5 CHAPTER 5
Conditional Arguments and Logical Proofs
In chapter 4, we learned how to translate and construct truth tables for three truth functional connectives. However, there is one more truth functional connective that we have not yet learned: the conditional. The English phrase that is most often used to express conditional statements is “if…then.” For example, this is a conditional statement: “If it is raining then the ground is wet.” Learning how to evaluate conditions can be a valuable tool in protecting ourselves from both relational and professional conflicts. Understanding conditionals is also an important part of learning how to reason and argue effectively.
5.1 Conditionals
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Like conjunctions and disjunctions, conditionals connect two atomic propositions. For example, in the statement in the first paragraph of this chapter, there are two atomic propositions:
It is raining. AND The ground it wet.
The proposition that follows the “if” is called the antecedent of the conditional and the proposition that follows the “then” is call the consequent of the conditional. The conditional statement above is not asserting either of these atomic propositions. Rather, it is telling us about the relationship between them. Let us symbolize “it is raining” as “R” and “the ground is wet” as “G.” Thus, our symbolization of the above conditional would be:
R 
G
The “
” symbol is called the “horseshoe” and it represents what is called the “material conditional.” A material conditional is defined as being true in every case except when the antecedent is true, and the consequent is false. Below is the truth table for the material conditional. Notice that, as just stated, there is only one scenario in which we count the conditional false: when the antecedent is true and the consequent false.
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p |
q |
p |
|
T |
T |
T |
|
T |
F |
F |
|
F |
T |
T |
|
F |
F |
T |
Let ussee how this applies to the above conditional, “if it is raining, then the ground is wet.” As before, we can think about the meaning of the truth functional connectives by asking whether the sentences containing those connectives would be true or false in the four possible scenarios. The first two are pretty easy. If I assert the above conditional “if it is raining then the ground is wet” when it is both raining and the ground is wet (i.e., the first line of the truth table below), then the conditional statement would be true in that scenario. However, if I assert it and it is raining but the ground is not wet (i.e., the second line of the truth table below), then my statement has been shown to be false. Why? Because I am asserting that any time it is raining, the ground is wet. But if it is raining but the ground is not wet, then this scenario is a counterexample to my claim—it shows that my claim is false. Now consider the scenario in which it is not raining but the ground is wet. Would this scenario show that my conditional statement is false? No, it would not. The reason is that the conditional statement R 
G is only asserting something about what is the case when it is raining. So this conditional statement is not asserting anything about those scenarios in which it is not raining. I am only saying that when it is raining, the ground is wet. But that does not mean that the ground could not be wet for other reasons (e.g., a sprinkler watering the grass). So the meaning of the material conditional should count a statement true whenever its antecedent is false. Thus, in a scenario in which it is neither raining nor the ground is wet (i.e., the fourth line of the truth table), the conditional statement should still be true. Would the fact of a sunny day and dry ground show that the conditional R
G is false? No. Thus, as we have seen, the material conditional is false only when the antecedent is true, and the consequent is false.
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R |
G |
R |
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T |
T |
T |
|
T |
F |
F |
|
F |
T |
T |
|
F |
F |
T |
It is sometimes helpful to think of the material conditional as a rule. For example, suppose that I tell my class:
If you pass all the exams, you will pass the course.
Let ussymbolize “you pass all the exams” as “E” and “you pass the course” as “C.” We would then symbolize the conditional as:
E 
C
Under what conditions would my statement E 
C be shown to be false? There are four possible scenarios:
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E |
C |
E |
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T |
T |
T |
|
T |
F |
F |
|
F |
T |
T |
|
F |
F |
T |
Suppose that you pass all the exams and pass the class (first row). That would confirm my conditional statement E 
C. Suppose, on the other hand, that although you passed all the exams, you did not pass the class (second row). This would make my statement is false (and you would have legitimate grounds for complaint!). How about if you do not pass all the exams and yet you do pass the course (third row)? My statement allows this to be true and it is important to see why. When I assert E 
C I am not asserting anything about the situation in which E is false. I am simply saying that one way of passing the course is by passing all of the exams; but that does not mean there are not other ways of passing the course. Finally, consider the case in which you do not pass all the exams and you also do not pass the course (fourth row). For the same reason, this scenario is compatible with my statement being true. Thus, again, we see that a material conditional is false in only one circumstance: when the antecedent is true, and the consequent is false.
