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Chapter 1 - Conditionals 1
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
CHAPTER 1 - CONDITIONALS
Here, you’ll learn:
How to understand complex sentences by
• symbolizing conditionals
• recognizing equivalent conditionals
• assessing conditionals
How to assess arguments for validity / invalidity by
• establishing that arguments are valid using the → O rule
• establishing that arguments are invalid by recognizing the fallacy of assuming the consequent
Symbolizing Conditionals with One Connector
I) Symbolizing with “If…. then….”
Let’s start our study of conditionals by considering the following argument:
1. If Max is a poodle then Max is a dog.
2. Max is a poodle.
3. Max is a dog.
(premise)
(premise)
(from 1 and 2)
Lines 1, 2, and 3 are all statements because each of them is either true or false and because that’s
just what statements are; statements are simply sentences that are either true or false, although
we might not always know which it is.i
Lines 2 and 3 of this argument are atomic statements because they have no parts that are
themselves statements. Instead, atomic statements are composed of a subject (what the statement
is about) and a predicate (what is being said about the subject). The subject of both of these
sentences is “Max.” The predicate in the sentence on line 2 is “is a poodle” and the predicate of
the sentence on line 3 is “is a dog.”
1. If Max is a poodle then Max is a dog.
2. Max is a poodle.
3. Max is a dog.
Subject
(premise)
(premise)
(from 1 and 2)
Predicate
We’ll study the logic of atomic statements in (appropriately enough) predicate logic.
i
“2+2=4,” “The President of the United States is a Martian,” and “There is intelligent life on other planets,” are all
statements, too. The first is true. The second is false. The third is either true or false, although I’m not sure which.
“What time is it?” and “Please pass the sugar,” on the other hand, are not statements because they aren’t really true
or false.
Chapter 1 - Conditionals 2
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
For now, we’ll represent atomic statements with capital letters. We can choose any letter we
want to represent an atomic statement, but we’ll make a practice of using the same letter for each
occurrence of the same statement in an argument and we won’t use the same letter to symbolize
different statements.
Letting “Max is a poodle” be “P,” and “Max is a dog” be “D,” we can rewrite our argument as
follows:
1. If P then D
2. P
3. D
(premise)
(premise)
(from 1 and 2)
Line 1 is a compound statement because it has parts that are themselves statements. Compound
statements are constructed by joining two or more constituent statements together with statement
connectors. We’ll study the logic of such statements in propositional logic.
A compound statement composed of two statements joined by the statement connector
“if…then…” is called a conditional. The statement (simple or compound) that follows the “if” is
called the antecedent. The statement (simple or compound) that follows the “then” is called the
consequent.
Conditional
“If [Antecendent] then [Consequent].”
We’ll symbolize conditionals using an arrow.
Rule: “If then ” is symbolized as “
→ .”
Completely symbolized, then, our argument looks like this:
1. P → D
2. P
3. D
(premise)
(premise)
(from 1 and 2)
Figuring Things Out
Symbolize the following conditionals and complete the following rule. The answers are in the
endnotes.
1. “If you’re a physics major then your major is physics.”1
2. “If you’re taking a physics class then you’re taking a science class.”2
Is this sentence true?3
Chapter 1 - Conditionals 3
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
3. “If you’re taking a science class then you’re taking a physics class.”4
Is this sentence true?5
Rule: → (= or ≠) → .6
II) Symbolizing with “If,” “Should,” and “Provided that”
Conditionals are extremely important – probably one of the most important kind of sentences
from a logical point of view.ii And because there are many different ways to express conditionals
besides the straightforward “If…then…,” we need to know how to recognize and symbolize
conditionals when they’re communicated through a variety of sentences.
Figuring Things Out
Symbolize the following conditionals and complete the following rules. The answers are in the
endnotes.
1. “If Jim is a pediatrician, he’s a doctor.”7
2. “Jim is a doctor, if he’s a pediatrician.”8
→
Rule: If , =
Rule: , if =
→
Rule: In general, whatever follows “if” becomes the
.
.
of the conditional. 9
3. “Should the movie be good, we’ll recommend it.”10
4. “We’ll recommend the movie, should it be good.”11
5. “Provided that it’s a nice day, we’ll go on a walk.”12
6. “We’ll go on a walk, provided that it’s a nice day.”13
Rule: “Should” and “provided that” work exactly like
becomes the
of the conditional.14
ii
and whatever follows them
Conditionals are important to logic because logic doesn’t give us any particular facts about the world. It doesn’t
tell us whether or not a certain animal is a poodle, for example, or whether or not a certain person is a bachelor.
What logic tells us is the relationships that would obtain between these facts if they were true. It tells us, for
example (and very roughly), that if an animals is a poodle then that animal is a dog, and that if a person is a bachelor
then that person is unmarried. We might go so far as to say that conditionals are the very heart of logic.
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Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Practice
Symbolize the following. The answers are in the endnotes.
1. “If C, D.”15
2. “A, if B.”16
3. “P, provided that Q.”17
4. “Provided that R, S.”18
5. “Should L, A.”19
6. “A, should P.”20
Rewrite the following conditionals by filling in the blanks:
1. “If the painting is expensive then he’ll like it.”21
“If
“
,
.”
, if
.”
2. “If the class is open then we’ll enroll.”22
“Should
“
,
.”
, should
.”
3. “If computers can think then minds are physical.”23
“Provided that
“
,
, provided that
.”
.”
Chapter 1 - Conditionals 5
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
III) Symbolizing with “Sufficient” and “Necessary”
There are two other important ways to express conditionals.
Figuring Things Out
Symbolize the following conditionals and complete the following rule. The answers are in the
endnotes.
1. “Getting 100% is a sufficient condition for passing the test.”24
(Hint: To say that something is sufficient is to say that it’s enough and maybe even more than
enough, as in “Having a trillion dollars is sufficient for being wealthy.” Certainly, having a
trillion dollars is enough – and more than enough - to count as wealthy so if someone has a
trillion dollars then that person is wealthy. Most wealthy people don’t have anywhere near a
trillion dollars. Same here. Getting 100% is enough – and more than enough – for passing the
test. Certainly anyone who got perfect score passed. And most people who passed probably
didn’t get a perfect score.)
2. “Getting 100% is sufficient for passing the test.”25
3. “To pass the test, it’s sufficient to get 100%.”26
Rule: Whatever is being described as sufficient becomes the
conditional.27
of the
4. “Completing the foreign language requirement is a necessary condition for obtaining your
degree.”28
(Hint: To say that something is necessary is to say that it’s required, although other things might
be required too, as in “Fuel is necessary for the car to go.” Fuel is required for the car to go, so if
the car is going then it must have fuel, although other things – like a working engine – are
required too. Same here. Completing the foreign language requirement is required for the degree,
although other things – like completing the science requirement – are required too.)
5. “Completing the foreign language requirement is necessary for obtaining your degree.”29
6. “To obtain your degree, it’s necessary to complete the foreign language requirement.”30
Rule: Whatever is being described as necessary becomes the
conditional.31
of the
Chapter 1 - Conditionals 6
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
(I once had a student who had great difficulty remembering how to symbolize sentences that
used the terms “sufficient” or “necessary,” until she thought of it this way:
Symbolizing these sentences is Super Nasty. In the expression “Super Nasty,” you have “S” first,
then “N.” As in “S → N.” As in “sufficient conditions are antecedents and necessary conditions
are consequents.” ☺)
Practice
Symbolize the following. The answers are in the endnotes.
1. “Q is sufficient for D.”32
2. “For E, it’s sufficient that O.”33
3. “L is a sufficient condition for Q.”34
4. “A is necessary for B.”35
5. “R is a necessary condition for C.”36
6. “For T, it’s necessary that D.”37
Rewrite the following conditionals by filling in the blanks:
1. “If Robin is a surgeon then she’s smart.”38
“
is sufficient for
.”
“
is necessary for
.”
2. “If the power was out then the clocks will be blinking.”39
“
is a sufficient condition for
.”
“
is a necessary condition for
.”
3. “If ghosts exist then minds are nonphysical.”40
“
is necessary for
.”
“
is sufficient for
.”
Chapter 1 - Conditionals 7
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
IV) Symbolizing with “Only if”
To ease into the logic of “only if,” suppose that George really likes shrimp. In fact, he likes
shrimp so much that “George will attend a party only if shrimp is being served” is true.
Now suppose we’re at a party. If we see George, can be we certain that there’s shrimp
somewhere? Is “if George is at a party then shrimp is being served” true? Yes, because George
attends parties only if shrimp is being served. George doesn’t attend any parties that don’t have
shrimp.
If we see shrimp, can we be certain that George is somewhere? Is “if shrimp is being
served then Georges is at the party” true? No, because maybe George wasn’t invited, or maybe
there are two simultaneous shrimp parties and George can’t attend both without bi-locating.
In summary, “George will attend a party only if shrimp is being served” is equivalent to
“If George is at a party then shrimp is being served.” It’s not equivalent to “If shrimp is being
served then Georges is at the party.”
Alternatively, let’s suppose that “Brenda is eligible to graduate only if she’s completed the math
requirement” is true.
Does it follow from this that “If Brenda is eligible to graduate then she’s completed the
math requirement” is true? Yes. She can graduate only if she’s completed the math requirement,
so if she can graduate then she must have completed the requirement.
Does it follow from this that “If Brenda has completed the math requirement then she’s
eligible to graduate?” No, because there are presumably many other graduation requirements,
and Brenda might not have met those yet.
In summary, “Brenda is eligible to graduate only if she’s completed the math
requirement” is equivalent to “If Brenda is eligible to graduate then she’s completed the math
requirement.” It’s not equivalent to “If Brenda has completed the math requirement then she’s
eligible to graduate.”
→
Rule: only if =
Rule: In general, whatever follows “only if” becomes the
conditional.41
.
of the
It can also be helpful to realize that “if” introduces sufficient conditions and “only if” introduces
necessary conditions. (Think about that for awhile. If you’re not thinking too hard, that should
make perfect sense. If we say “She’ll accept the job offer only if benefits are included,” we’re
saying that the inclusion of benefits are necessary for her accepting the job offer.) As we’ve seen
above, necessary conditions become the consequents of conditionals. Therefore, the sentence
introduced by “only if” becomes the consequent of a conditional.
Note: The rules for “if” and “only if” refer to what follows “if” and “only if.” The rules for
“sufficient” and “necessary” refer to what is being described as sufficient and necessary.
Chapter 1 - Conditionals 8
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Practice
Symbolize the following. The answers are in the endnotes.
1. “Democracy will work only if the citizens are well-informed.” 42
2. “Only if the citizens are well-informed, can democracy work.”43
3. “You can join the club only if you pay the dues.”44
4. “Only if you play the dues, can you join the club.”45
5. “P only if Z.”46
6. “P, if Z.”47
7. “If L, M.”48
8. “Only if L, M.”49
Rewrite the following conditionals by filling in the blanks:
1. “If Jill enjoyed the movie then it must be a foreign film.”50
“
only if
.”
“
, if
.”
2. “If the cookie is good for you then it contains raisins.”51
“
, if
.”
“
only if
.”
3. “If Hank won then the election was rigged.”52
“
only if
.”
“
, if
.”
