Logic: A Brief Introduction Ronald L. Hall, Stetson University

Chapter 8 - Sentential Truth Tables and Argument Forms

8.1 Introduction

The truth-value of a given truth-functional compound proposition depends on the truth-values of each of its components. One and the same proposition may be true if its components are all true and false if its components are all false. For example, the propositions, “The cat is on the mat and the dog is in the yard” (“C•D”) is true if both the “C” and the “D” are true, but false if either “C” or “D” is false or if both are false.

A complete interpretation of this proposition will track every possible combination and permutation of truth-values. Interpreting compound propositions that are not very complex is fairly easy. When these propositions become complex, interpreting them becomes more difficult. Of any given proposition, the logician should be able to say one of the following: that the proposition is (1) false under every possible interpretation; (2) true under every possible interpretation; or (3) true under some interpretations and false under others.

Logicians have given names to these three possible complete interpretations of propositions. A complete interpretation of a proposition will determine that it is a tautology, a contradiction, or a contingent proposition. We define these terms as follows:

Tautology : A proposition that is true under every possible interpretation. Contradiction : A proposition that is false under every possible interpretation. Contingent Proposition : A proposition that is true under some interpretations and false under others.

The reason that it is important to know how to recognize these kinds of propositions is because knowing this can be of great help in evaluating arguments.

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Accordingly, logicians have developed a technique for interpretation that will insure that every possible combination and permutation of truth-values a given proposition can have is considered. This is the technique of interpreting propositions with truth tables. We have already introduced them informally in the last chapter. We used these simple tables to spell out the truth-conditions for conjunctive, disjunctive, conditional, and biconditional propositional forms. Now we need to explain this technique more formally so that we can use this method of tracking truth-values to provide us with reliable and exhaustive interpretations of any given proposition or propositional form.

8.2 Constructing Truth Tables

The technique for constructing a truth table is rather simple and mechanical. There are basically three steps to follow. The first step in this construction process is to determine the number of columns of “T’s” and “F’s” that the table will have. To the far left of each truth table there will be a column of “T’s” and “F’s” for each of the simple propositions in the compound proposition that is being interpreted. To the right of this we place one column under each of the elements of the proposition that is being interpreted, that is, one column for each of the truth-functional connectives in the proposition and one column for each of the component propositions. For example if we are interpreting the proposition, “(C ...D) ^~(C ^~D)” we put one column of “T’s” and “F’s” for “C” to the far left, and one column of “T’s” and “F’s” for “D” to the right of this. To the right of the column for "D" we put one column for each of the component propositions and one for each of the truth-functional connectives in the proposition being interpreted. So far then your table should look like this:

C D (C ...D) ^ ~ (C ^~D)

It is very important in this process that you keep in mind which truth-functional connective is the main one, that is, the one that determines what kind of proposition it is that is being interpreted. In our example, the column that is highlighted in red indicates that this proposition is a disjunction. A complete truth table will give us the complete interpretation of this disjunction, that is, every possible truth-value it can have.

The second step is to determine the number of rows of “T’s” and “F’s” that our truth table will have. To do this all we need to do is count the number of different simple propositions (that is, the number of different propositional letters) in the proposition or propositional form that we are interpreting and then plug that number into the following formula: 2n (where n is the number of different propositional letters in the proposition and “ 2” represents the possible truth-values, and of course there are only two, that is, true and false). For example, if our proposition has 3 different propositional letters, our truth

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table will have 2 3 rows, that is, 8 rows (2 4===16; 2 5===32; 2 6===64, and so on). In our example, our table should look like this:

C D (C . D) ^ ~ (C ^ ~ D)

The third step is simply a matter of plugging in values to determine what truth-value the proposition has in each row of interpretation. To accomplish this, we must fill in all the possible combinations and permutations of truth-values that our simple propositions can have. In our example, the first two columns to the far left must represent every possible combination and permutation of T's and F's for "C" and "D". There is a mechanical procedure for insuring that we cover every possibility. Starting with "D" we simply alternate "T's" and "F's" as follows:

C D (C . D) ^ ~ (C ^ ~ D)

T

F

T

F

Moving to the left, that is, under "C," we simply double the "T's" and "F's" as follows:

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C D (C . D) ^ ~ (C ^ ~ D)

T T

T F

F T

F F

If the compound proposition that we are interpreting has three different simple propositions, our truth table will have 8 rows of “T’s” and “F’s. In this case we simply keep the mechanical doubling procedure going as follows:

C D G

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

And of course, if we had 4 simple propositions we would have one more column of “T’s” and “F’s” and 16 rows. The procedure for filling in the rows however stays the same: alternate, then double, then double again, and so forth.

