus, there are
4 x 4 x 4 x 4 = 256 possible varieties of SFCS.

Let’s look at another of the 256 varieties of SFCSs, an OAO-3.

EX 4.3-d

Some of **M**y favorite things are not **P**ink things

All of **M**y favorite things are **S**alamanders

-----------------------------------------

Some **S**alamanders are not **P**ink

Here the first premise of the argument (containing the P or predicate term of
the conclusion) is an O statement, the second premise is an A statement and
the conclusion is again an O statement. Hence, the mood of the argument is OAO.
If we look at the arrangement of the M or middle terms we see that the figure
of the argument is figure 3 or 3rd figure. Thus, this argument has the form
of an OAO-3 SFCS.

But while Aristotle’s work provides a basis for distinguishing certain
common varieties of syllogistic arguments, none of this by itself helps us to
determine whether these arguments are “good arguments.” That is,
how do we know which of these 256 syllogistic forms will be the forms of valid
arguments and which ones won’t?

If we had been studying Aristotelian logic in Europe 1000 years ago, the next
step in our instruction would be to learn a Latin poem that listed all of the
valid argument forms from the 256. Just for fun, here is the first line of the
poem: “Barbara, Celarent, Darii, Ferioque prioris.” What we have
here are a series of Latin names that were chosen so that the first three vowels
of each name spelled out the mood of a valid first figure (“prioris”)
argument -- AAA-1, EAE-1, AII-1, EIO-1. Through intuition, inspection and argument,
Aristotle and his followers found which of the 256 argument forms were valid
forms and students merely took their word for it and learned the poem. (In some
respects, not too different from what students are often asked to do today!)

But let’s try to determine the validity of these kinds of arguments in
a more intuitive fashion by using diagrams. Consider first how we might try
to diagram the logical information in simple categorical statements.

For A statements, everything that is in S is also in P. For E statements,
there is no overlap between S and P. I statements say that there is some x that
is found in both of S and P while O statements say that there is some x to be
found in S that is not in P.

Diagrams for SFCSs, however, are more complicated since there are three terms
that must be considered. We will begin all of our SFCS diagrams with a rectangle
to symbolize the M or middle term. Given this, let’s consider “Barbara,”
an AAA-1, probably the most common valid form of SFCS.

Looking back at EX. 4.3-a, we begin our diagram by drawing a rectangle to symbolize
the M-kind or class “My favorite things.” Next we look at the first
premise of the argument which says that “All of My favorite things are
Pink” (All M is P). To symbolize this we circumscribe the M rectangle
with the P circle. Next, we diagram our second premise “All salamanders
are among My favorite things” (All S is M) by placing the S circle inside
of the M rectangle. Thus we have diagramed both of our premises and our diagram
is complete.

But now comes an inspection of the diagram in accord with the “key principle”
on which this process for determining the validity of arguments depends. Here
is the “key principle:”

Given this, after diagramming the premises of an argument as above, if the
argument is valid, we should expect to see the conclusion already diagrammed.
The conclusion of the argument under consideration is “All Salamanders
are Pink.” Is this categorical statement already diagrammed? Yes, the
“S-circle” is inside of the “P-circle” in our diagram
and thus this AAA-1 SFCS, and, indeed, any argument having the AAA-1 form, is
a valid argument.

Here is a diagram we might give for the invalid AAA-2 argument expressed in
EX. 4.1-f.

The first premise is “All mammals are warm blooded” (All P is M),
and thus we inscribe the P circle in the M rectangle. For the second premise,
“All dogs are mammals” (All S is M) we must do the same for the
S circle but the difficulty is we have no information about how the S and P
kinds are related. Hence we assume as little as possible. That is, we assume
they are wholly distinct and thus No S is P. Given that we have diagrammed the
premises, we now look to see whether the conclusion is satisfied. The conclusion,
“All dogs are mammals” (All S is P), is not already depicted in
the diagram and hence this AAA-2 SFCS is invalid. (And again, any SFCS having
this form is invalid.)

We conclude this section with one more diagram based on the OAO-3 SFCS from
EX. 4.3-d.

We know that to diagram our first premise, “Some of My favorite things
are not Pink” (Some M is not P), we must put an x in the M rectangle that
is not in the P circle. The problem is again that we do not know how to situate
the P circle in the diagram. Again our solution is to suppose that the M rectangle
and the P circle have nothing in common, they are wholly distinct. To diagram
the second premise, “All of My favorite things are Salamanders”
(All M is S) we circumscribe the M rectangle with the S circle. We now look
to see whether our conclusion is satisfied and, yes, there is depicted in the
diagram something that is an S but not a P and hence our argument is valid.

There are many discussions of more complete and comprehensive accounts of diagramming
techniques for determining the validity of SFCSs. For our purposes it is enough
to see that this is possible and that validity can most often be ascertained
in a reasonably intuitive fashion. In the next two sections we look at a different
variety of deductive reasoning along with some new techniques for determining
validity.