There are other English phrases that are commonly used to express conditional statements.
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Here are some equivalent ways of expressing the conditional, “if it is raining then the ground is wet”:
It is raining only if the ground is wet
The ground is wet if is raining
Only if the ground is wet is it raining
That it is raining implies that the ground is wet
That it is raining entails that the ground is wet
As long as it is raining, the ground will be wet
So long as it is raining, the ground will be wet
The ground is wet, provided that it is raining
Whenever it is raining, the ground is wet
If it is raining, the ground is wet
All of these conditional statements are symbolized the same way, namely R 
G. The antecedent of a conditional statement always lays down what logicians call a sufficient condition. A sufficient condition is a condition that suffices for some other condition to obtain. To say that x is a sufficient condition for y is to say that any time x is present, y will thereby be present. For example, a sufficient condition for dying is being decapitated; a sufficient condition for being a U.S. citizen is being born in the U.S. The consequent of a conditional statement always lays down a necessary condition. A necessary condition is a condition that must be in present in order for some other condition to obtain. To say that x is a necessary condition for y is to say that if x were not present, y would not be present either. For example, a necessary condition for being President of the U.S. is being a U.S. citizen; a necessary condition for having a brother is having a sibling. Notice, however, that being a U.S. citizen is not a sufficient condition for being President, and having a sibling is not a sufficient condition for having a brother. Likewise, being born in the U.S. is not a necessary condition for being a U.S. citizen (people can become “naturalized citizens”), and being decapitated is not a necessary condition for dying (one can die without being decapitated).
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Review the difference between necessary and sufficient conditions by watching the Critical Thinking Academy’s video, Propositional Logic: Necessary and Sufficient |
5.2 “Unless” and “If and only if”
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Learn about the aspects of arguing with “Unless” by watching the Critical Thinking Academy’s video, Propositional Logic: A unless B. |
The English term “unless” can be tricky to translate. For example,
The Reds will win unless their starting pitcher is injured.
If we use the constant “R” to stand for the atomic proposition, “the Reds will win” and “S” to stand for the atomic proposition, “the Reds’ starting pitcher is injured,” how would we translate this sentence using truth functional connectives? Think about what the sentence is saying (think carefully). Is the sentence asserting that the Reds will win? No; it is only saying that
The Reds will win as long as their starting pitcher is not injured.
“As long as” denotes a conditional statement. In particular, what follows the “as long as” phrase is a sufficient condition, and as we have seen, a sufficient condition is always the antecedent of a conditional. But notice that the sufficient condition also contains a negation. Thus, the correct translation of this sentence is
~S 
R
One simple trick you can use to translate sentences which use the term “unless” is just substitute the phrase
“if it is not the case that” for the “unless.” But another trick is just to substitute an “or” for the “unless.” Although it may sound strange in English, a disjunction will always capture the truth functional meaning of “unless.” Thus, we could also correctly translate the sentence like this:
S v R
Another conditional statement is expressed when people use the phrase “if and only if.” In logic, this is called a “biconditional.” The thing to remember about biconditionals is that the statement is unless true WHEN both atomic propositions share the same truth value. The statement is false when the truth values are different. The biconditional is represented using the symbol “≡” which is called a “tribar.”
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p |
q |
p ≡ q |
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T |
T |
T |
|
T |
F |
F |
|
F |
T |
F |
|
F |
F |
T |
Some common ways of expressing the biconditional in English are with the phrases “if and only if” and “just in case.” If you have been paying close attention (or do from now on out) you will see me use the phrase “if and only if” often. It is most commonly used when one is giving a definition, such as the definition of validity and also in defining the “material equivalence” in this very section. It makes sense that the biconditional would be used in this way since when we define something we are laying down an equivalent way of saying it.
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5.3. Tautologies, Contradictions And Contingent Statements
Can you think of a statement that could never be false? How about a statement that could never be true? It is harder than you think unless you know how to utilize the truth functional operators to construct a tautology or a contradiction. A tautology is a statement that is true in virtue of its form. Thus, we do not even have to know what the statement means to know that it is true. In contrast, a contradiction is a statement that is false in virtue of its form. Finally, a contingent statement is a statement whose truth depends on the way the world actually is. Thus, it is a statement that could be either true or false—it just depends on what the facts actually are. In contrast, there is an important sense in which the truth of a tautology or the falsity of a contradiction does not depend on how the world is. As philosophers would say, tautologies are true in every possible world, whereas contradictions are false in every possible world. Consider a statement like:
Matt is either 40 years old or not 40 years old.