Now we are ready to interpret a proposition. To do this we simply make substitutions of truth-values in the columns of the truth table to the right. After we have made these substitutions of truth-values, we have a complete interpretation of the proposition and can tell if the proposition is a tautology, a

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contradiction or simply a contingent proposition. Again, this is determined by looking at the column of “T’s” and “F’s” under the main truth-functional connective.

In the process of making correct substitutions of truth-values you must keep in mind what the truth- conditions are for all of our various compound propositions. That is, you are going to have to remember when conjunctions, disjunctions, conditionals, and so forth are true, and when they are false.

Using our example, (C ... D) ^~ (C ^~ D), let's go through each substitution, one at a time. In the first row of interpretation we know that “C” is true and that “D” is true. Hence, knowing what the truth- conditions are for a conditional proposition, we know that “C ...D” is true. So we place a “T” under that column as follows:

C D (C ... D) ^ ~ (C ^ ~ D)

T T T T T

T F

F T

F F

Next, since we know that when both “C” and “D” are true on this first row, we know that “C ^~D” must be also be true, even though ~D is false. (Remember just one true disjunct “C” is sufficient to make a disjunction true.) But this disjunction is negated, so the value of the negation (the right disjunct of our main disjunction) is “F.” So we place an “F” in the column under the right hand negation.

C D (C ... D) ^ ~ (C ^ ~ D)

T T T T T F T T F T

T F

F T

F F

Now all we have to do is to fill in the column under the main truth-functional connective. Since one side of this wedge is true (and this is sufficient to know that the whole disjunction is true on the first row of interpretation) and the right hand negation is false, we know that the whole disjunction is true. So we place a “T” in the column under the main truth-functional connective (which is highlighted in our table). 137

C D (C ... D) ^ ~ (C ^ ~ D)

T T T T T TT F T T F T

T F

F T

F F

We have now finished the first row of interpretation and know (from the column highlighted) that when “C” is true and when “D” is true, this disjunction is true.

Now we simply repeat this process for each row of interpretation. Doing so, our complete table will look as follows:

C D (C ... D) ^ ~ (C ^ ~ D)

T T T T T TT F T T F T

T F T F F FF F T T T F

F T F T T TT T F F F T

F F F T F TT F F T T F

Clearly, we see from the column under the main truth-functional connective (highlighted) that the only case in which this disjunction is false is when C is true and D is false. On the other rows of interpretation, this disjunction is true. This means that the complete interpretation of this proposition shows that it is not a tautology or a contradiction but a contingent proposition. That is, it is true on some interpretations and false on others. (Recall that a tautology has all “T’s” in the column under the main truth-functional connective, and a contradiction has all “F’s.” When we have a mix, we have a contingent proposition, as we do in this case.)

In the Exercises you will be asked to construct some truth tables and interpret the possible truth values for given propositions.

8.3 Testing for Validity with Truth-Tables

One of the most important concepts that we can learn in this course, and perhaps the most difficult, is that of validity. As we have said over and over in one way or another, an argument is valid if and only if 138

it is impossible for the premises of that argument to be true and the conclusion false. With the introduction of the various propositional forms, we are now ready to see that this notion of validity has an interesting relation to the conditional propositional form.

The form of a conditional proposition, namely its “if/then” structure, exactly parallels the structure of an argument. Indeed, we read arguments as asserting that “if” the premises were true, “then” the conclusion must be true. So every argument has a kind of “if/then” or conditional structure.

There is something further to notice in this parallel. The only time that a conditional proposition is false is when the antecedent is true and the consequent is false. If we take the “if” part of an argument to parallel the antecedent of a conditional proposition and the “then” part to parallel the conclusion, we notice that the only case where a conditional proposition is false exactly parallels the only case in which an argument cannot be valid, that is, when its “if” part is true and its “then” part is false. Noticing these parallels allows us to come up with the following method for testing the validity of arguments with truth tables.

We will say that every argument may be expressed as a conditional proposition. (We must be careful here: a conditional proposition is not an argument, but it may express one.) We express an argument as a conditional proposition by making the antecedent of that proposition a conjunction of the premises of the argument that it is expressing. (If there is only one premise in the argument that we are expressing, then, of course, the antecedent of that conditional proposition will not be a conjunction.) Next, we make the conclusion of the argument that we are expressing the consequent of the conditional proposition.

Since no valid argument can have true premises and a false conclusion, and no true conditional proposition can have a true antecedent and a false conclusion, we can see that if an argument is valid, then the conditional proposition that expresses it must be a tautology. This gives us the following rule:

An argument is valid if and only if the conditional proposition that expresses it is tautological.