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Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
V) Learning how to Symbolize: “Meaning” vs. “Rules”
We’ve developed rules for symbolizing conditionals expressed as
“If ___ then ___”
“If ___ , ___” and “___, if ___”
“Should ___, ___” and “___, should ___”
“Provided that ___, ___” and “___, provided that ___”
“___ is a sufficient condition for ___” and “For ___ it’s sufficient that ___”
“___ is a necessary condition for ___” and “For ___ it’s necessary that ___”
“___ only if ___”, and “Only if ___, ___”
At this point, it’s natural to wonder whether it’s better to learn how to symbolize conditionals by
memorizing the rules or by understanding the meaning of the sentences. That’s a good question,
and the answer to it partly depends upon what you mean by “understanding the meaning of the
sentences.”
Generally speaking, it’s not a good idea to symbolize a compound sentence by thinking too much
about the meaning of its component sentences (the sentences that go in the blanks above). This is
because thinking too much about the component sentences can easily lead us to symbolize the
sentence in way that reflects what we think the sentence should say, rather than in a way that
reflects what the sentence actually says.
Suppose, for example, that we’re presented with the sentence “Being about to carry on an
intelligent conversation is necessary for having a mind.” If we reflect too much upon the
relationship between taking being able to carry on an intelligent conversation and having a mind,
we might think in the following mistaken way:
“I’m going to let ‘C’ be ‘Something can carry on an intelligent conversation’ and ‘M’ be ‘That
thing has a mind.’ Now I don’t think it’s the case that if something has a mind then it can carry
on an intelligent conversation. After all, I think that some nonhuman animals have minds even
though they can’t carry on a conversation. Therefore, I don’t think that ‘M → C’ is right. What
about ‘C → M?’ That says that if something can carry on an intelligent conversation then it must
have a mind. Yes! I agree with that. It would be impossible to carry on an intelligent
conversation without a mind. So ‘C → M’ must be the right answer. ‘Being able to carry on an
intelligent conversation is necessary for having a mind,” is symbolized as ‘C → M.’
Can you see what’s going wrong here? By thinking about the meaning of the component ideas,
we symbolized the relationship that we believe actually obtains between those ideas and not the
relationship that the sentence asserts to obtain between those ideas. (We might say that we’re
reading into the sentence, rather than reading the sentence as it stands.)
If we follow the rule for symbolizing sentences that employ the expression “necessary,” on the
other hand, we’d remember that the thing described as necessary becomes the consequent of the
conditional and we’d correctly symbolize ‘Being able to carry on an intelligent conversation is
necessary for having a mind,” as ‘M → C.’
Chapter 1 - Conditionals 10
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
The moral of this story is that it’s better (i.e. more reliable) to symbolize a sentence by following
the rules than it is to symbolize a sentence by thinking about the meaning of the component
sentences.
But it can be unsatisfying to think that you need to memorize rules instead of understanding the
meaning or the reason behind the rules. After all, aren’t we always told that we should
understand rather than memorize? There are four responses to this concern, and I think that all of
them are right.
1) Remember that the symbolization rules didn’t just drop from the sky. Instead, we symbolized
sentences using the connector expressions and then generalized the rules from those
symbolizations.
This means that if you understood how to symbolize the original sentences from which we
generalized when formulating the rules, and if you saw how the rules simply represent those
generalizations, then you do understand the rules. You don’t need to reinvent the wheel each
time by figuring out the rule all over again each time you’re faced with a particular connector. Of
course you can reinvent the wheel, and you might want to, which brings us to…
2) We can symbolize sentences by understanding meanings rather than by memorizing rules, as
long as we remember that the meaning at issue is the meaning of the connector expressions, like
“necessary,” “sufficient,” “should,” “provided that,” and “only if,” and not the meaning of the
component sentences.
So, for example, one way to symbolize a sentence like “A only if B” is to think about how “only
if” works, possibly by remembering a favorite sentence like “Max is a poodle only if Max is a
dog.” You can then think about how you’d symbolize that sentence, conclude that it would be
symbolized as “P → D,” and transfer that insight to the sentence you need to symbolize. That’s a
perfectly acceptable approach to symbolization. It might take a little longer than simply applying
the rules, but if you’re more comfortable with this approach then by all means take it. Don’t,
however, adopt this approach to symbolizing because you think that memorizing the rules
somehow conflicts with understanding the connectors, because…
3) The meaning of the connectors just is how they work to ‘connect up’ ideas. Since the rules
express how the connectors work, to a very large extent understanding the meaning of a
connector just is knowing the symbolization rule. Understanding and memorization are not easily
separable here.
Let’s think again about how we might symbolize “A only if B” by working through a favorite
sentence like “Max is a poodle only if Max is a dog.” When we reflect on this sentence,
concluding that it would be symbolized as “M → D,” we’re using that favorite sentence to
remind us of how “only if” connects ideas. This is exactly the information that’s given in the rule.
And finally, relying on symbolization rules is acceptable because…
Chapter 1 - Conditionals 11
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
4) Although we can memorize things without understanding them, and so memorization and
understanding are not always the same thing, it’s frequently a mistake to think of memorization
as the enemy of understanding because sometimes we can fully understand something only after
we’ve memorized it.
If that makes perfect sense to you because you can remember some occasions in which your
learning worked that way, you can stop reading this section and proceed to practice symbolizing.
If, however, you’re skeptical about this, think for a moment about how you learned to multiply
26 by 3. I’ll bet you learned to do it like this: First you multiply 6 by 3 to get 18. You then “carry
the 1” which you add to what you get when you multiply 2 by 3. And this gives you the answer
of 78. It looks like this:
1
26
x 3
78
You memorized this procedure, I’ll wager, without really understanding why you carry the 1,
and you might follow the procedure now without ever thinking about why you carry the 1.
Of course, there is a reason. Here it is: 26 means that you have 2 groups of ten and 6 groups of
one. When you multiply the 6 groups of one by 3, you have 18 groups of one. In other words,
you have 1 group of ten and 8 groups of one. Remember that because we’ll refer to it later. Now
you need to multiple your 2 groups of ten by 3. This gives you 6 groups of ten. But you still have
that 1 group of ten from before, so in all you have 7 groups of ten and the 8 groups of one. 7
groups of ten and 8 groups of one is 78. We might represent it like this:
26 x 3 =
(2 tens + 6 ones) x 3 =
(2 tens x 3) + (6 ones x 3) =
(2 tens x 3) + (18 ones) =
(2 tens x 3) + (1 ten and 8 ones) =
(6 tens) + (1 ten and 8 ones) =
7 tens + 8 ones =
78
Carrying the 1 is just a way of keeping track of the fact that we get one extra unit of ten when we
multiple 6 by 3. It’s a book-keeping technique.
Sometimes children are taught how to multiply in this way. In fact, I’ll bet that your math
textbook had an explanation something like the one I just gave you. It was probably
accompanied by pictures of blocks. And there might be children who completely understand that
explanation and think about the reasoning behind carrying the 1 (or the 2 or the 3) each and
every time they do it. But I don’t think that most children are like that. I suspect that most
children find the explanation confusing and just learn to carry the 1 (or the 2 or the 3). If, many
Chapter 1 - Conditionals 12
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
years later, they take a course in college designed for elementary school teachers, they might
learn the explanation for why people carry the 1 and have a pleasurable “Ah-ha” moment. Why
didn’t they have that moment when they were learning how to multiply in elementary school?
It’s because understanding why they carry the 1 makes a lot more sense after they’ve memorized
that they should carry 1 and have internalized the procedure with a lot of examples. The same
applies here. If you don’t completely understand a symbolization rule, don’t let that deter you
from memorizing it. You might find yourself understanding it quite well after you’ve completely
internalized the rule through memorization and lots of practice.
Practice
Symbolize the following sentences, letting B be “Behaviorism is right” and D be “Mental states
are dispositions to behave.” The answers are in the endnotes.
1. “If behaviorism is right then mental states are dispositions to behave.”53
2. “Behaviorism is right only if mental states are dispositions to behave.”54
3. “For behaviorism to be right, it’s sufficient that mental states be dispositions to behave.”55
4. “Should behaviorism be right, mental states are dispositions to behave.”56
5. “Behaviorism is right, provided that mental states are dispositions to behave.”57
6. “Mental states being dispositions to behave is a necessary condition for behaviorism being
right.”58
7. “Behaviorism is right, if mental states are dispositions to behave.”59
8. “Only if mental states are dispositions to behave, is behaviorism right.”60
9. “Behaviorism is right, should mental states be dispositions to behave.”61
10. “For mental states to be dispositions to behave, it’s necessary that behaviorism is right.”62
Recognizing Equivalent Conditionals
We’ve seen that a sentence like “If Max is a poodle then Max is a dog” is not the same as a
sentence like “If Max is a dog then Max is a poodle.” (The first sentence is true because all
poodles are dogs. The second sentence is false because not all dogs are poodles.) Order matters
very much when we have conditionals.
“If Max is a poodle then Max is a dog” is equivalent to many other sentences, though, including
• “Max is a dog, if Max is a poodle.”
• “Max is a poodle only if Max is a dog.”
• “Should Max be a poodle, Max is a dog.”
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Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
•
•
•
“Provided that Max is a poodle, Max is a dog.”
“Max being a poodle is sufficient for Max being a dog.”
“Max being a dog is necessary for Max being a poodle.”
To see that all of these sentences are equivalent, we can notice that they’re all symbolized in the
same way. They’re all symbolized as “P → D.”
In general, that’s a good way to see if two sentences say the same thing. Just symbolize the
sentences and see if the symbolizations are the same (or, later on, see if the symbolizations are
equivalent to each other). If so, then the sentences say the same thing. If not, then they don’t.
Practice
Which of the following sentences are equivalent to
“If our actions have moral value then we have free will.”? The answers are in the endnotes.
1. “Our actions have moral value only if we have free will.”63
Equivalent / Not equivalent
2. “Our actions have moral value, if we have free will.”64
Equivalent / Not equivalent
3. “Our actions have moral value, provided that we have free will.”65
Equivalent / Not equivalent
4. “Our actions having moral value is a sufficient condition for our having free will.”66
Equivalent / Not equivalent
5. “Our having free will is a necessary condition for our actions having moral value.”67
Equivalent / Not equivalent
Which of the following sentences are equivalent to
“Having a shape is a necessary condition for having a color”? The answers are in the endnotes.
1. “Something has a shape only if it has a color.”68
Equivalent / Not equivalent
2. “Something has a color only if it has a shape.”69
Equivalent / Not equivalent
3. “Something has a shape if it has a color.”70
Equivalent / Not equivalent
4. “Having a color is a sufficient condition for having a shape.”71
Equivalent / Not equivalent
5. “Something has a color, provided that it has a shape.”72
Equivalent / Not equivalent
Chapter 1 - Conditionals 14
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Assessing Conditionals
I) What Makes a Conditional False
When is a conditional true and when is it false? What are the truth conditions for conditionals?
Philosophers and logicians treat all conditionals as material conditionals. Material conditionals
are like promises. The conditional is false when (and only when) the promise is broken.
Consider the following (pseudo!) promise: “If you clean my house then you’ll get an A in my
class.” What would need to happen for this promise to be broken? Well, you’d have to clean my
house, holding up your end of the bargain, without me giving you an A in return. That’s the only
way that I could break my promise. If you do clean my house and I give you A, obviously I
haven’t broken my promise. And if you don’t clean my house then, in some sense, the promise
doesn’t apply because I only said what I’d do if you did clean. If you don’t clean my house, I’m
free to give you an A or not. Neither decision will break my promise, because the promise isn’t
in force, so to speak. We can represent this information in the following chart:
You clean my house.
You got an A.
Yes
Yes
Yes
No
If you clean my house then
you’ll get an A in the class.
Kept
Broken
No
Yes
Kept
No
No
Kept
And we can generalize from this to represent the truth conditions for a conditional in a truth
table.