That statement is a tautology, and it has a particular form, which can be represented symbolically like this:
p v ~p
In contrast, consider a statement like:
Matt is both 40 years old and not 40 years old.
That statement is a contradiction, and it has a particular form, which can be represented symbolically like this:
p ⋅ ~p
Finally, consider a statement like:
Matt is either 39 years old or 40 years old
That statement is a contingent statement. It does not have to be true (as tautologies do) or false (as contradictions do). Instead, its truth depends on the way the world is. Suppose that Matt is 39 years old. In that case, the statement is true. But suppose he is 37 years old. In that case, the statement is false (since he is neither 39 or 40). We can use truth tables to determine whether a statement is a tautology, contradiction, or contingent statement. In a tautology, the truth table will be such that every row of the truth table under the main operator will be true. In a contradiction, the truth table will be such that every row of the truth table under the main operator will be false. And contingent statements will be such that there is mixture of true and false under the main operator of the statement.
The following two truth tables are examples of tautologies and contradictions, respectively.
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A |
B |
(A ⊃ B) v A |
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T |
T |
T T |
|
T |
F |
F T |
|
F |
T |
T T |
|
F |
F |
T T |
|
A |
B |
(A v B) ⋅ (~A ⋅ ~B) |
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T |
T |
T F F F F |
|
T |
F |
T F F F T |
|
F |
T |
T F T F F |
|
F |
F |
F F T F T |
Notice that in the second truth table, I had to do quite a lot of work before I could figure out what the truth values of the main operator were. I had to first determine the left conjunct (A v B) and then the right conjunct (~A ⋅ ~B), but in order to figure out the truth values of the right conjunct (which is itself a conjunct), I had to determine the negations of A and B. Constructing truth tables can sometimes be a chore, but once you understand what you are doing (and why), it certainly is not very difficult.
5.4 Proofs And The 8 Valid Forms Of Inference
Although truth tables are our only formal method of deciding whether an argument is valid or invalid in propositional logic, there is another formal method of proving that an argument is valid: the method of proof. Although you cannot construct a proof to show that an argument is invalid, you can construct proofs to show that an argument is valid. The reason proofs are helpful, is that they allow us to show that certain arguments are valid much more efficiently than do truth tables. For example, consider the following argument:
(R v S) ⊃ (T ⊃ K)
~K
R v S /∴ ~T
(Note: in this section I will be writing the conclusion of the argument to the right of the last premise—in this case premise 3. As before, the conclusion we are trying to derive is denoted by the “therefore” sign, “∴”.) We could attempt to prove this argument is valid with a truth table, but the truth table would be 16 rows long because there are four different atomic propositions that occur in this argument, R, S, T, and K. If there were 5 or 6 different atomic propositions, the truth table would be 32 or 64 lines long! However, as we will soon see, we could also prove this argument is valid with only two additional lines. That seems a much more efficient way of establishing that this argument is valid. We will do this a little later—after we have introduced the 8 valid forms of inference that you will need in order to do proofs. Each line of the proof will be justified by citing one of these rules, with the last line of the proof being the conclusion that we are trying to ultimately establish. I will introduce the 8 valid forms of inference in groups, starting with the rules that utilize the horseshoe and negation.
The first of the 8 forms of inference is “modus ponens” which is Latin for “way that affirms.” Modus ponens has the following form:
p ⊃ q
p
∴ q
What this form says, in words, is that if we have asserted a conditional statement (p ⊃ q) and we have also asserted the antecedent of that conditional statement (p), then we are entitled to infer the consequent of that conditional statement (q). For example, if I asserted the conditional, “if it is raining, then the ground is wet” and I also asserted “it is raining” (the antecedent of that conditional) then I (or anyone else, for that matter) am entitled to assert the consequent of the conditional, “the ground is wet.”