So the first step in testing an argument with truth tables is to express that argument as a conditional proposition. Let’s see how this works with the following argument:

If I go to the movies then I will see Jane. I did go to the movies. Therefore, I saw Jane.

When we translate this argument into sentential symbolization, we have the following expression:

M...J M Therefore, J

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There are two premises in this argument. So to express this argument as a conditional proposition, we must conjoin these two premises and make them the antecedent of that proposition and make the conclusion the consequent of that proposition. Our expression of this argument as a conditional proposition then looks like this: [(M ...J)•M] ...J.

Now all we have to do is to construct a truth table to give a complete interpretation of this conditional proposition. That table would look like this:

M J [(M ... J) • M] ... J

T T

T F

F T

F F

Now we simply fill in the columns with appropriate “T’s” and “F’s.” The last column is the easiest since it is the same as the second column. The conclusion of this argument “J” is true when “J” is true and false when “J” is false. The next thing we do is to fill in the column under the first implication sign " ...." You must remember the truth-conditions for the conditional proposition to do this correctly.

After you have done this, you must move on to filling in the column under the dot "•”. The value of this conjunction is the value of the antecedent of the larger conditional proposition that we are interpreting. We know that a conjunction is true only when both conjuncts are true. As such, we know that whenever “M” is false, the whole conjunction of “(M ...J) •M” is false. We also know that when “M” is true and “J” is false, the conditional proposition “M ...J” is false, thus making the conjunction of it and “M” false. When both the “M” and the “J” are true, “M ...J” is true, and hence the conjunction (“M ...J) •M” must be true.

All that is left to do in order to get a complete interpretation of this proposition is to fill in the column under the main truth-functional connective. In the first row, the antecedent is true and the consequent is true, so the conditional is true. In the next three rows, the antecedent is false, so the conditional is true regardless of the truth-value of the consequent. So our column under the main truth-functional connective contains nothing but “T’s.” With the column under the main truth-functional connective highlighted in red, the final truth table then looks as follows:

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M J [(M ... J) • M] ... J

T T T T T T T TT T

T F T F F F T TT F

F T F T T F F TT T

F F F T F F F TT F

As we know from our definitions above, this table shows that this conditional proposition is a tautology.

Having determined that the conditional proposition that expresses the argument we are testing is tautological, we know that the argument is valid.

So we now have a mechanical procedure for testing validity. There are three steps in the procedure. All we have to do is:

(1) Express the argument we are testing as a conditional proposition; (2) interpret it with a truth table; (3) determine whether or not it is a tautology (it is, if and only if, there are all “T’s” in the main truth- functional column).

If the conditional proposition is a tautology, the argument it expresses is valid; if it has even one “F” in the column under the main truth-functional connective, that is, if the conditional proposition is a contradiction or a contingent proposition, the argument it expresses is invalid.

Let's apply these three steps to the following argument:

If I go to the movies then I will see Jane. I did see Jane. Therefore, I went to the movies.

Your truth table should show that the conditional proposition that expresses this argument is not a tautology. It should look like this:

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M J [(M ... J) • J] ... M

T T T T T T T TT T

T F T F F F F TT T

F T F T T T T FF F

F F F T F F F TT F

Clearly, the row under the main truth-functional connective (highlighted) shows that this argument is invalid, for it shows that the argument the conditional proposition expresses is not tautological.

8.4 A Short-Cut Test for Validity

The truth table method of testing for validity is fine so long as the number of different proposition letters and truth-functional connectives is limited. In complicated arguments that involve many different propositional letters, the truth table method of testing for validity can become unwieldy. If, for example, we have an argument that involves 6 different propositional letters, our truth table will have 2 6===64 rows of “T’s” and “F’s.” Of course the method will work in such complicated tables, but we might prefer a less cumbersome method if one is available. And fortunately one is available. We will call it the short-cut method.

If we correctly understand why an argument is valid if the conditional proposition that expresses it is tautological, then we can readily see how the short-cut method works. The only time that a conditional proposition is false is when the antecedent is true and the consequent is false. That combination of “T’s” and “F’s” is not possible if the conditional proposition is tautological. For if that combination did exist, then there would be an “F” in the interpretation of that proposition and it would not be tautological. With these things in mind, then our short-cut method is as follows:

(1) Simply assign the consequent of the conditional proposition that expresses the argument we are testing the truth-value of “F.” (Now whatever the values are that you use to make the consequent false, these same values must be used when we make assignment to the antecedent.)

(2) After we have assigned the consequent of the conditional proposition “F,” we then see if there is any way to make the antecedent true.