T
T
T
F
→
T
F
F
T
T
F
F
T
Rule: A conditional is false when and only when the antecedent is
and the consequent is
.73
By the way, the truth table defines the meaning of “→” and, in so doing, it gives us what is
called “the material conditional.” There is some “slippage” between the material conditional and
many “if…then…” sentences that we encounter in real life. For example, “If the Pope won the
lottery then he’d spend the money on fast cars” is intuitively false (because fast cars aren’t the
sort of thing we’d expect the Pope to spend his money on) but it’s a true material conditional
(because the antecedent is false; the Pope didn’t win the lottery). For our purposes, however, we
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Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
don’t need to worry about this too much because the material conditional captures enough of the
logical features of normal conditionals to allow us to properly the evaluate arguments that
contain conditionals.
II) Assessing Conditionals
The ability to identify the “falsification conditions” for conditional statements is very useful
because many important claims are conditional in nature. This is especially true in philosophy.
By understanding how conditional claims can be proven false, we’re in a much better position to
level appropriate objections to such claims.
For example, let’s consider the sentence “If we’re hallucinating all the time then we can’t know
anything.”
This sentence is false if the antecedent, “We’re hallucinating all the time,” can be true
while the consequent, “We can’t know anything,” is false. Of course, if it’s false that we can’t
know anything, then it must be true that we can know something. (If that confuses you, you’re
just thinking too hard.)
Thus, the falsification condition for “If we’re hallucinating all the time then we can’t
know anything” is “We are hallucinating all the time (true antecedent) + We can know
something (false consequent),” and this condition can be expressed in a number of ways, such as
“Even if we’re hallucinating all the time, we might know some things, like math,” and “We can
know certain things, like logical truths, even if we’re always hallucinating.” (It doesn’t matter
whether we mention the true antecedent or the false consequent first. All that matters is that
we’re describing a situation in which the antecedent is true and the consequent is false.)
It’s easy to think “If we’re hallucinating all the time then we can’t know anything” would
also be proven false by “Maybe we don’t know anything even if we’re not hallucinating all the
time” but this is mistaken because that circumstance would be one in which the antecedent of the
conditional is false and the consequent is true. (Don’t believe me? Check it out for yourself.
Better yet, check it out for yourself even if you do believe me.)
Now let’s consider “The existence of God is a necessary condition for a well-designed universe.”
Because this conditional isn’t expressed in straightforward “if… then…” form, the first thing we
should do is symbolize this sentence in order to make sure that we correctly identify the
antecedent and the consequent.
Letting “G” be “God exists,” and “W” be “The universe is well-designed,” “The
existence of God is a necessary condition for a well-designed universe” is symbolized as
“W → G.” This allows us to see that “The universe is well-designed” is the antecedent of this
conditional and “God exists,” is the consequent.
Thus, the falsification condition for “The existence of God is a necessary condition for a
well-designed universe” is “The universe is well-designed (true antecedent) + God doesn’t exist
(false consequent),” and we can express this in a number of ways, including “The universe might
be well-designed even if God doesn’t exist,” and “Purely naturalistic forces could account for the
design that we see in the universe.” (It’s true that this second way of expressing the falsification
condition doesn’t explicitly say “God doesn’t exist,” but it accounts for how the universe could
be well-designed even if God didn’t exist so “God doesn’t exist” is still there, sort of hovering in
the background.)
Chapter 1 - Conditionals 16
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
If someone said, “Why would God create a well-designed universe?” however, she
wouldn’t be leveling a relevant objection to the conditional because this question alludes to the
fact that the antecedent might be false while the consequent is true.
It’s very easy to be mistaken about what would, and what wouldn’t, show a conditional to be
false, and many people get it wrong more often than not. This is why the ability to correctly
identify falsification conditions is a skill worth obtaining. Knowing how to properly object to
conditional statements, how to recognize improper objections, will give you a real advantage in
many contexts.
Practice
For each of the claims below identify the objection that, if true, would show that claim to be false.
1. “If we can’t know anything for certain then knowledge isn’t worth seeking.”74
a. Although we can’t know anything for certain, knowledge is worth seeking.
b. Although we can know some things for certain, knowledge isn’t worth seeking.
2. “If a belief is useful then we’re justified in holding it.”75
a. We’re justified in holding some beliefs even though they’re not useful.
b. We’re not justified in holding some beliefs even though they are useful.
3. “If euthanasia maximizes happiness then it’s ethical.”76
a. Some things that maximize happiness aren’t ethical.
b. Some things that are ethical don’t maximize happiness.
4. “Lying is unethical only if it violates our culture’s moral code.”77
a. Lying is unethical even if it doesn’t violate our culture’s moral code.
b. Lying violates our culture’s moral code even if it isn’t unethical.
5. “Provided that some beliefs are justified without argument, religious faith is philosophically
acceptable.”78
a. Even though no beliefs are justified without argument, religious faith is philosophically
acceptable.
b. Even though some beliefs are justified without argument, religious faith isn’t philosophically
acceptable.
6. “God doesn’t know our future actions, if we have free will.”79
a. “God knows our future actions but we have free will anyway.”
b. “God doesn’t know our future actions but we don’t have free will anyway.”
Chapter 1 - Conditionals 17
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
7. “We have good reason to believe in God, provided that we have good reason to believe in
miracles.”80
a. Even though we don’t have good reason to believe in God, we do have good reason to believe
in miracles.
b. Even though we have good reason to believe in God, we don’t have good reason to believe in
miracles.
8. “Cross-cultural agreement about ethics is a necessary condition for the objectivity of
morality.”81
a. Although morality is objective, there can still fail to be cross-cultural agreement about ethics.
b. Even if there is cross-cultural agreement about ethics, morality can still fail to be objective.
9. “For an action to be moral it’s sufficient that it satisfy Kant’s Categorical Imperative.”82
a. An action can be moral without satisfying Kant’s Categorical Imperative.
b. An action can satisfy Kant’s Categorical Imperative without being moral.
Symbolizing Conditionals with Multiple Connectors
Consider the following assignments to sentence letters.
M = You’re a philosophy major.
S = You’ll take symbolic logic.
E = You’ll enjoy the course.
What do the following sentences mean? Do they mean the same thing?
(M → S) → E
M → (S → E)
I find it pretty hard to get a grip on what these sentences are saying, so the surest way for me to
decide if these sentences mean the same thing is to construct their truth tables and see if they
differ.
M
T
T
S
T
T
E
T
F
T
F
T
f
T t
t
T
t
T
F
F
T
F
T
f
t
T f
T t
t T
f T
t
t
F
T
F
t
F f
f T
f
F
F
T
t
T t
f T
t
F
F
F
t
F t
f
t
(M → S) → E
t
T t
t
F f
M → (S → E)
t T
t
t F f
T
Chapter 1 - Conditionals 18
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Because these sentences do differ in truth value on the sixth and eighth lines, (M → S) → E isn’t
the same thing as M → (S → E), which means that M → S → E is genuinely ambiguous. We’ll
always need to provide grouping symbols in a sentence like that.
The main connector in a symbolized sentence is the connector that stands outside the grouping
symbols. A well formed formula (wff) must be such that, when it’s decomposed around its main
connector, the remaining parts have only one main connector.
For example, the connectives in the following formula are ranked like this:
1. Main Connector
2. Next to Main
{A → [(B → C) → D]} → F
3. Next to Next to Main Connector
4. Least Main Connector
Of course, it’s one thing to identify the main connector in an already-symbolized sentence. It’s
quite another thing to identify the main connector in a natural-language sentence. That can be
very tricky indeed, until you get the hang of it. Being able to recognize the main connector in
natural language sentences is an important skill, though, because it can really help you to
understand the logically complex sentences that you find in places like philosophy and law.
How to find the main connector in an English sentenceiii:
i) Identify all of the connectors. Underline or circle them, if you wish.
ii) For each connector, see what it connects.
iii) If there is nothing left over, that connector may be the main connector. If there is part of the
sentence left over, that connector isn’t the main connector.
Once we’ve identified the main connector, we symbolize “around” that main connector.
Let’s see how this works by doing some examples.
iii
If you encounter significant difficulties identifying the main connectors, here’s another approach for you to try:
i) Identify all of the connectors. Underline or circle them, if you wish.
ii) Read around each connector. Don’t leave out any words in the sentence but don’t say the connector you’re testing.
iii) If the things connected are sensible statements on their own (possibly after grammatical changes from the
subjunctive or gerund), then that connector may be the main connector. If the things connected aren’t sensible
statements on their own, then that connector isn’t the main connector.
Chapter 1 - Conditionals 19
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Example 1
“If this novel worth is reading then having an attractive cover is sufficient for being a good
book.”
First note the connectors:
“If this novel worth is reading then having an attractive cover is sufficient for being a good
book.”
Then identify the main connector:
“If… then…” connects “this novel worth is reading” and “having an attractive cover is sufficient
for being a good book.” Because there is nothing left over, that’s the main connector. (In contrast,
“is sufficient for” connects “has an attractive cover” with “is a good book” and “this novel is
worth reading” is left over.)
“If this novel worth is reading then having an attractive cover is sufficient for being a good
book.”
Finally, symbolize around the main connector:
“If (this novel worth is reading) then (having an attractive cover is sufficient for being a good
book.)”
“(this novel worth is reading) → (having an attractive cover is sufficient for being a good
book.)”
“R→ (having an attractive cover is sufficient for being a good book.)”
“R→ (C is sufficient for B.)”
“R→ (C → B)”
Chapter 1 - Conditionals 20
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Example 2
“Being a good bowler is necessary for membership only if it’s a bowling club.”
First note the connectors:
“Being a good bowler is necessary for membership only if it’s a bowling club.”
Then identify the main connector:
“Only if” connects “Being a good bowler is necessary for membership” with “it’s a bowling
club.” There’s nothing left over, so that’s the main connective.
“Being a good bowler is necessary for membership only if it’s a bowling club.”
Finally, symbolize around the main connector:
“(Being a good bowler is necessary for membership) only if (it’s a bowling club).”
“(Being a good bowler is necessary for membership) → C.”
“(B is necessary for M) → C.”
“(M → B) → C.”
Chapter 1 - Conditionals 21
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Example 3
“George will be admitted to the club if he seeks admission, provided that being a good bowler is
sufficient for membership.”
First note the connectors:
“George will be admitted to the club if he seeks admission, provided that being a good bowler is
sufficient for membership.”
Then identify the main connector:
“George will be admitted to the club if he seeks admission, provided that being a good bowler is
sufficient for membership.”
Finally, symbolize around the main connector:
“(George will be admitted to the club if he seeks admission,) provided that (being a good bowler
is sufficient for membership).”
“(G if A,) provided that (B is sufficient for M)”
“(B is sufficient for M) → (G if A)”
“(B → M) → (G if A)”
“(B → M) → (A → G)”
Chapter 1 - Conditionals 22
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Practice
Symbolize the following complex conditionals using the assigned sentence letters. The answers
are in the endnotes.
1. “If it’s the case that computers deserve moral consideration if they can think, then the ability
to think is the source of our moral value.”83
C = Computers can think
S = The ability to think is the source of our moral value.
M = Computers deserve moral consideration.
2. “If computers can think then provided that the ability to think is the source of our moral value,
computers deserve moral consideration.”84
C = Computers can think
S = The ability to think is the source of our moral value.
M = Computers deserve moral consideration.
3. “Provided that being able to carry on a conversation is sufficient for being able to think, the
Turing Test is an accurate indicator of artificial intelligence.”85
C = x can carry on an intelligent conversation.
T = x can think.