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To learn more, watch the Critical Thinking Academy’s video, Modus Ponens |
As with any valid forms of inference in this section, we can prove that modus ponens is valid by constructing a truth table. As you see from the truth table below, this argument form passes the truth table test of validity (since there is no row of the truth table on which the premises are all true and yet the conclusion is false).
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p |
q |
p ⊃ q |
p |
q |
|
T |
T |
T |
T |
T |
|
T |
F |
F |
T |
F |
|
F |
T |
T |
F |
T |
|
F |
F |
T |
F |
F |
Thus, any argument that has this same form is valid. For example, the following argument also has this same form (modus ponens):
(A ⋅ B) ⊃ C
(A ⋅ B)
∴ C
In this argument we can assert C according to the rule, modus ponens. This is so even though the antecedent of the conditional is itself complex (i.e., it is a conjunction). That does not matter. The first premise is still a conditional statement (since the horseshoe is the main operator) and the second premise is the antecedent of that conditional statement. The rule modus ponens says that if we have that much, we are entitled to infer the consequent of the conditional.
We can actually use modus ponens in the first argument of this section:
(R v S) ⊃ (T ⊃ K)
~K
R v S /∴ ~T
T ⊃ K Modus ponens, lines 1, 3
What I have done here is I have written the valid form of inference (or rule) that justifies the line I am deriving, as well as the lines to which that rule applies, to the right of the new line of the proof that I am deriving. Here I have derived “T ⊃ K” from lines 1 and 3 of the argument by modus ponens. Notice that line 1 is a conditional statement and line 3 is the antecedent of that conditional statement. This proof is not finished yet, since we have not yet derived the conclusion we are trying to derive, namely, “~T.” We need a different rule to derive that, which we will introduce next.
The next form of inference is called “modus tollens,” which is Latin for “the way that denies.” Modus tollens has the following form:
p ⊃ q
~q
∴ ~p
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To learn more, watch the Critical Thinking Academy’s video, Modus Tollens |
What this form says, in words, is that if we have asserted a conditional statement (p ⊃ q) and we have also asserted the negated consequent of that conditional (~q), then we are entitled to infer the negated antecedent of that conditional statement (~p). For example, if I asserted the conditional, “if it is raining, then the ground is wet” and I also asserted “the ground is not wet” (the negated consequent of that conditional) then I am entitled to assert the negated antecedent of the conditional, “it is not raining.” It is important to see that any argument that has this same form is a valid argument. For example, the following argument is also an argument with this same form:
C ⊃ (E v F)
~(E v F)
∴ ~C
In this argument we can assert ~C according to the rule, modus tollens. This is so even though the consequent of the conditional is itself complex (i.e., it is a disjunction). That does not matter. The first premise is still a conditional statement (since the horseshoe is the main operator) and the second premise is the negated consequent of that conditional statement. The rule modus tollens says that if we have that much, we are entitled to infer the negated antecedent of the conditional.
The next form of inference is called “hypothetical syllogism.” This is what ancient philosophers called “the chain argument” and it should be obvious why in a moment. Here is the form of the rule:
p ⊃ q
q ⊃ r
∴ p ⊃ r
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To learn more, watch the Critical Thinking Academy’s video, Hypothetical Syllogism |
As you can see, the conclusion of this argument links p and r together in a conditional statement. We could continue adding conditionals such as “r ⊃ s” and “s ⊃ t” and the inferences would be just as valid. And if we lined them all up as I have below, you can see why ancient philosophers referred to this valid argument form as a “chain argument”:
p ⊃ q
q ⊃ r
r ⊃ s
s ⊃ t
∴ p ⊃ t
Notice how the consequent of each preceding conditional statement links up with the antecedent of the next conditional statement in such a way as to create a chain. The chain could be as long as we liked, but the rule that we will cite in our proofs only connects two different conditional statements together. As before, it is important to realize that any argument with this same form is a valid argument. For example,
(A v B) ⊃ ~D
~D ⊃ C
∴ (A v B) ⊃ C
Notice that the consequent of the first premise and the antecedent of the second premise are exactly the same term, “~D”. That is what allows us to “link” the antecedent of the first premise and the consequent of the second premise together in a “chain” to infer the conclusion. Being able to recognize the forms of these inferences is an important skill that you will have to become proficient at in order to do proofs.