(3) If it is not possible to make the antecedent true when the consequent is false, the argument is valid. If it is possible to make the antecedent true when the consequent is false, then the argument is invalid.

OK, let’s see how this short-cut method works. Consider the argument that we ended our last section with. I asked you to use the truth-table method to show that it was an invalid argument. That was fairly 142

easy since that argument had only 4 rows of “T’s” and “F’s.” Now we can show that it is invalid with our short-cut method of testing for validity. Recall the argument (which is, by the way, a form of the fallacy we spoke of earlier, and we will address again in the last section of this chapter, called the fallacy of affirming the consequent):

If I go to the movies then I will see Jane. I did see Jane. Therefore, I went to the movies. Expressing this argument as a conditional proposition yields the following symbolic sentential proposition: “[(M ...J) •J] ...M.” The conclusion of our argument is here expressed as the consequent of this corresponding conditional proposition. Following our short-cut method, we simply assign this consequent, that is, “M” the truth-value “F.” Now we see if there is any way that we can make the antecedent true when “M” is false. If so, the argument is invalid, if not, it is valid. What if we make “J” true? If we do then “M ...J” will be true, and so the conjunction “(M ...J) •J” will be true when the consequent “M” is false. This shows that the argument is invalid, for it shows that it is possible for the antecedent to be true when the consequent is false.

This short method is particularly useful when we have an argument with more than two or three propositional letters. Consider the following argument:

If I go the movies then I will see Jane. If I go to the races, then I will see Sally. I will either go to the movies or to the races. Therefore, I will either see Jane or Sally.

To express this argument as a conditional proposition we must first symbolize the three premises and conjoin them to make the antecedent of the conditional proposition that will express the argument. That antecedent would be as follows: “[(M ...J) • (R ...S)] • (M ^R).”

Now we make this expression the antecedent of the conditional proposition that expresses this argument, and the conclusion its consequent, and we get this expression:

“{[(M ...J)•(R ...S)]•(M ^R)} ... (J ^S)”

Following the procedure for the short-cut method, we assign the consequent of this conditional proposition the truth-value of “F,” and then try to make the antecedent true. As it turns out, the only way to make the consequent false in this case is to make both “J” and “S” false, since this consequent is a disjunction and for a disjunction to be false, both disjuncts must be false.

Having made this assignment, our only choices now are to assign truth-values to “M” and “R.” Remember, we are trying to make the antecedent true when the consequent is assigned the value “F.” Since the antecedent is a conjunction, it can be true only if all of its conjuncts are true. Given that “J” is assigned the truth-value of “F,” the only way to make the first conjunct of the first conjunct true is to make “M” false. If “M” were true, then the conditional “M ...J” would be false, making the conjunction that it is part of false, and hence the entire conjunction which is the antecedent, false. But we want to make the antecedent true. So our only choice is to make “M” false. 143

The same reasoning works with the second conditional “R...S.” Given that “S” is assigned the value of false, the only to way to make “R ...S” true is to assign “R” the truth-value of false.

Having made these two assignments to “M” and “R” the final conjunct “M ^R” of the antecedent, which is itself a disjunction, becomes false. So we see that there is no possible substitution of truth-values for the simple propositions in this conditional proposition that would make it false. Hence this conditional proposition is tautological, and hence the argument that it expresses is valid.

With practice, this short-cut method can be a handy tool for testing validity.

8.5 Argument Forms

A particular argument contains particular propositions (as its premises and its conclusion). Such particular propositions are about this or that, e.g., cats on mats, dogs in yards, and sealing wax, and are symbolized with upper case letters that remind us of their content. We call these propositional letters . By contrast, an argument form contains propositional forms (as its premises and its conclusion). These propositional forms have no particular content and are symbolized by lower case letters from p-z and are called propositional variables . These propositional forms can stand for any particular propositions that are instances of that form. Consider the following particular argument that we will call “Argument A” and its sentential expression (Here we are introducing the following symbol ∴

to stand for “therefore.”):

Argument A

If I go to the movies, then I will see Jane. I went to the movies. Therefore I saw Jane.

Sentential Symbolic Expression of Argument A

M...J M ∴J

Now, using propositional variables, we can express as follows the form that this argument is an instance of:

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Sentential Symbolic Expression of the Form of Argument A

p...q p ∴q One important thing to notice here is that there is a one-to-one correspondence between the upper case propositional letters in “Argument A” and the lower case propositional variables in its corresponding argument form (“p” here is standing for “M” and “q is standing for “J”). To express this one-to-one correspondence of propositional variables and the proposition letters, we say that the particular argument that is symbolized as:

(M ...J) M ∴J has

p...q p 444q

as its specific form .