**4.4 Deductive Reasoning -- Sentence Connectives
and Truth Tables
**

The logic that grows from Aristotle’s work is a deductive logic that is based on the relations that hold between kinds or classes. But in the generations that followed Aristotle, there appeared a kind of deductive logic that was formulated not in terms of classes but rather in terms of the relations between sentences (‘propositions’ was the technical term). Eventually, through the work of nineteenth century logicians G. Frege, C. S. Peirce, and others, this sentential logic developed into our contemporary symbolic logic. But while contemporary symbolic logic can be quite complicated, the essence of sentential logic is not, and, indeed, sentential logic retains a practical usefulness that is at least equal to its Aristotelian cousin. We lay the groundwork for sentential logic in this section and complete our short introduction in the next.

What is this essence of sentential logic? Simply it is the fact that most all natural languages (English, Chinese, Swahili, etc.) either explicitly or implicitly have special sentence connectives -- words/techniques for joining sentences together -- that are truth-functional. And, truth-functional sentence connectives are those that allow us to calculate the truth values of the new “complex sentence” that is produced based on the truth values of the simpler sentences on which the truth-functional sentence connectives operate (join).

Sounds complicated but some examples should help. Consider two “simple sentences:”

p: SpongeBob is here.

q: Sandy Cheeks is here.

Now suppose it is true that “SpongeBob is here” and suppose it is also true that “Sandy Cheeks is here.” Given this, what is the truth value of the complex sentence “SpongeBob is here and Sandy Cheeks is here”? Obviously it is true. We understand that for any sentences ‘P’ and ‘Q’, if ‘P’ is true and ‘Q’ is true, then it is likewise true that ‘P and Q’. Moreover, we also understand that if either ‘P’ is false or ‘Q’ is false, then the complex sentence ‘P and Q’ is also false. We can chart this feature of the truth-functional sentence connective ‘and’ in the following truth table:

On the left hand side of the truth table we have listed the possible combinations
of “truth and falsity” that ‘P’ and ‘Q’
can take on. On the first line of the truth table, both ‘P’ and
‘Q’ are true, on the second ‘P’ is true and ‘Q’
false, on the third ‘P’ is false and ‘Q’ true and on
the fourth line both are false.) The complex sentence form ‘P and Q’
is true only on the first line of the truth table. On lines 2, 3, and 4, ‘P
and Q’ is false.

It is traditional in sentential logic to identify five truth functional sentence
connectives.

Here are the truth tables for each of these five connectives.

Following the truth table for ‘P and Q’ we come to ‘Not P’
or, to state its longer version, ‘It is not the case that P’. In
this case the “sentence connective” ‘Not’ operates on
just one sentence and it reverses the truth value of the sentence on which it
operates. The truth table for ‘P or Q’ captures the ordinary language
notion of “one or the other or both.” We can capture the “exclusive
or” that says “one or the other but not both,” with the complex
sentence ‘(P or Q) and (Not (P and Q))’.

We stated above that sentential logic is based on the fact that natural languages
have truth-functional sentence connectives. So far, the match-up between “ordinary
language” and the truth tables has been reasonably straightforward. Not
so, however, for the sentence connective “If __, then __.” “Yes,”
if we consider the first two lines of the truth table (line one, when ‘P’
and ‘Q’ are both true and line two, when ‘P’ is true
and ‘Q’ is false), it does seem to agree with our common sense understanding
of “If P, then Q.” However, lines three and four are a puzzle. In
ordinary language, we are not quite sure what to say when the “if-part”
of the “if __, then __” is false. But in order to avoid gaps in
our truth tables we interpret “If P, then Q” as expressing the claim
“It is not the case that both P is true and Q is false. Thus we claim
that the truth table for “If P, then Q” is identical to the following:

For example, when we say that “If SpongeBob is here, then Sandy Cheeks
is here,” we claim that this is equivalent to saying that “It is
not the case that both SpongeBob is here and Sandy Cheeks isn’t.”

The final truth-functional sentence connective that is traditionally discussed
is the less often employed “if and only if.” We interpret this as
saying “If P, then Q AND If Q, then P.” The upshot of this is that
if we join simpler sentences by means of the connective “if and only if,”
the resulting complex sentence will be true whenever the two simpler sentences
have the same truth value and it will be false when they do not.

**4.5 Deductive Reasoning -- Sentential Logic
**

Thus far we have introduced the foundations of sentential logic but we have yet to show how we can employ these foundations to test for the validity of arguments. In this section we remedy this lack and in the process look at some of the most common forms of valid arguments expressed in terms of sentential logic.

We begin with what is probably “the” most common valid argument form expressed in ordinary language, modus ponens (the “mode of putting” the truth of the first part (antecedent) of the “if __, then __”). Here is an oft’ used example: If it is raining, then the streets are wet. It is raining. Therefore, the streets are wet. If we make “sentence assignments” as follows:

p: It is raining.

q: The streets are wet.

then we can symbolize this argument as:

If p, then q

p

-------------

q

Most people will intuitively conclude that this is a valid argument. That is,
they “see” that this is an argument of such a form that if the premises
are true, then the conclusion must be true. Now, however, we can employ the
truth tables discussed in the previous section to “prove” that this
is in fact a valid argument. Here is the proof:

We give the truth tables of each of the premises, “If P, then Q’,
and ‘P’ and we follow this by giving the truth table of ‘Q’.
Note that when we give the truth tables of ‘P’ and ‘Q’,
we are merely repeating the initial assignments of truth and falsity to ‘P’
and ‘Q’ that we employ to generate truth tables.