The next four forms of inference we will introduce utilize conjunction, disjunction and negation in different ways. We will start with the rule called “simplification,” which has the following form:
p ⋅ q
∴ p
What this rule says, in words, is that if we have asserted a conjunction then we are entitled to infer either one of the conjuncts. This is the rule that I introduced in the first section of this chapter.
The next rule we will introduce is called “conjunction” and is like the reverse of simplification. (Do not confuse the rule called conjunction with the type of complex proposition called a conjunction.) Conjunction has the following form:
p
q
∴ p ⋅ q
What this rule says, in words, is that if you have asserted two different propositions, then you are entitled to assert the conjunction of those two propositions. As before, it is important to realize that any inference that has the same form as conjunction is a valid inference. For example,
A ⊃ B
C v D
∴ (A ⊃ B) ⋅ (C v D)
is a valid inference because it has the same form as conjunction. We are simply conjoining two propositions together; it does not matter whether those propositions are atomic or complex. In this case, of course, the propositions we are conjoining together are complex, but as long as those propositions have already been asserted as premises in the argument (or derived by some other valid form of inference), we can conjoin them together into a conjunction.
The next form of inference we will introduce is called “disjunctive syllogism” and it has the following form:
p v q
~p
∴ q
In words, this rule states that if we have asserted a disjunction and we have asserted the negation of one of the disjuncts, then we are entitled to assert the other disjunct. Once you think about it, this inference should be pretty obvious. If we are taking for granted the truth of the premises—that either p or q is true; and that p is not true—then is has to follow that q is true in order for the original disjunction to be true. (Remember that we must assume the premises are true when evaluating whether an argument is valid.) If I assert that it is true that either Bob or Linda stole the diamond, and I assert that Bob did not steal the diamond, then it has to follow that Linda did. That is a disjunctive syllogism. As before, any argument that has this same form is a valid argument. For example,
~A v (B ⋅ C)
~~A
∴ B ⋅ C
is a valid inference because it has the same form as disjunctive syllogism. The first premise is a disjunction (since the wedge is the main operator), the second premise is simply the negation of the left disjunct, “~A”, and the conclusion is the right disjunct of the original disjunction. It may help you to see the form of the argument if you treat “~A” as the p and “B ⋅ C” as the q. Also notice that the second premise contains a double negation. Your English teacher may tell you never to use double negatives, but as far as logic is concerned, there is absolutely nothing wrong with a double negation. In this case, our left disjunct in premise 1 is itself a negation, while premise 2 is simply a negation of that left disjunct.
The next rule we will introduce is called “addition.” It is not quite as “obvious” a rule as the ones we have introduced above. However, once you understand the conditions under which a disjunction is true, then it should be obvious why this form of inference is valid. Addition has the following form:
p
∴ p v q
What this rule says, in words, is that that if we have asserted some proposition, p, then we are entitled to assert the disjunction of that proposition p and any other proposition q we wish. Here is the simple justification of the rule. If we know that p is true, and a disjunction is true if at least one of the disjuncts is true, then we know that the disjunction p v q is true even if we do not know whether q is true or false. Why? Because it does not matter whether q is true or false, since we already know that p is true. As before, is it important to realize that any argument that has this same form, is a valid argument. For example,
A v B
∴ (A v B) v (~C v D)
is a valid inference because it has the same form as addition. The first premise asserts a statement (which in this case is complex—a disjunction) and the conclusion is a disjunction of that statement and some other statement. In this case, that other statement is itself complex (a disjunction). But an argument or inference can have the same form, regardless of whether the components of those sentences are atomic or complex.
The final of our 8 valid forms of inference is called “constructive dilemma” and is the most complicated of them all. It may be most helpful to introduce it using an example. Suppose I reasoned thus:
The killer is either in the attic or the basement. If the killer is in the attic then he is above me. If the killer is in the basement then his is below me. Therefore, the killer is either _________________ or _________________.
Can you fill in the blanks with the phrases that would make this argument valid? I am guessing that you can. It should be pretty obvious. The conclusion of the argument is the following:
The killer is either above me or below me.