While “p” may stand for the simple proposition “M,” it need not. Remember the propositional form “p” may stand for any proposition whatsoever. So “p” may stand for the compound proposition: “R ...S.” By the same token “q” may stand for any proposition whatsoever, for instance, “ M ^B.” If we make this substitution for “p” and “q” throughout the argument form, we would have the following argument:

(R ...S) ...(M ^B) R...S ∴M^B Even though this argument does not have:

p...q p ∴q as its specific form, it is what we call a substitution instance of this form. It is a substitution instance of our argument form since the same particular proposition is substituted for each propositional variable consistently throughout the form (“p” ===“R ...S” and “q” ===“M ^B”).

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Clearly, an argument form can have many substitution instances, but only one of these substitution instances will have this form as its specific form.

Consider the following argument form:

(p•q) ^ [(p•q) ...q] ~ (p•q) ∴ (p•q) ...q

This is an argument form with two premises, the first premise is any disjunction, one of whose disjuncts is a conjunction of two simple propositions, and the other disjunct is a conditional proposition, the antecedent of which is the same as the left hand disjunct and the consequent is the same as the right hand conjunct of each of the conjunctions. The second premise is the negation of the left hand disjunct in the first premise (which happens to be a conjunction) and the conclusion is the right hand disjunct of the first premise standing alone. Now it should be clear that the following argument is not only a substitution instance of this argument form, but also has this form as its specific form (“p” === “M” and “q” === “R”).

(M•R) ^ [(M•R) ...R] ~ (M•R) ∴ (M•R) ...R Just as clearly, it should be obvious that the following symbolic expression of a particular argument does not have the above form as its specific form, but nevertheless is a substitution instance of that form:

[“p”= “ ~ Z” and “q”= “M •R”].

(~Z•(M•R)) ^ [( ~Z• (M•R)) ...(M•R)] ~ (~Z• (M•R)) ∴ (~Z•(M•R)) ...(M•R) The import of this discussion is enormous. If we determine that a particular argument is a substitution instance of a given argument form, and we know that this given form is valid, then we know that the particular argument under assessment is also valid. The import of this can be generalized as follows:

Any particular argument is valid if it is a substitution instance of an argument form that is valid.

This is very helpful since, as it happens, there are, surprisingly, only a few valid argument forms that we are likely to encounter amongst the thousands of different arguments that we commonly hear, read, and/or construct. In fact, we will consider only five of these valid forms in this chapter. If we become familiar with these forms, we will be able to assess thousands of particular arguments as valid if we develop our skill at recognizing these arguments as substitution instances of any of these valid argument forms. This should become clearer in the following discussion of five of these forms. 146

1. Modus Ponens

One of the most common valid argument forms is called Modus Ponens (in the mode of affirming). That form is as follows:

p...q p ∴q

It is quite easy to see that this is a valid argument form. All we have to do to see this is to express this argument as a conditional propositional form and construct a truth table to provide a complete interpretation of that conditional form. Remember, to express an argument as a conditional proposition or as a conditional propositional form we simply conjoin its premises and make this conjunction the antecedent of that conditional proposition or propositional form and then make the conclusion the consequent of that conditional proposition or propositional form. Our conditional then should be as follows: [(p ...q)•p] ...q

Our truth table looks like this:

p q [(p ... q) • p] ... q

T T T T T T T TT T

T F T F F F T TT F

F T F T T F F TT T

F F F T F F F TT F

What this table shows is that the conditional propositional form that expresses this argument form is tautological (it is true under every possible interpretation). This shows the argument form to be valid.

Once we know that a particular argument form is valid, we know that any argument that is a substitution instance of that argument form is also valid. For example, Argument A above is such an example:

If I go to the movies, then I will see Jane. I went to the movies. Therefore I saw Jane. More generally, we can read Modus Ponens as a rule of inference with a very wide application potential. We read that rule as follows: an argument is a substitution instance of the valid argument form known as Modus Ponens, and hence valid, if one of its premises is a conditional proposition, and one of its premises is the assertion of the antecedent of that conditional proposition, and its conclusion is the consequent of that conditional proposition. 147

Be sure to note that an argument can be a substitution instance of a valid argument form without being of that specific form. Of course if an argument has the specific form of Modus Ponens, then it is a substitution instance of that form, but an argument can be a substitution instance of Modus Ponens, and hence valid, without having that specific form. Argument A above has Modus Ponens as its specific form, and hence is a substitution instance of it, and hence is valid. However, consider the following argument that does not have Modus Ponens as its specific form but nevertheless is a substitution instance of that form:

If I go to the movies, then I will see Jane or Sally. I went to the movies. Therefore I saw Jane or Sally.