We are now in a position to judge whether the argument we have symbolized is
valid. First, remember that in sentential logic we employ truth-functional sentence
connectives that allow us to calculate the truth values that complex sentences
take on based on the truth values of their simpler-sentence-parts. Because we
consider all possible combinations that the “simpler parts” can
take on, this in turn allows us to give a complete accounting of the truth values
of the truth-functionally-connected complex sentences in which they participate.
Thus, by giving truth tables of the premises and conclusion of an argument in
sentential logic, we can look to see whether there is any initial assignment
of truth/falsity that makes each of the premises of the argument true and the
conclusion false. IF THERE ARE NO ASSIGNMENTS THAT MAKE EACH OF THE PREMISES
TRUE AND THE CONCLUSION FALSE, THEN (since we have considered all of the possibilities)
THE ARGUMENT IS TRUTH-FUNCTIONALLY VALID. For modus ponens, since there is no
assignment of truth values to ‘P’ and ‘Q’ (no line on
the truth table) that makes the premises ‘If P, then Q’ and ‘P’
both true, and the conclusion ‘Q’ false, we have shown that this
is a valid argument form and that any particular argument that has this form
is valid.

Compare this with an intuitively invalid argument: If it is raining, then the
streets are wet. The streets are wet. Therefore, it is raining. In symbols:

If p, then q

q

-------------

p

With a bit of thought we see that it is possible that both premises of this
argument are true and yet the conclusion is false. (The streets may become wet
because of a leaky hydrant.) Hence, the argument is invalid. Here is the proof.

Inspection reveals that in the list of initial assignments to ‘P’
and ‘Q’, when ‘P’ is false and ‘Q’ is true
(third line of the truth table), each of the premises of the argument is true
while the conclusion is false. THIS IS IMPOSSIBLE FOR A VALID ARGUMENT and hence
this argument is invalid. (Indeed, we say that this argument commits the logical
fallacy of “affirming the consequent” (consequent -- second or “then-part”
of the “If __, then __”).

We conclude this section with a brief catalogue of some of the most common valid
argument forms (in addition to modus ponens) that we employ in our ordinary,
everyday reasoning. We leave it to the interested reader to employ truth table
techniques to prove that these argument forms are valid.

Modus tollens (the “mode of taking” away the truth of the “then-part”
of the “if __, then __”): If it is raining, then the streets are
wet. The streets aren’t wet. Therefore, it’s not raining.

If p, then q

Not q

-------------

Not p

Hypothetical Syllogism. Note that arguments often contain more than two simple
sentences and this entails that there are more combinations of truth and falsity
that must be considered in a truth table representation of the argument. (For
three simple sentences, eight combinations of truth/falsity; four simple sentences,
sixteen combinations; N simple sentences, 2N combinations.) Here is an example
of a hypothetical syllogism: If SpongeBob is here, then Sandy Cheeks is here.
If Sandy Cheeks is here, then Squidward is here. Therefore, if SpongeBob is
here, then Squidward is here.

If p, then q

If q, then r

-------------

If p, then r

Disjunctive Syllogism. Either SpongeBob is here or Sandy Cheeks is here. Sandy
Cheeks isn’t here. Therefore, SpongeBob is here.

p or q

Not q

-------------

p

Dilemma. SpongeBob is here or Squidward is here. If SpongeBob is here, then
Sandy Cheeks is here. If Squidward is here, then Patrick Starfish is here. Therefore,
either Sandy Cheeks is here or Patrick Starfish is here.

p or q

If p, then r

If q, then s

-------------

r or s

Reductio ad absurdum. This is a kind of valid deductive argument that sounds
unusual but is actually quite common. We often argue as follows: “If what
you say is true, then something “completely nutso” follows. Therefore,
what you say must be false.” In formal logic, the “most nutso”
claim one might make is a “self contradictory statement,” a statement
that must be false. In sentential logic, the most common form of such a statement
is “P and not P.” Thus the sentential logic version of reductio
ad absurdum might look like this.

p

If p, then (q and not q)

q and not q

-------------

Not p

Note that in sentential (formal) logic, “anything and everything”
follows from a contradiction. This is because if there is a self contradictory
statement among the premises, then it will always be impossible for “all
the premises to be true and the conclusion false.” And this is because
it will always be impossible for there to be “all true premises.”

**4.6 Deductive Reasoning -- Summing Up
**

In this Chapter we have looked at a form of reasoning employed in problem solving that has been the subject of research for well over two thousand years. Deductive reasoning is notable for its seeming precise and conclusive nature as well as the simple techniques we have developed to distinguish good deductive reasoning from bad. It is rare, however, that deductive reasoning by itself can solve any of our problems of action/reflection. More help is needed and to an account of another kind of reasoning we now turn.