That this argument is valid should be obvious (can you imagine a scenario where all the premises are true and yet the conclusion is false?). What might not be as obvious is the form that this argument has. However, you should be able to identify that form if you utilize the tools that you have learned so far. The first premise is a disjunction. The second premise is a conditional statement whose antecedent is the left disjunct of the disjunction in the first premise. And the third premise is a conditional statement whose antecedent is the right disjunct of the disjunction in the first premise. The conclusion is the disjunction of the consequents of the conditionals in premises 2 and 3. Here is this form of inference using symbols:
p v q
p ⊃ r
q ⊃ s
∴ r v s
Important Note: It is highly doubtful that in your day-to-day life you will ever use proofs as outlined above. I am quite sure that your friend or spouse will not slow down and allow you to write out their argument using symbolic logic symbols while in the midst of a heated discussion. So, what is the value of learning this information? The answer is that it is important for you to understand how arguments are constructed so that you can evaluate those that are presented to you. This will become even more obvious in chapter 6 when we begin to look at mistakes in arguments (called Fallacies).
5.5 Short Review Of Propositional Logic
In chapter 4, we learned a formal method for determining whether a certain class of arguments (i.e., those that utilize only truth functional operators) are valid or invalid. That method is the truth table test of validity. In this chapter, we learned another formal method for proving that arguments are valid or invalid (the method of proof). The other important skill we have learned in these chapters is translating sentences into propositional logic. Thus, there are three different skills that you should know how to do:
Translate sentences from English into propositional logic
Construct truth tables in order to determine whether an argument is valid or invalid
Construct proofs to prove an argument is valid
It is important to reiterate that truth tables are the only formal method that allow us to determine whether an argument is valid or invalid; proofs can only show that an argument is valid, but not that it is invalid. You might think that you can use proofs to show that an argument is invalid—for example, if you are unable to construct a proof for an argument, that means that the argument is invalid. However, this does not follow. There could be many reasons why you are unable to construct a proof, including that you just are not skilled enough to construct proofs. But the fact that you are not skilled enough to find a proof for an argument would not mean that the argument is invalid, it would just mean that you weren’t skilled enough to show that it is valid! So, we cannot use one’s inability to construct a proof for an argument to establish that the argument is invalid. Again, only the truth table test of validity can establish that an argument is invalid.
The study of propositional logic has given us a way of understanding what “formal” means in the phrase, “formal logic.” We can see this clearly with the truth table test of validity. After we translate an argument into propositional logic using constants and the truth functional connectives, we do not need to know what the constants mean in order to know whether the argument is valid or invalid. We simply have to fill out the truth table in the mechanical way we have learned and then apply the truth table test of validity (which is also a mechanical procedure). Thus, once an argument has been translated into propositional logic, determining whether an argument passes the truth table test of validity is something a computer could easily do. The translation from English to symbolic format is not as easy for a computer to do because successfully doing so depends on understanding the nuances of English. Although today there are computer programs that are pretty good at doing this, it has taken many years to get there. In contrast, any simple computer program from half a century ago could easily construct and evaluate a truth table using the truth table test of validity because this does not take any understanding—it is simply a mechanical procedure. There are many different programs, many of which are readily available on the web, that allow you to construct and evaluate truth tables.
In contrast, the informal test of validity (from chapter 1) requires that we understand the meaning of the statements involved in the argument in order for us to be try to imagine the premises as true and the conclusion as false. Since this test requires the use of our imagination, it clearly also requires that we understand the meanings of the statements in the argument. The truth table test of validity does not require any of this. Since the truth table method does not require understanding of the meaning of the statements involved in the argument, but only an awareness of their logical form, we refer to it as a formal logic. Formal logic is a kind of logic that looks only at the form, rather than the content (meaning) of the statements. We can easily see this by constructing an argument where the atomic propositions use silly, made-up words, such as those from Lewis Carroll’s “Jabberwocky”:
If toves are slithy, then the borogoves are mimsy
Borogoves are not mimsy
Therefore, toves are not slithy
If we translate “toves are slithy” at “T” and “borogoves are mimsy” as “B” then the form of this argument is clearly modus tollens, which is one of the 8 valid forms of inference:
T ⊃ B
~B
∴ ~T
We can thus see that this argument is valid even though we have no idea what “toves” or “borogoves” are or what “slithy” and “mimsy” mean. Thus, propositional logic, which includes the truth table test of validity, is a kind of formal logic, whereas the informal test of validity is not.
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Chapter Review |
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Review what you have learned in this chapter by watching A Crash Course in Formal Logic Pt 8c: Conditional |