If we express this argument symbolically, we get the following:

M. (J ^S) M ∴J^S The form of this argument is not identical to the form of Modus Ponens. Accordingly, we say that this argument does not have Modus Ponens as its specific form. The specific form of the argument is as follows:

p...(q ^s) p ∴q^s However it should be clear that the argument above is a substitution instance of Modus Ponens. The argument does have a conditional proposition as one of its premises: “M ...(J ^S)”; as well, the argument has the antecedent of that conditional proposition (“M”) as a separate premise; and finally, its conclusion “J ^S” is the consequent of that conditional proposition (in this case it happens to be a disjunction). Because this argument is a substitution instance of a valid argument form, it is valid.

Notice that what we are developing here is a proof of validity. If any particular argument is a substitution instance of one of the five valid argument forms that we are going to discuss in this chapter, then it is valid. However, if we run across an argument that is not a substitution instance of one of these five valid argument forms we do not know that it is invalid. Indeed, it may be valid or invalid. We can use our short-cut method to determine whether it is valid or not.

You might be wondering why we are making all this fuss about the difference between an argument being a substitution instance of a valid argument form and its having that specific form. We can see why this is important when we look at an argument form that is often confused with Modus Ponens but is in fact an invalid argument form. I note this because it is such common a mistake in reasoning, almost as common as its corresponding valid Modus Ponens form. This invalid argument form is called The Fallacy of Affirming the Consequent. An example of an argument of this form is as follows:

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If I go to the movies I will see Jane. I saw Jane. Therefore, I went to the movies.

The form of this argument, that is, the invalid form of affirming the consequent, is as follows:

p...q q ∴p We can see that this argument form is invalid if we express it as a conditional propositional form and interpret it with a truth table. That interpretation would be as follows:

p q [(p ... q) • q] ... p

T T T T T T T TT T

T F T F F F F TT T

F T F T T T T FF F

F F F T F F F TT F

We see from the last highlighted row that the corresponding conditional propositional form that expresses the form of this argument is not tautological (it has an “F” in its complete interpretation). Hence the argument that this conditional propositional form expresses is invalid. That is, its premises can be true when its conclusion is false.

We must take careful note here, however, that this argument has the specific form of the invalid form known as affirming the consequent. This allows us to generalize the following rule: if an argument has the specific form of an invalid form, it will be invalid. However, if an argument is merely a substitution instance of an invalid form, and does not have that form as its specific form, then we do not know whether it is valid or not. That is, substitution instances of invalid forms may be valid or invalid. Consider the following example of an argument that is a substitution instance of the invalid form known as affirming the consequent, but is nevertheless a valid argument:

If I go to the movies or go to the races, then I will go to the movies and go to the races. I go to the movies and I go to the races. Therefore, I either go to the movies or go to the races.

We can symbolize this argument as follows:

(M ^R) ...(M•R) M•R ∴M^R

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It should be clear to you that this argument is in fact a substitution instance of the invalid argument form of affirming the consequent, even though it does not have this specific form. However, in this case, the argument is valid, and we can easily show this with our short-cut method of determining validity. To use this method we express the argument as a conditional proposition as follows:

{[(M ^R) ...(M•R)]•(M • R)} ...(M ^R)

Next we assign the truth-value of “F” to the consequent and try to make the antecedent true. If we can, the argument is invalid, if we cannot it is valid. In order to make the consequent here false, we must make both “M” and “R” false since the only way a disjunction can be false is for both disjuncts to be false. So we have to make the same assignments to the “M” and the “R” in the antecedent. Clearly the conjunction “M•R” is false. However, this conjunction is a conjunct of a larger conjunction. One false conjunct, however, is sufficient to make a conjunction false. Hence when the consequent of this conditional proposition is false, the antecedent cannot be true. Hence the argument that this conditional proposition expresses is valid. As it happens then, even though this argument is a substitution instance of the invalid form of affirming the consequent, it is nevertheless a valid argument.

2. Modus Tollens

A second very common valid argument form is called Modus Tollens(in the mode of taking away or denying). That form is as follows:

p...q ~q ∴~p

Again, it is quite easy to demonstrate with a truth table that this is a valid argument form. Our truth table looks like this:

p q [(p ... q) • ~ q] ... ~ p

T T T T T F F T TT F T

T F T F F F T F TT F T

F T F T T F F T TT T F

F F F T F T T F TT T F

What this table shows is that the conditional propositional form that expresses this argument form is tautological (it is true under every possible interpretation). This shows the argument form to be valid. Once we know that a particular argument form is valid, we know that any argument that is a

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substitution instance of that argument form is also valid. It should be clear then that the following argument is valid:

If I go to the movies, then I will see Jane. I do not see Jane. Therefore I did not go to the movies.