A = The Turing Test is an accurate indicator of artificial intelligence.
4. “Being able to carry on an intelligent conversation is necessary for having a mind only if many
people don’t have minds.”86
C = x can carry on an intelligent conversation.
M = x has a mind.
D = Many people don’t have minds.
5. “If our being able to think is sufficient for our having moral value, then if computers can think,
they deserve moral consideration.”87
T = We can think.
V = We have moral value.
C = Computers can think.
M = Computers deserve moral consideration.
6. “Being able to think is necessary for having moral value only if being able to think is
sufficient for having a soul.”88
T = x can think.
V = x has moral value.
S = x has a soul.
Chapter 1 - Conditionals 23
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
7. “Should being able to carry on an intelligent conversation be sufficient for having a mind, then
an answering machine would have a mind if it could answer back.”89
C = x can carry on an intelligent conversation.
M = x has a mind.
A = An answering machine has a mind.
B = An answering machine can answer back.
8. “If a computer could carry on an intelligent conversation then provided that being able to carry
on an intelligent conversation is sufficient for having a mind, the computer would have a mind.”
C = A computer can carry on an intelligent conversation.90
M = x has a mind.
I = x can carry on an intelligent conversation
H = A computer has a mind.
9. “Neither side can avoid circularity. This is disturbing if a non-question-begging way of
mutually resolving basic disagreements over the overall force of a body of evidence is a
necessary condition of the possibility of being rational in these matters.” (William J. Wainwright,
Reason and the Heart, Cornel University Press, 1995, p. 123)91
Symbolize the second sentences in this quotation, using the following sentence letters:
D = It’s disturbing that neither side can avoid circularity.
N = There is a non-question-begging way of mutually resolving basic disagreements over the
overall force of a body of evidence.
R = One can be rational in matters involving basic disagreements over the overall force of a body
of evidence
INFERENCE RULE: → O
Now that we know quite a bit about conditionals, we can study their logic by reflecting upon
what can, and what can’t, be inferred from them. Let’s start by reflecting upon one of my
favorite arguments:
1. If Max is a poodle then Max is a dog. (premise)
2. Max is a poodle. (premise)
3. Max is a dog. (from 1 and 2)
The inference in this argument is valid because if the premises are true then the conclusion must
be true as well. We can see this using the Bob Method. If Bob believes “If Max is a poodle then
Max is a dog” and if Bob believes “Max is a poodle,” Bob will be forced to believe “Max is a
dog.”
Now, what if instead of “Max,” we used the name “Leo?” Would the inference still be valid?
Sure.
And what if instead of “poodle,” we said “schnauzer?” Would the inference still be valid? Of
course.
Chapter 1 - Conditionals 24
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
And what if instead of “Max is a poodle” we said “The book is over 500 pages” and instead of
“Max is a dog” we said “Sam will never finish reading it.” Would the inference still be valid?
Yes.
What makes this inference valid is the form of the sentences, not the content of the sentences.
In general, any inference of the following form will be valid:
1.
(Premise)
→
2.
(Premise)
3.
(from 1 and 2)
Since this kind of inference is both valid and very common, it’s worthwhile to take it as an
inference rule. Inference rules are argument forms that specify correct inferences. They tell us
what we may correctly infer, given certain kinds of information, and so they’re very useful.
We’ll call this inference rule “Arrow Out” (abbreviated “→O”) because it takes us from (or out
of) an arrow statement. Arrow Out tells us that from a line of the form “ → ” and a line of
the form “” we can infer a line of the form “.”
Sometimes it will be convenient to present an argument horizontally, with all of the premises on
the left, separated by commas, and the conclusion on the right, separated from the premises with
the symbol “|-.” For instance, the argument “ → . . Therefore ,” can be written like this:
“ → , |- .” Because this way of representing arguments is so visually clean, I’ll use it to
present the inference forms. Arrow Out, for example, is presented as follows:
→O: → , |- Before we start to actually use this rule, there are a couple of things that I want to mention.
•
Why We’ll Use Shapes
First, many logic books use letters like “P” and “Q” when they present argument forms, instead
of using shapes like “” and “.” Such a book would present →O as “P → Q, P |- Q”.
I’ll be using shapes when I give you inference rules because I want to stress that the parts of the
compound sentence at issue (in this case the “” and the “” in the conditional “ → ”)
don’t need to be atomic statements symbolized by one letter but can, themselves, be compound
statements. We can think of these statements being written “inside” the shapes.
Chapter 1 - Conditionals 25
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
So, Arrow Out doesn’t just tell us that from P
It also tells us that from (A→B)
→
(C→D)
→
and
Q
and
(A→B)
P
Q
.
(C→D)
.
we can infer
we can infer
Thinking in terms of shapes in which smaller formulas can be written helps us to develop the
habit of proper “mental clumping.” In the argument above, for instance, it allows us to see
“A → B” and “C → D” as basic clumps of information. This mental clumping makes logic much
easier.
•
Primitive vs. Derived Inference Rules
Second, Arrow Out is what we’ll call a primitive inference rule. Many textbooks justify
primitive inference rules by appealing to a truth table. To justify Arrow Out in this way, we
construct a truth table that represents the premises and the conclusion of the argument. Then we
note that all of the lines of this truth table on which the premises are true must also represent the
conclusion as true. Like this:
Basic Term
T
T
F
F
Basic Term
T
F
T
F
Premise
→
T
F
T
T
Premise
T
T
F
F
Conclusion
T
F
T
F
The first row is the only row in which both premises are true, and in that row the conclusion is
true as well. This demonstrates that whenever the premises are true, the conclusion must be true.
In other words, it demonstrates that the argument is valid.
Now I don’t know about you, but I don’t find this appeal to truth tables to be very helpful. I
understand it, but psychologically speaking, it doesn’t add anything. It doesn’t scratch any itch
or clear anything up because I don’t need a truth table to demonstrate to me that Arrow Out is
valid. Of course, it’s valid. You can just see that it’s valid. If we know that P gets us Q (i.e. if we
know “P→Q”) and if we know that we have P, then we’re bound to know that we have Q as well.
If you didn’t already think this way, you certainly would have perished long ago. Somewhere
along the line, you’d have thought something like “If the truck is coming then I should stand on
the curb,” and “The truck is something,” but you’d have failed to infer “I should stand on the
curb.” And if you failed to draw that inference, you’d have stepped out into traffic and you
wouldn’t be reading this now. We might say that the validity of Arrow Out is self evident.
Indeed, if someone didn’t understand Arrow Out, it would be practically impossible to find
anything simpler in terms of which Arrow Out could be explained to him. To that extent, this
inference rule is psychologically primitive.
Chapter 1 - Conditionals 26
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
We’ll be getting other primitive inference rules as we go along, but I won’t bother to justify them
with truth tables because once we understand these rules it will be fairly easy for us to see that
they’re valid. Indeed, the fact that the validity of a rule is fairly easy to spot is what leads us to
take it as a primitive rule in the first place.
Not all valid inference rules are like this, though. Some valid inferences might not seem valid to
us at first. In this case, we’ll justify the validity of the rule by showing how it follows from other
rules that we already know to be valid. Inference rules that are justified in this way are called
derived inference rules.
In summary, then, some of our inference rules (the primitive rules) will be taken for granted
because they’re obviously valid, and the rest of our inference rules (the derived rules) will be
justified by appeal to rules that we already know to be valid. iv
ESTABLISHING VALIDITY: FORMAL PROOFS
Now that we have an valid inference rule in hand, we can take a look at the heart of symbolic
logic – formal proofs!
The basic idea is pretty simple. If we can proceed from the premises of an argument to its
conclusion by a series of valid inference rules (so far, we have only one inference rule), then the
argument as a whole is valid because all of its little steps are valid. This is the method of natural
deduction, otherwise known as the construction of formal proofs or derivations, for proving
validity.
A proof for an argument will symbolize all sentences in the argument and record them in a
numbered list, with the premises at the top and the conclusion at the bottom. (I usually make a
note of the conclusion to the far right of the last premise, as “want…,” to remind me of what I
want to prove. This isn’t really part of the proof, and it isn’t strictly necessary, but it will be very
helpful later on!) Each line in a proof is justified in the justification column on the right.
Premises get their justification for free and are noted with “P.” The justification for a derived
line and refers to the lines used and the rule employed in the derivation of that line. Here’s a little
baby proof of the argument P→Q, P |- Q:
1. P → Q
2. P
3. Q
P
P - want Q
1, 2 →O
(Isn’t it cute?)
iv
“Fairly easy to see as valid” will be a necessary condition for being a primitive rule but not a sufficient condition.
In other words, if we take an inference as primitive then it must be fairly easy to see as valid (generally speaking),
but it isn’t the case that every inference that we can easily see as valid will be taken as primitive. Some fairly “easy”
inference rules will derived instead. This is because logicians generally try to take as few primitive inference rules as
possible. There’s more “elegant simplicity” to a system that takes two primitive inference rules and derives the rest
than there is to a system that takes, say 27 inference rules as primitive.
Chapter 1 - Conditionals 27
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Before you start constructing proofs of your own, there are some things that we should note:
Note: The order in which the premises appear never matters.
So this proof is fine.
1. P → Q
2. P
3. Q
P
P - want Q
1, 2 →O
And this proof is fine, too:
1. P
2. P → Q
3. Q
P
P - want Q
1, 2 →O.
We don’t have to have the “→” premise before we have the “” premise.
Note: The order in which we write the down the numbers in the justification column never
matters.
Look at the proof we’ve just seen, we can write “1, 2 →O” in the justification column for line 3,
like so:
1. P
2. P → Q
3. Q
P
P - want Q
1, 2 →O.
Or we can write “2, 1 →O” in the justification column for line 3, like this:
1. P
2. P → Q
3. Q
P
P - want Q
2, 1 →O.
All that matters is that we refer to the correct lines, in this case lines 1 and 2; the order in which
we refer to these lines is immaterial. (I will follow the convention of always referring to the lines
in the order that they apper in the proof, so I’ll always write down “1, 2” rather than “2, 1,” but
that’s just my convention and nothing hangs on it.)
Chapter 1 - Conditionals 28
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Note: → O can be used only on two lines at a time. It’s a “two line rule.”
For example, here’s how we’ll construct a proof for this argument:
A, A → B, B → C, C → D | - D
First we’ll write down the premises, noting the conclusion that we want:
1. A
2. A → B
3. B → C
4. C → D
P
P
P
P - want D
Next, we’ll conclude B from lines 1 and 2 using →O:
1. A
2. A → B
3. B → C
4. C → D
5. B
P
P
P
P - want D
1, 2 →O
Then we’ll conclude C from lines 3 and 5 using →O:
1. A
2. A → B
3. B → C
4. C → D
5. B
6. C
P
P
P
P - want D
1, 2 →O
3, 5 →O
And finally, we’ll conclude D from lines 4 and 6 using →O:
1. A
2. A → B
3. B → C
4. C → D
5. B
6. C
7. D
P
P
P
P - want D
1, 2 →O
3, 5 →O
4, 6 →O
It might be tempting to put lines 1, 2, 3 and 4 together all at once, but that’s against the rules.
→O puts sentences together two at a time, as we did above.
Chapter 1 - Conditionals 29
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Note: Any statement, simple or compound, may be the antecedent, , or the consequent, .
For example, we can construct a proof for this argument: P → Q, (P → Q) → (R → S) |- R → S
1. P → Q
2. (P → Q) → (R → S)
3. R → S
P
P – want R → S
1, 2 →O
Here’s where mental clumping comes in handy. If you have trouble following that proof,
imagine the shapes around the components, like so:
1.
P→Q
P
2. (P → Q) → (R → S)
P – want R → S
3. R → S
1, 2 →O
See? No big deal. Just think about the “P→Q” and the “R→S” as one unit.
Note: You can apply →O only to the main connector! (This will be true of all “out rules.”)
This point is incredibly important, and failing to fully appreciate it is the cause of many incorrect
proofs.
For example, consider the argument (A → C) → (B → D), A → C, B |- D
The following proof of this argument is incorrect.
1. (A → C ) → (B → D)
2. A → C
3. B
4. D
P
P
P - want D
1, 3 →O
Can you see the problem? The proof fails on line 4 because →O is incorrectly applied. This
proof “arrows out” on the arrow between B and D, but that arrow isn’t the main connector on
line 1.
1. (A → C ) → (B → D)
2. A → C
3. B
4. D
P
P
P - want D
1, 3 →O
Chapter 1 - Conditionals 30
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
The arrow between “A → C” and “B → D” is the main connector on line 1, so here’s a correct
proof of the argument:
1. (A → C ) → (B → D)
2. A → C
3. B
4. B → D
5. D
P
P
P - want D
1, 2 →O
3, 4 →O
Can you see how this proof uses →O correctly? Each instance of →O is applied only to the main
connector of the conditional.
By the way, if you look at that proof and get all confused, you’re probably trying to take in the
whole thing at once. Follow it through one line at a time. When you look at line 4, see how it
follows from lines 1 and 2. Don’t bother with line 3 because line 3 isn’t used to get line 4. Then
when you look at line 5, see how it follows from lines 3 and 4. Don’t bother with line 2 because
line 2 isn’t used to get line 5. Never swallow a proof whole. Chew thoroughly. Like this:
1. (A → C ) → (B → D)
2. A → C
3. B
4. B → D
5. D
P
P
P - want D
1, 2 →O (See how this line follows from 1 and 2?)
3, 4 →O
1. (A → C ) → (B → D)
2. A → C
3. B
4. B → D
5. D
P
P
P - want D
1, 2 →O
3, 4 →O (See how this line follows from 3 and 4?)
Note: There can be more than one correct proof of an argument.
Consider, for example, the following two proofs of the same argument:
A → (B → C), D → B, A, D |- C
1. A → (B → C)
2. D → B
3. A
4. D
5. B → C
6. B
7. C
Proof 1
P
P
P
P – want C
1, 3 →O
2, 4 →O
5, 6 →O
Proof 2
1. A → (B → C)
2. D → B
3. A
4. D
5. B
6. B → C
7. C
P
P
P
P – want C
2, 4 →O
1, 3 →O
5, 6 →O
Chapter 1 - Conditionals 31
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Both of these proofs are correct; they simply draw the inferences in a different order. In Proof 1,
“B → C” is inferred before “B.” In Proof 2, “B” is inferred before “B → C.” Because both
proofs use →O correctly, however, both proofs are fine.
In general, as arguments become more complex, they will be open to a greater variety of proofs.
Different proofs might employ the same rules in a different order, as we’ve seen above, or
(especially as the set of rules gets richer) they might employ slightly different rules. All of these
different proofs will be correct, as long as they properly use the inference rules that they employ.
What this means in practical terms, of course, is that if your proof disagrees with mine, your
proof can still be fine – as long as you’ve correctly used the inference rules.
Note: The same line can be used in multiple inferences.
We don’t “use lines up” when we deploy them in an inference. For example, let’s walk through
the proof for this argument:
P → (Q → R), L → P, L, (L → P) → Q |- R
1. P → (Q → R)
2. L → P
3. L
4. (L → P) → Q
P
P
P
P - want R
To start the proof, we could either use →O on lines 2 and 3
1. P → (Q → R)
2. L → P
3. L
4. (L → P) → Q
P
P
P
P - want R
or we could use →O on lines 2 and 4.
1. P → (Q → R)
2. L → P
3. L
4. (L → P) → Q
P
P
P
P - want R
It doesn’t matter which we decide to do first. I’ll do →O on lines 2 and 3 just because that’s
just what I decided.
1. P → (Q → R)
2. L → P
3. L
4. (L → P) → Q
5. P
P
P
P
P - want R
2, 3 →O
Chapter 1 - Conditionals 32
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Now, I’ll do →O on lines 2 and 4.
1. P → (Q → R)
2. L → P
3. L
4. (L → P) → Q
5. P
6. Q
P
P
P
P - want R
2, 3 →O
2, 4 →O
Note that line 2 was used twice: one in the inference to “P” on line 5 and again in the inference
to “Q” on line 6. That’s just fine. We don’t use lines up.
At this point, we can use →O on lines 1 and 5 (which might be hard to see at first only because
lines 1 and 5 are so far away from each other in the proof).
1. P → (Q → R)
2. L → P
3. L
4. (L → P) → Q
5. P
6. Q
7. Q → R
P
P
P
P - want R
2, 3 →O
3, 4 →O
1, 5 →O
And then we can complete the proof by using →O on lines 6 and 7.
1. P → (Q → R)
2. L → P
3. L
4. (L → P) → Q
5. P
6. Q
7. Q → R
8. R
P
P
P
P - want R
2, 3 →O
3, 4 →O
1, 5 →O
6, 7 →O
(By the way, that was a reasonably complex proof so don’t worry if you feel like you couldn’t
have done it on your own. After all, you’ve only seen a handful of proofs at this point. If you
were able to follow along, step by step, as I did the proof, you’re doing just fine.)
Finally, the construction of proofs is very much like solving puzzles. Some proofs, like some
puzzles, are easy; some proofs, like some puzzles, are hard. But, just as there are often hints, or
rules of thumb, that we can use to solve puzzles more easily, there are hints, or rules of thumb,
that we can use to make our way from the premises to the conclusion more easily. Forward hints
tell us what to do given what we have. Backward hints tell us what to do given what we want.
Each inference rule has its own hints. Here are the hints for arrow out.
Chapter 1 - Conditionals 33
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Forward Hint for →O: If you have a line of the form → , see if you have or can get a line of
the form . If so, use →.
Backward Hint for →O: If you want , see if you have or can get a line of the form → . If
so, see if you have or can get a line of the form and use →O.
The backward hint is particularly helpful because it basically tells us to see if we can find what
we want sitting as a consequent of something that we already have. If we can, then we should
feel happy because we might be able to get what we want using →O.
For example, suppose we want to prove the following argument:
[(P → Q) → R] → T, T → (L → M), (P → Q) → R |- L → M
1. [(P → Q) → R] → T
2. T → (L → M)
3. (P → Q) → R
P
P
P - want L → M
The first thing we might notice is that our conclusion is sitting as the consequent on line 2.
1. [(P → Q) → R] → T
2. T → (L → M)
3. (P → Q) → R
P
P
P - want L → M
If we could get the antecedent of line 2, T, then we could infer L→M using →O.
1. [(P → Q) → R] → T
2. T → (L → M)
3. (P → Q) → R
P
P
P - want L → M
And T is sitting as the consequent of the conditional on line 1.
1. [(P → Q) → R] → T
2. T → (L → M)
3. (P → Q) → R
P
P
P - want L → M
So if we could get the antecedent of the conditional on line 1, we could conclude T using →O.
And guess what? We already have the antecedent of the conditional on line 1. It’s just the
statement on line 3.
1. [(P → Q) → R] → T
2. T → (L → M)
3. (P → Q) → R
P
P
P - want L → M
Chapter 1 - Conditionals 34
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
So there we have it! If we do →O on lines 1 and 3, we can infer T. Then we can do →O on line 2,
to give us our conclusion. The proof looks like this:
1. [(P → Q) → R] → T
2. T → (L → M)
3. (P → Q) → R
4. T
5. L → M
P
P
P - want L → M
1, 3 →O
2,4 →O
Mental Proofs
One last word before presenting some practice arguments: Although constructing written proofs
is a handy skill and can be rather fun, in most contexts outside a logic classroom, it’s much more
useful to be able to construct a proof in your head, mentally tracing the inferences.
For example, let’s consider this argument: L→M, L, M→(A→B) |- A→B
Can you see that it’s valid by doing the proof in your head?
Obviously, I can’t write down exactly what I see in my own head, but roughly speaking, here’s
how it goes:
The argument is: L→M, L, M→(A→B) |- A→B.
Well, from L→M and L we get M. And from M
and M→(A→B) we get A→B.
So this argument is valid.
Of course, to be really useful, we need to be able to do this with arguments written in English. I
suppose that we should, ideally, be able to symbolize and prove mentally, but I don’t do that. I
find it much easier to actually write out the symbolization, especially if the sentences are written
in anything other than standard “if…then…” form, as they often are. I can then do the proof in
my head (unless, of course, the proof is very difficult).
For example, let’s consider this argument: “If all of the parties to the contract are competent then
the contract is binding, provided that all of the signers had an opportunity to read it. All of the
signers had an opportunity to read the contract. And all of the parties to the contract are
competent. Therefore the contract is binding.”
Can you sketch out a symbolization of this argument on paper and then prove it in your head?
This is roughly how I do it:
Chapter 1 - Conditionals 35
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
“If all of the parties to the contract are competent then
the contract is binding, provided that all of the signers
had an opportunity to read it. All of the signers had an
opportunity to read the contract. And all of the parties
to the contract are competent. Therefore the contract is
binding.”
Okay, that’s a bit confusing. I’d better symbolize it.
If all of the parties to the contract are competent then the contract is binding,
provided that all of the signers had an opportunity to read it.
C → (R → B)
All of the signers had an opportunity to read the contract.
R
And all of the parties to the contract are competent.
C
Therefore the contract is binding.”
B
That’s better! Looking that this symbolization, I see
that C → (R → B) and the C gives us R →B. And we
already have R, so that gives us B.
The argument is valid.
Because in order for symbolic logic to be as useful to us as possible we need to be able to do
some proofs in our head, I’ll be asking you to try to do some proofs mentally. Don’t worry at all
if this is difficult for you at first. If you can’t do a proof mentally, do a written proof for the
argument, leave the argument for five minutes or so, and then come back and see if you can do
the proof again, this time entirely in your head. With practice, it gets much easier.
Chapter 1 - Conditionals 36
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Practice
Prove the following arguments valid. The answers follow in the endnotes.
1) A →B, A, B →C |- C92
2) R → (T →S), R, T |- S93
3) M →N, (M →N) →Q, Q →T |- T94
4) P →Q, (P →Q) → (A →F), A |- F95
5) C, D→(F→H), (A→B)→(C→D), A→B |- F→H96
6) Do this proof mentally: R→S, R, S → (A → B) |- A → B97
7) Do this proof mentally: R→S, R, S → (A → B), A |- B98
8) Do this proof mentally: R→S, R, S → (A → B), A, B→C |- C99
9) B→(L→R), [B→(L→R)]→B |- L→R100
10) A→B, A, (A→B)→(B→D) |- D101
11) (W → B) → [C → (A →W)], C, A, W → B |-B102
12) P→Q, {B→[(P→Q)→R]}→(A→B), A, B→[(P→Q)→ R] |- R103
13) Do this proof mentally: “If God exists then miracles are possible. And God does exist.