More generally, we can read Modus Tollens as a rule of inference with a very wide application potential. We read that rule as follows: an argument is a substitution instance of the valid argument form known as Modus Tollens, and hence valid, if one of its premises is a conditional proposition, and one of its premises is the negation of the consequent of that conditional proposition, and its conclusion is the negation of the antecedent of that conditional proposition.

Now let’s consider an argument form that is often confused with Modus Tollens but is in fact an invalid argument form. This invalid argument form is called the Fallacy of Denying the Antecedent . An example of an argument of this invalid form is as follows:

If I go to the movies I will see Jane. I did not go the movies. Therefore, I did not see Jane. The form of this argument, that is, the invalid form of denying the antecedent, is as follows:

p...q ~p ∴~q

We can see that this is invalid if we express it as a conditional propositional form and interpret it with a truth table. That interpretation would be as follows:

p q [(p . q) • ~ p] ... ~ q

T T T T T FF F T TT FF T

T F T F F FF F T TT TT F

F T F T T TT T F FF FF T

F F F T F TT T F TT TT F

We see from the table that the corresponding conditional propositional form that expresses the form of this argument is not tautological (it has an “F” in its complete interpretation). Hence the argument that this conditional propositional form expresses is invalid. That is, its premises can be true when its conclusion is false.

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Again we can generalize these findings: if an argument has the specific form of this invalid form of denying the antecedent, then it is invalid. But we must be careful to note that if an argument is merely a substitution instance of this invalid form, and does not have this form as its specific form, then we do not know whether it is valid or not. That is, substitution instances of the form of denying the antecedent may be valid or invalid. Consider the following rather strange argument that is an example of an argument that is a substitution instance of the invalid form known as denying the antecedent, and nevertheless is valid:

If I do not go to the movies, then I do not go to the movies. I do go to the movies. Therefore, I do go to the movies.

We can symbolize this argument as follows:

~M... ~M M ∴M

It should be clear to you that this argument is in fact a substitution instance of the invalid argument form of denying the antecedent, even though it does not have this specific form. However, in this case, the argument is valid, since it is also a substitution instance of Modus Tollens. You could also show that the argument is valid with our short-cut method for testing for validity. To use this method we express the argument as a conditional proposition as follows: (( ~ M...~ M)•M) ...M.

Next we assign the truth-value of “F” to the consequent and try to make the antecedent true. If we can, the argument is invalid, if we cannot it is valid. In order to make the consequent here false, we must make “M” false. But once we make “M” false in the consequent, it must also be false in the antecedent. However, since “M” is one of the conjuncts of the antecedent, then the conjunction it is part of must also be false. That is, there is no way to make the antecedent here true when the consequent is false. Hence the argument that this conditional proposition expresses is valid. As it happens then, even though this argument is a substitution instance of the invalid form of denying the antecedent, it is nevertheless a valid argument.

3. Disjunctive Syllogism

A third very common valid argument form is called disjunctive syllogism. This valid argument form has two logically equivalent expressions. They are as follows: “p” follows from “p ^q” and “ ~ q” and “q” follows from “p ^q” and “ ~ p.”

It should be obvious that if a disjunction is true and we know that one of its disjuncts is false, it follows the other disjunct must be true for otherwise the disjunction could not be true. But if we need to verify

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the validity of this form with a truth table, this is easy to do for either of these expressions of this argument form.

If you construct a truth table you will see that the conditional propositional form that expresses this argument form is tautological (it is true under every possible interpretation). This shows the argument form to be valid.

Once we know that a particular argument form is valid, we know that any argument that is a substitution instance of that argument form is also valid. Clearly this argument is a substitution instance of disjunctive syllogism, indeed, it has disjunctive syllogism as its specific form. As such we know that it is valid.

Either I will go to the movies, or I will see Jane. I do not see Jane. Therefore I went to the movies.

More generally, we can read disjunctive syllogism as a rule of inference with a very wide application potential. We read that rule as follows: an argument is a substitution instance of the valid argument form known as disjunctive syllogism, and hence valid, if one of its premises is a disjunction, and one of its premises is the negation of one of the disjuncts of that disjunction, and its conclusion is the assertion of the other disjunct.

4. Hypothetical Syllogism

A fourth common valid argument form is called hypothetical syllogism. The following argument is a substitution instance of this form. It should be clear not only that this particular argument is valid, but also that every argument that is exactly like it is valid as well.