Therefore miracles are possible.”104
14) Do this proof mentally: “Intelligent origin is necessary for an ordered universe. If the
universe has intelligent origin then God exists. The universe is ordered. Therefore, God
exists.”105
15) Do this proof mentally: “People have an innate sense of right and wrong. People can have an
innate sense of right and wrong only if God exists. And provided that God exists, evil-doers will
be punished. Therefore, evil-doers will be punished.”106
16) Do this proof mentally: “If utilitarianism is right then lying can be morally permissible if it
sometimes serves to promote the greatest happiness for the greatest number of people. In fact,
utilitarianism is right. And lying does sometimes serve to promote the greatest happiness for the
greatest number of people. Therefore, lying can be morally permissible.”107
Chapter 1 - Conditionals 37
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
17) “If Kant is right then ethical acts must pass the categorical imperative. Kant is right. Lying is
ethically wrong if lying is unable to become the universal law, provided that ethical acts must
pass the categorical imperative. If lying would make communication and society impossible then
it’s unable to become a universal law. Lying would make communication and society impossible.
Therefore lying is ethically wrong.”108
ESTABLISHING INVALIDITY
If we can construct a proof for an argument then we know that the argument is valid. However, if
we can’t construct a proof for an argument, we don’t thereby know that the argument is invalid.
This is because there are lots of reasons why we might not be able to construct a proof for an
argument. Maybe the argument is invalid, but maybe the argument is valid and we can’t prove it
because we’re tired, or because we haven’t learned all of the rules we need yet.
So, how can we show that an argument is invalid? We’ll see two ways.
•
The Truth Table Test
First, we can exploit the fact that in a valid argument, whenever the premises are true the
conclusion must be true as well. In other words, in a valid argument, the premises can’t be true
and the conclusion false at the same time. To show that an argument is invalid, then, all we need
to do is demonstrate that the premises can be true and the conclusion false at the same time, and
in order to do that we’ll assign truth values to sentence letters.v
This is sometimes called “the Truth Table Test” and we can summarize it as follows:
The Truth Table Test for Validity and Invalidity:
An argument is valid. = It’s impossible for the premises to be true and the conclusion to be false.
An argument is invalid. = It’s possible for the premises to be true and the conclusion to be false.
-SoIf we can’t make the premises true and the conclusion false then the argument is valid.
If we can make the premises true and the conclusion false then the argument is invalid.
Get it? A few of examples will probably make this a lot clearer.
v
Since we can show that an argument is invalid by demonstrating that the premises can all be true and the
conclusion false at the same time, can we show that an argument is valid by demonstrating that the premises can all
be true and the conclusion true at the same time? No. If we know that an argument has true premises and a true
conclusion, we don’t know that the argument is valid because it may have simply gotten lucky (e.g. “Many women
are over 5 feet tall. Therefore Madison is the capitol of Wisconsin.”) However, if we know that an argument has true
premises and a false conclusion then we know that the argument must be invalid because we know that the truth of
the premises doesn’t “force” the truth of the conclusion.
The key question isn’t, “If this argument has true premises, can it have a true conclusion?” The key
question is, “If this argument has true premises, must it have a true conclusion?”
Chapter 1 - Conditionals 38
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Let’s start by considering the valid argument form we’ve been studying:
P → Q,
P
|-
Q
And let’s see if we can make the premises true and the conclusion false at the same time.
Well, if the premises are true, then P must be true because P is the second premise.
P → Q,
t
P
T
|-
Q
And if the conclusion is false then Q must be false because Q is the conclusion.
P → Q,
t
f
P
T
|-
Q
F
But now, what about the first premise? If P is true and Q is false, then P→Q is false.
P → Q,
t F f
P
T
|-
Q
F
We’ve seen that we can’t make the premises true and the conclusion false at the same time. This
means that whenever the premises are true, the conclusion must be true as well. And this, in turn,
means that the argument is valid (which is hopefully what you expected all along).
Now let’s turn to another argument:
P → Q,
Q→R
|-
P→R
Once again, we’ll see if we can make the premises true and the conclusion false at the same time.
There are a number of ways that the premises can be true, but there’s only one way for the
conclusion to be false: P must be true and Q must be false.
P → Q,
t
Q→R
f
|-
P→R
t F f
Now, if the premise P→Q is to be true, Q has to be true.
P → Q,
t T t
Q→R
t
f
|-
P→R
t F f
But this makes the second premise, Q → R, false.
P → Q,
t T t
Q→R
t F f
|-
P→R
t F f
Chapter 1 - Conditionals 39
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Once again, we’ve seen that we can’t make the premises true and the conclusion false and so this
argument is valid even though we can’t construct a proof for this argument yet. (We will be able
to construct a proof for this argument later on.vi)
Finally, let’s look at this argument:
P → Q,
Q
|-
P
Can the premises be true and the conclusion false? Well, if the premises are true then Q must be
true because Q is a premise.
P → Q,
t
Q
T
|-
P
And of course, P must be false because that’s the conclusion.
P → Q,
f
t
Q
T
|-
P
F
But now, what about the premise P→Q? That premise is true! (Remember that a conditional is
true if the antecedent is false and the consequent is true.)
P → Q,
f T t
Q
T
|-
P
F
This shows that the premises can be true and the conclusion false at the same time.
In other words, the truth of the premises doesn’t force the truth of the conclusion.
This argument, then, is invalid.
•
Spotting Formal Fallacies
The invalid argument form that we’ve just seen is very important because it shows up a lot and
can fool the unwary. After all, it’s so temptingly close to Arrow Out, a valid argument form!
→O - Valid
P→Q
P
Therefore Q.
Invalid
P→Q
Q
Therefore P.
To see the difference between these argument forms more clearly, it helps to use the “Poodle /
Dog” example.
vi
Since we can use this method to establish validity, why bother with proofs? We’ll continue to focus extensively
upon proofs because a proof corresponds – at least roughly - to the thought process behind an argument and because
we want to use logic to think through arguments. (Besides, in predicate logic truth tables won’t work anymore, but
proofs still will.)
Chapter 1 - Conditionals 40
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Invalid
P→Q
Q
Therefore P.
If Max is a poodle then Max is a dog.
Max is a dog.
Therefore Max is a poodle.
→O - Valid
P→Q
P
Therefore Q.
If Max is a poodle then Max is a dog.
Max is a poodle.
Therefore Max is a dog.
The first argument works because, given the fact that all poodles are dog (premise 1), if we’re
told that Max is a poodle (premise 2), we can conclude that Max is a dog.
The second argument fails because, given the fact that all poodles are dog (premise 1), if we’re
told that Max is a dog (premise 2), we can’t conclude that Max is a poodle. Max could be a
terrier, or a schnauzer, or a Chihuahua.
Here’s another way to think about this that many people find helpful.
Consider the following map:
HIGHWAY P
Hi!
This map shows us that there are
at least three ways to reach Q-City.
If we take Highway P then we’ll
get to Q-City, so “P → Q” is true.
HIGHWAY R
Q-CITY
If we take Highway R then we’ll
get to Q-City, so “R → Q” is true.
HIGHWAY S
And if we take Highway S then
we’ll get to Q-City, so “S → Q” is
true.
Now, suppose that we run into a friend in Q-City. Can we conclude that he took Highway P to
get there? No, because he could have taken Highway R or Highway S instead.
In other words, although it’s true that if our friend took Highway P then he’d be in Q-City
(i.e. “P → Q” is true), and although it’s true that our friend is in Q-City (i.e. “Q” is true), it does
not follow that our friend took Highway P (i.e. “P” doesn’t follow).
“P→Q, Q |- P” is invalid.
Chapter 1 - Conditionals 41
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Since this kind of faulty inference is so very common, it’s worthwhile to take it as a formal
fallacy. Formal fallacies are argument forms that specify common but incorrect inferences and
they’re very useful to know. This formal fallacy is called in the “Fallacy of Assuming the
Consequent,” because it involves assuming that the consequent of a conditional is true and
inferring that the antecedent of the conditional is true.
→O - Valid
P→Q
P
Therefore Q.
If Max is a poodle then Max is a dog.
Max is a poodle.
Therefore Max is a dog.
Fallacy of Assuming the Consequent - Invalid
P→Q
Q
Therefore P.
If Max is a poodle then Max is a dog.
Max is a dog.
Therefore Max is a poodle.
FAC (Fallacy of Assuming the Consequent): → , |- .
No! Don’t do this! It doesn’t work! This is an invalid inference! Beware!
It’s good to be familiar with formal fallacies because it’s much more common to recognize that
an argument is invalid by seeing that it commits a formal fallacy than it is to recognize that an
argument is invalid by employing the truth table method. In fact, the only time I use truth tables
is when I’m teaching formal logic; in “real life,” I look for formal fallacies instead. If every
proof for an argument depends upon the commission of a formal fallacy then I know that the
argument is invalidvii.
For example, consider the argument: A, A → (R → S), S | - R. Is this argument valid or invalid?
Can we prove it using only valid inferences, or does any proof depend upon the commission of a
fallacy? Let’s see:
1. A
2. A → (R → S)
3. S
4. R → S
5. R
P
P
P – want R
1,2 MP
3,4 FAC!
Because this proof compels us to commit the Fallacy of Assuming the Consequent, this argument
is invalid. The conclusion does not logically follow from the premises.
vii
Technically speaking, any “proof” that commits a formal fallacy isn’t a real proof because a real proof contains
only valid inferences. But I’ll speak in this looser way because it’s less cumbersome and more natural and because I
don’t think that this relaxed use of the term will cause any serious misunderstandings.
Chapter 1 - Conditionals 42
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Mental Evaluation
As with recognizing that arguments are valid by constructing proofs for them, the ability to
recognize that an argument is valid by detecting that it commits a formal fallacy is the most
useful when we can carry it out mentally.
For example, let’s consider this argument: L→M, L, (A→B)→M |- A→B
Can you see that it’s invalid by attempting to do the proof in your head?
Here’s roughly how I think about it:
The argument is: L→M, L, (A→B)→M |- A→B.
Well, from L→M and L we validly get M. But in order
to go from M and (A→B)→M to A→B, we’d need to
commit the Fallacy of Assuming the Consequent.
So this argument is invalid.
No let’s consider this English argument: “If all of the parties to the contract are competent, then
the contract is binding only if all of the signers had an opportunity to read it. All of the signers
had an opportunity to read the contract. And all of the parties to the contract are competent.
Therefore the contract is binding.”
Can you sketch out a symbolization of this argument on paper and then try to prove it in your
head?
This is roughly how I do it:
“If all of the parties to the contract are competent, then
the contract is binding only if all of the signers had an
opportunity to read it. All of the signers had an
opportunity to read the contract. And all of the parties
to the contract are competent. Therefore the contract is
binding.”
That’s a bit confusing. I’d better symbolize it.
Chapter 1 - Conditionals 43
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
If all of the parties to the contract are competent, then the contract is binding
only if all of the signers had an opportunity to read it.
C → (B → R)
All of the signers had an opportunity to read the contract.
R
And all of the parties to the contract are competent.
C
Therefore the contract is binding.”
B
That’s better! Looking that this symbolization, I see
that C → (B → R) and the C gives us B →R. But in
order to go from R and B→R to B, we’d need to
commit the Fallacy of Assuming the Consequent.
This argument is invalid.
In order to help you learn how to assess validity in your head, I’ll be asking you to try to do some
evaluations mentally. Remember, don’t worry at all if this is difficult for you at first. If you can’t
evaluate an argument mentally, attempt a written proof first. Then leave the argument for five
minutes or so, come back to it again, and see if you can evaluate it entirely in your head. With
practice, it gets much easier.
Practice
Determine whether the arguments are valid or invalid by attempting to prove them. The answers
follow in the endnotes.
1. (P → Q) → (R → S), R, P→Q, |- S109
2. (P → Q) → (R → S), S, P→Q, |- R110
3. R, B→A, A→C, C→(R→S), B |- S111
4. L → (P → Q), S→R, P, L, Q→R |- S112
Chapter 1 - Conditionals 44
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
5. Assess mentally: L→M, (L→M) → (A→B), B | A113
6. Assess mentally: L→M, (L→M) → (A→B), A | B114
7. Write the symbolization and assess mentally: “If God exists then the world would be welldesigned. And the world is well designed. Therefore God exists.”115
8. Write the symbolization and assess mentally: “If the world is well-designed then God exists.
And the world is well designed. Therefore God exists.”116
9. Write the symbolization and assess mentally: “People have an innate sense of right and wrong.
People can have an innate sense of right and wrong only if God exists. And provided that God
exists, evil-doers will be punished. Therefore, evil-doers will be punished.”117
10. Write the symbolization and assess mentally: “People have an innate sense of right and
wrong. People can have an innate sense of right and wrong if God exists. And provided that God
exists, evil-doers will be punished. Therefore, evil-doers will be punished.” 118
11. “If lying can be morally permissible provided that it sometimes serves to promote the
greatest happiness for the greatest number of people then utilitarianism is right. In fact,
utilitarianism is right. And lying does sometimes serve to promote the greatest happiness for the
greatest number of people. Therefore, lying can be morally permissible.”119
12. “If utilitarianism is right then lying can be morally permissible provided that it sometimes
serves to promote the greatest happiness for the greatest number of people. In fact, utilitarianism
is right. And lying does sometimes serve to promote the greatest happiness for the greatest
number of people. Therefore, lying can be morally permissible.”120
WHERE NEXT?
Where next? Well, we’ve seen one valid inference form, Arrow Out, which tells us that from
P →Q, and P, we may infer Q. Arrow Out has another name, modus ponens, which is Latin for
“the way of affirming.” (This name makes sense because Arrow Out involves affirming the
antecedent of a conditional and then affirming the consequent of the conditional.)
At this point, I’m tempted to introduce the inference rule modus tollens, which tells us that from
P→Q, and Not-Q, we may infer Not-P. The name “modus tollens” is Latin for “the way of
denying,” (This name makes sense because modus tollens involves denying the consequent of a
conditional and then denying the antecedent of the conditional.)
It would be nice to discus modus tollens immediately after modus ponens because modus tollens
and modus ponens form a “matched set,” so to speak.
Chapter 1 - Conditionals 45
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Modus Ponens
Modus Tollens
P → Q, P |- Q
P → Q, Not-Q |- Not-P
There is, however, an excellent reason to not introduce modus tollens right now: unlike modus
ponens, which is obviously valid, modus tollens can really puzzle people. (If you find modus
tollens to be just as obvious as modus ponens, good for you. You’re in the minority.) Because
people can be unsure that modus tollens is valid, we won’t want to take it as a primitive inference
rule; we’ll want to derive it. But the best way to derive modus tollens is to use a method called
“proof by contradiction,” and in order to carry out a proof by contradictions, we need to be
familiar with conjunctions, conditional proofs, and negations. Our next order of business, then,
will be to learn about these topics, and about all of the other incredibly useful things associated
with them.
Chapter 1 - Conditionals 46
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
CHAPTER SUMMARY
UNDERSTANDING COMPLEX SENTENCES
If then = → •
•
•
•
•
→ is false if and only if can be true and can be false at the same time.
Whatever follows “if” (“should” and “provided that”) becomes the antecedent.
Whatever follows “only if” becomes the consequent.
Whatever is being described as sufficient becomes the antecedent.
Whatever is being described as necessary becomes the consequent.
If the sentence has more than one connector, we identify the main connector by
i) Identifying all of the connectors.
ii) Seeing what each connector connects.
iii) If there is nothing left over, that connector may be the main connector. If there is part of the
sentence left over, that connector isn’t the main connector.
Once we’ve identified the main connector, we symbolize “around” that main connector
To see if two sentences say the same thing, symbolize the sentences and see if the
symbolizations are the same or equivalent to each other (see the equivalence rules below). If so,
then the sentences say the same thing. If not, then they don’t.
ASSESSING ARGUMENTS FOR VALIDITY / INVALIDITY
If we can proceed from the premises of an argument to its conclusion by a series of valid
inference rules then the argument as a whole is valid. If we can proceed from the premises of an
argument to its conclusion only by committing a formal fallacy then the argument as a whole is
invalid.
Valid Inference Rules
→O
→, |- Formal Fallacies
FAC
→ , |- Chapter 1 - Conditionals 47
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
ANSWERS
Symbolizing Conditionals with One Connector
I) Symbolizing with “If…. then….”
Figuring Things Out
1
1. “If you’re a physics major then your major is physics.”
P→P
We use “P” for the antecedent and the consequent because “You’re a physics major” and
“Your major is physics” is the same idea.
2
2. “If you’re taking a physics class then you’re taking a science class.”
P→S
3
P→S is true.
Physics classes are science classes so if you’re taking physics, you’re taking science.
4
3. “If you’re taking a science class then you’re taking a physics class.”
S→P
5
S→P isn’t true.
You could be taking a biology or chemistry class instead.
6
→ ≠ → .
This is an important logic fact. Order matters when we have conditionals.
II) Symbolizing with “If,” “Should,” and “Provided that”
Figuring Things Out
7
1. “If Jim is a pediatrician, he’s a doctor.”
P→D
8
2. “Jim is a doctor, if he’s a pediatrician.”
P→D
9
Rule: If , = → .
Rule: , if = →
.
Rule: In general, whatever follows “if” becomes the antecedent
10
3. “Should the movie be good, we’ll recommend it.”
of the conditional.
Chapter 1 - Conditionals 48
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
M→R
11
4. We’ll recommend the movie, should it be good.”
M→R
12
5. “Provided that it’s a nice day, we’ll go on a walk.”
N→W
13
6. “We’ll go on a walk, provided that it’s a nice day.”
N→W
14
Rule: “Should” and “provided that” work exactly like “if” and whatever follows them
becomes the antecedent of the conditional.
Practice
15
1. “If C, D.”
C→D
16
2. “A, if B.”
B→A
17
3. “P, provided that Q.”
Q→P
18
4. “Provided that R, S.”
R→S
19
5. “Should L, A.”
L→A
20
6. “A, should P.”
P→A
21
1. “If the painting is expensive then he’ll like it.”
“If the paining is expensive, he’ll like it.”
“He’ll like the paining, if it’s expensive.”
22
2. “If the class is open then we’ll enroll.”
“Should the class be open, we’ll enroll.”
“We’ll enroll, should the class be open.”
23
3. “If computers can think then minds are physical.”
“Provided that computers can think, minds are physical.”
“Minds are physical, provided that computers can think.”
Chapter 1 - Conditionals 49
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
III) Symbolizing with “Sufficient” and “Necessary”
Figuring Things Out
24
1. “Getting 100% is a sufficient condition for passing the test.”
H→P
25
2. “Getting 100% is sufficient for passing the test.”
H→P
26
3. “To pass the test, it’s sufficient to get 100%.”
H→P
27
Rule: Whatever is being described as sufficient becomes the
conditional.
28
4. “Completing the foreign language requirement is a necessary condition for obtaining your
degree.”
D→F
29
5. “Completing the foreign language requirement is necessary for obtaining your degree.”
D→F
30
6. “To obtain your degree, it’s necessary to complete the foreign language requirement.”
D→F
31
Rule: Whatever is being described as necessary becomes the consequent of the conditional.
Practice
32
1. “Q is sufficient for D.”
Q→D
33
2. “For E, it’s sufficient that O.”
O→E
34
3. “L is a sufficient condition for Q.”
L→Q
35
4. “A is necessary for B.”
B→A
36
5. “R is a necessary condition for C.”
C→R
antecedent
of the
Chapter 1 - Conditionals 50
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
37
6. “For T, it’s necessary that D.”
T→D
38
1. “If Robin is a surgeon then she’s smart.”
“Robin being a surgeon is sufficient for her being smart.”
“Robin being smart is necessary for her being a surgeon.”
39
2. “If the power was out then the clocks will be blinking.”
“The power being out is a sufficient condition for the clocks blinking.”
“The clocks blinking is a necessary condition for the power having been out.”
40
3. “If ghosts exist then minds are nonphysical.”
“Minds being nonphysical is necessary for ghosts existing.”
“Ghosts existing is sufficient for minds being nonphysical.”
IV) Symbolizing with “Only if”
41
Rule: only if = → .
Rule: In general, whatever follows “only if” becomes the
consequent
Practice
42
1. “Democracy will work only if the citizens are well-informed.”
D→W
43
2. “Only if the citizens are well-informed, can democracy work.”
D→W
44
3. “You can join the club only if you pay the dues.”
J→P
45
4. “Only if you play the dues, can you join the club.”
J→P
46
5. “P only if Z.”
P→Z
47
6. “P, if Z.”
Z→P
48
7. “If L, M.”
L→M
49
8. “Only if L, M.”
of the conditional.
Chapter 1 - Conditionals 51
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
M→L
50
1. “If Jill enjoyed the movie then it must be a foreign film.”
“Jill enjoyed the movie only if it’s a foreign film.”
“It must be a foreign film, if Jill enjoyed it.”
51
2. “If the cookie is good for you then it contains raisins.”
“The cookie contains raisins, if it’s good for you.”
“The cookie is good for you only if it contains raisins.”
52
3. “If Hank won then the election was rigged.”
“Hank won only if the election was rigged.”
“The election was rigged, if Hank won.”
V) Learning how to Symbolize: “Meaning” vs. “Rules”
Practice
53
1. “If behaviorism is right then mental states are dispositions to behave.”
B→D
54
2. “Behaviorism is right only if mental states are dispositions to behave.”
B→D
55
3. “For behaviorism to be right, it’s sufficient that mental states be dispositions to behave.”
D→B
56
4. “Should behaviorism be right, mental states are dispositions to behave.”
B→D
57
5. “Behaviorism is right, provided that mental states are dispositions to behave.”
D→B
58
6. “Mental states being dispositions to behave is a necessary condition for behaviorism being
right.”
B→D
59
7. “Behaviorism is right, if mental states are dispositions to behave.”
D→B
60
8. “Only if mental states are dispositions to behave, is behaviorism right.”
B→D
61
9. “Behaviorism is right, should mental states be dispositions to behave.”
D→B
Chapter 1 - Conditionals 52
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
62
1 0. “For mental states to be dispositions to behave, it’s necessary that behaviorism is right.”
D→B
Recognizing Equivalent Conditionals
Practice
Are the following sentences equivalent to
“If our actions have moral value then we have free will.”?
63
1. “Our actions have moral value only if we have free will.”Equivalent
64
2. “Our actions have moral value, if we have free will.” Not equivalent
65
3. “Our actions have moral value, provided that we have free will.” Not equivalent
66
4. “Our actions having moral value is a sufficient condition for our having free will.”
Equivalent
67
5. “Our having free will is a necessary condition for our actions having moral value.”
Equivalent
Are the following sentences equivalent to
“Having a shape is a necessary condition for having a color.”?
68
1. “Something has a shape only if it has a color.” Not equivalent
69
2. “Something has a color only if it has a shape.” Equivalent
70
3. “Something has a shape if it has a color.” Equivalent
71
4. “Having a color is a sufficient condition for having a shape.” Equivalent
72
5. “Something has a color, provided that it has a shape.” Not equivalent
Assessing Conditionals
I) What Makes a Conditional False
73
Rule: A conditional is false when and only when the antecedent is true and the consequent is
false.
II) Assessing Conditionals
Chapter 1 - Conditionals 53
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Practice
For each of the claims below identify the objection that, if true, would show that claim to be false.