If I go to the movies then I will see Jane. If I see Jane, then I will see Sally. Therefore, if I go to the movies then I will see Sally.

This form of this argument is as follows:

p...q q...r ∴p...r

Notice that in this form, the premises and the conclusion are all conditional propositional forms. Now notice the pattern of this argument form. One thing that stands out is that there is one element that is common to both of the premises (“q”); it is the consequent of one of the conditionals and the antecedent of the other. The conclusion then is formed by making the antecedent of one of the premises that has the common element, in this case “p,” the antecedent of the conclusion, and the consequent of one of the premises with the common element, in this case “r,” the consequent of the conclusion. The truth table for this argument, which is quite long, (8 rows) shows that the conditional

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propositional form that expresses this argument form is tautological (it is true under every possible interpretation). This shows the argument form to be valid.

Once we know that this argument form is valid, we know that any argument that is a substitution instance of it is also valid. Clearly the argument above is a substitution instance of this syllogism, indeed, it has hypothetical syllogism as its specific form. As such we know that it is valid.

More generally, we can read hypothetical syllogism as a rule of inference with a very wide application potential. We read that rule as follows: an argument is a substitution instance of the valid argument form known as hypothetical syllogism, and hence valid, if both its premises and its conclusion are conditional propositions, and the consequent of one of the premises is the antecedent of the other, and the conclusion is a conditional proposition that has as its antecedent the antecedent of the premise with the common element, and as its consequent the consequent of the premise with the common element.

5. Constructive Dilemma

The last common valid argument form that we will consider in this chapter is called constructive dilemma. The following argument is a substitution instance of this form.

If I go to the movies then I will see Jane. If I go to the races then I will see Sally. I will either go to the movies or go to the races. Therefore, I will either see Jane or Sally.

This form of this argument is as follows:

(m ... j ) • (r ... s ) m ^ r ∴j ^ s In this form, two of the premises are conditional propositional forms, and one of the premises and the conclusion are disjunctive propositional forms. Again, there is a precise pattern in this form. What is definitive of the form is that if we assume the disjunction of the antecedents of two assumed conditional propositions, we can validly deduce the disjunction of the consequents of the two conditionals. A truth table will show that this form is valid.

Once we know that this argument form is valid, we know that any argument that is a substitution instance of it is also valid. Clearly the argument above is a substitution instance of this syllogism; indeed, it has constructive dilemma as its specific form. As such we know that it is valid.

More generally, we can read constructive dilemma as a rule of inference with a very wide application potential. We read that rule as follows: an argument is a substitution instance of the valid argument form known as constructive dilemma, and hence valid, if it has two premises that are conditional propositions and one premise that is the disjunction of the antecedents of these two conditionals, and a conclusion that is the disjunction of the consequents of the two conditional propositions.

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One reason this argument form is called a dilemma, is that it is often used to set up a conclusion that one can’t escape. The way this is done is to make the antecedents of the two conditional propositions such that their disjunction is an exclusive one, that is, one in which it is impossible for both disjuncts to be true. The easiest way to do this is to make the antecedents of the two conditional propositions negations of one another (for example, p and ~ p) and the consequents of both of them the same. Consider the following dilemma:

If I get married, then I will regret it. If I do not get married, then I will regret it. I will either get married or not get married. Therefore, I will either regret it or regret it.

The form of this argument is as follows:

M ... R ~M ... R M ^ ~ M ∴R ^ R As we will find out in the next chapter, “R ^R” is logically equivalent to “R.” Hence, we have an argument that forces us to accept a conclusion, if the premises hold, to the effect that we are “dammed if we do, and damned if we don’t.” Of course, not every dilemma has to have such a damning conclusion. We could certainly construct one with a more pleasant consequent. For example, simply replace the “R” above with “H,” and let “H” stand for “you will be happy.”

Time to practice; the exercises below will help you master the ideas presented in this chapter.

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Study Guide for Chapter 8

Tautology : A proposition that is true under every possible interpretation. Contradiction : A proposition that is false under every possible interpretation. Contingent Proposition : A proposition that is true under some interpretations and false under others.

Valid Argument Forms

1. Modus Ponens (MP) p ...q, p, therefore, q

2. Modus Tollens (MT) p...q, ~q, therefore, ~p

3. Disjunctive Syllogism (DS) p ^ q, ~p, therefore, q

4. Hypothetical Syllogism (HS) p...q, q ...r, therefore, p ...r

5. Constructive Dilemma (CD)

(p ...q) • (r ...s), p ^r, therefore, q ^s

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