74
1. “If we can’t know anything for certain then knowledge isn’t worth seeking.”
a. Although we can’t know anything for certain, knowledge is worth seeking.
b. Although we can know some things for certain, knowledge isn’t worth seeking.
75
2. “If a belief is useful then we’re justified in holding it.”
a. We’re justified in holding some beliefs even though they’re not useful.
b. We’re not justified in holding some beliefs even though they are useful.
76
3. “If euthanasia maximizes happiness then it’s ethical.”
a. Some things that maximize happiness aren’t ethical.
b. Some things that are ethical don’t maximize happiness.
77
4. “Lying is unethical only if it violates our culture’s moral code.”77 U → V
a. Lying is unethical even if it doesn’t violate our culture’s moral code.
b. Lying violates our culture’s moral code even if it isn’t unethical.
78
5. “Provided that some beliefs are justified without argument, religious faith is philosophically
acceptable.”
a. Even though no beliefs are justified without argument, religious faith is philosophically
acceptable.
b. Even though some beliefs are justified without argument, religious faith isn’t
philosophically acceptable.
79
6. “God doesn’t know our future actions, if we have free will.” Free will → God doesn’t know
a. “God knows our future actions but we have free will anyway.”
b. “God doesn’t know our future actions but we don’t have free will anyway.”
80
7. “We have good reason to believe in God, provided that we have good reason to believe in
miracles.” M → G
a. Even though we don’t have good reason to believe in God, we do have good reason to
believe in miracles.
b. Even though we have good reason to believe in God, we don’t have good reason to believe
in miracles.
81
8. “Cross-cultural agreement about ethics is a necessary condition for the objectivity of
morality.” O → A
a. Although morality is objective, there can still fail to be cross-cultural agreement about
ethics.
b. Even if there is cross-cultural agreement about ethics, morality can still fail to be objective.
Chapter 1 - Conditionals 54
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
82
9. “For an action to be moral it’s sufficient that it satisfy Kant’s Categorical Imperative.”
K→M
a. An action can be moral without satisfying Kant’s Categorical Imperative.
b. An action can satisfy Kant’s Categorical Imperative without being moral.
Symbolizing Conditionals with Multiple Connectors
Practice
83
1. “If it’s the case that computers deserve moral consideration if they can think, then the ability
to think is the source of our moral value.”
C = Computers can think
S = The ability to think is the source of our moral value.
M = Computers deserve moral consideration.
(C → M) → S
84
2. “If computers can think then provided that the ability to think is the source of our moral
value, computers deserve moral consideration.”
C = Computers can think
S = The ability to think is the source of our moral value.
M = Computers deserve moral consideration.
C → (S → M)
85
3. “Provided that being able to carry on a conversation is sufficient for being able to think, the
Turing Test is an accurate indicator of artificial intelligence.”
C = x can carry on an intelligent conversation.
T = x can think.
A = The Turing Test is an accurate indicator of artificial intelligence.
(C → T) → A
86
4. “Being able to carry on an intelligent conversation is necessary for having a mind only if
many people don’t have minds.”
C = x can carry on an intelligent conversation.
M = x has a mind.
D = Many people don’t have minds.
(M → C) → D
Chapter 1 - Conditionals 55
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
87
5. “If our being able to think is sufficient for our having moral value, then if computers can
think, they deserve moral consideration.”
T = We can think.
V = We have moral value.
C = Computers can think.
M = Computers deserve moral consideration.
(T → V) → (C → M)
88
6. “Being able to think is necessary for having moral value only if being able to think is
sufficient for having a soul.”
T = x can think.
V = x has moral value.
S = x has a soul.
(V → T) → (T → S)
89
7. “Should being able to carry on an intelligent conversation be sufficient for having a mind,
then an answering machine would have a mind if it could answer back.”
C = x can carry on an intelligent conversation.
M = x has a mind.
A = An answering machine has a mind.
B = An answering machine can answer back.
(C → M) → (B → A)
90
8. “If a computer could carry on an intelligent conversation then provided that being able to
carry on an intelligent conversation is sufficient for having a mind, the computer would have
a mind.”
C = A computer can carry on an intelligent conversation.
M = x has a mind.
I = x can carry on an intelligent conversation
H = A computer has a mind.
C → ((I → M) → H )
91
9. “Neither side can avoid circularity. This is disturbing if a non-question-begging way of
mutually resolving basic disagreements over the overall force of a body of evidence is a
necessary condition of the possibility of being rational in these matters.” (William J.
Wainwright, Reason and the Heart, Cornel University Press, 1995, p. 123)
Chapter 1 - Conditionals 56
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Symbolize the second sentences in this quotation, using the following sentence letters:
D = It’s disturbing that neither side can avoid circularity.
N = There is a non-question-begging way of mutually resolving basic disagreements over the
overall force of a body of evidence.
R = One can be rational in matters involving basic disagreements over the overall force of a
body of evidence
(R → N) → D
INFERENCE RULE: → O
ESTABLISHING VALIDITY: FORMAL PROOFS
Practice
92
1. A →B, A, B →C |- C
1. A →B
2. A
3. B →C
4. B
5. C
93
2. R → (T →S), R, T |- S
1. R → (T →S)
2. R
3. T
4. T →S
5. S
94
P
P
P- want S
1, 2 →O
3, 4 →O
3. M →N, (M →N) →Q, Q →T |- T
1. M →N
2. (M →N) →Q
3. Q →T
4. Q
5. T
95
P
P
P - want C
1, 2 →O
3, 4 →O
P
P
P - want T
1, 2 →O
3, 4 →O
4. P →Q, (P →Q) → (A →F), A |- F
1. P →Q
P
Chapter 1 - Conditionals 57
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
2. (P →Q) → (A →F)
3. A
4. A →F
5. F
96
5. C, D→(F→H), (A→B)→(C→D), A→B |- F→H
1. C
2. D→(F→H)
3. (A→B)→(C→D)
4. A→B
5. C → D
6. D
7. F→H
97
P
P - want F
1, 2 →O
3, 4 →O
P
P
P
P - want F→H
3,4 →O
1,5 →O
2,6 →O
6. Do this proof mentally: R→S, R, S → (A → B) |- A → B
R→S and R gives us S. And S and S→(A→B) gives us A→B.
98
7.Do this proof mentally: R→S, R, S → (A → B), A |- B
R→S and R gives us S. S and S→(A→B) gives us A→B. And A→B and A gives us B.
99
8. Do this proof mentally: R→S, R, S → (A → B), A, B→C |- C
R→S and R gives us S. S and S→(A→B) gives us A→B. A→B and A gives us B. And B
and B→C gives us C.
100
9. B→(L→R), [B→(L→R)]→B |- L→R
1. B→(L→R)
2. [B→(L→R)]→B
3. B
4. L →R
P
P – want L→R
1,2 →O
2,3 →O
Note: Line 2 was used twice. That’s okay. We never use lines up.
101
10. A→B, A, (A→B)→(B→D) |- D
1. A→B
2. A
3. (A→B)→(B→D)
4. B
P
P
P - want D
1,2 →O
Chapter 1 - Conditionals 58
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
5. B→D
6. D
102
1,3, →O
4,5 →O
11. (W → B) → [C → (A →W)], C, A, W → B |-B
1. (W →B) → [C → (A →W)]
2. C
3. A
4. W →B
5. C → (A →W)
6. A →W
7. W
8. B
103
12. P→Q, {B→[(P→Q)→R]}→(A→B), A, B→[(P→Q)→ R] |- R
1. P→Q
2. {B→[(P→Q)→R]}→(A→B)
3. A
4. B→[(P→Q)→ R]
5. A→B
6. B
7. (P→Q) → R
8. R
104
P
P
P
P – want B
1, 4 →O
2, 5 →O
3, 6 →O
4, 7 →O
P
P
P
P - want R
2, 4 →O
3, 5 →O
4, 6 →O
1, 7 →O
13. Do this proof mentally: “If God exists then miracles are possible. And God does exist.
Therefore miracles are possible.”
G → M, G |- M
From G→M and G, M follows directly.
105
14. Do this proof mentally: “Intelligent origin is necessary for an ordered universe. If the
universe has intelligent origin then God exists. The universe is ordered. Therefore, God
exists.”
O → I, I → G, O |- G
From O→I and O, we get I. Then from I and I →G, we get G.
Written out, the proof looks like this:
1. O → I
2. I → G
A
A
Chapter 1 - Conditionals 59
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
3. O
4. I
5. G
106
A - want G
1, 3 →O
2, 4 →O
15. Do this proof mentally: “People have an innate sense of right and wrong. People can have
an innate sense of right and wrong only if God exists. And provided that God exists, evildoers will be punished. Therefore, evil-doers will be punished.”
I, I→G, G→P |- P
From I and I→G we get G. Then from G and G→P, we get P
Written out, the proof looks like this:
1. I
2. I→G
3. G→P
4. G
5. P
107
A
A
A – want P
1,2 →O
3,4 →O
16. Do this proof mentally: “If utilitarianism is right then lying can be morally permissible if
it sometimes serves to promote the greatest happiness for the greatest number of people. In
fact, utilitarianism is right. And lying does sometimes serve to promote the greatest
happiness for the greatest number of people. Therefore, lying can be morally permissible.”
U → (G → L), U, G |- L
From U→(G→L) and U we get G→L. Then from G and G→L, we get L.
Written out, the proof looks like this:
1. U → (G → L)
2. U
3. G
4. G → L
5. L
108
P
P
P - want L
1,2 →O
3,4 →O
17. “If Kant is right then ethical acts must pass the categorical imperative. Kant is right. Lying
is ethically wrong if lying is unable to become the universal law, provided that ethical acts
must pass the categorical imperative. If lying would make communication and society
impossible then it’s unable to become a universal law. Lying would make communication
and society impossible. Therefore lying is ethically wrong.”
K → C, K, C → (U → W), I → U, I |- W
Chapter 1 - Conditionals 60
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
1. K → C
2. K
3. C → (U → W)
4. I → U
5. I
5. C
6. U → W
7. U
8. W
P
P
P
P
P – want W
1,2 →O
3,5 →O
4,5 →O
6,8 →O
ESTABLISHING INVALIDITY
109
1. (P → Q) → (R → S), R, P→Q, |- S
1. (P → Q) → (R → S)
2. R
3. P→Q
4. R→S
5. S
P
P
P - want S
1,3 →O
2,3 →O
Valid
110
2. (P → Q) → (R → S), S, P→Q, |- R
1. (P → Q) → (R → S)
2. S
3. P→Q
4. R→S
5. R
P
P
P - want R
1,3 →O
2,3 FAC!
Invalid
111
3. R, B→A, A→C, C→(R→S), B |- S
1. R
2. B→A
3. A→C
4. C→(R→S)
5. B
6. A
7. C
8. R → S
P
P
P
P
P – want S
2,5 →O
3,6 →O
4,7 →O
Chapter 1 - Conditionals 61
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
9. S
1,8 →O
Valid
112
4. L → (P → Q), S→R, P, L, Q→R |- S
1. L → (P → Q)
2. S→R
4. P
5. L
6. Q → R
7. P → Q
8. Q
9. R
10. S
P
P
P
P
P - want S
1,5 →O
4,7 →O
6,8 →O
2,9 FAC!
Invalid
113
5. Assess mentally: L→M, (L→M) → (A→B), B | A
L→M and (L→M)→(A→B) gives us A→B. But to go from A→B and B to A is to commit
the Fallacy of Assuming the Consequent.
Invalid.
114
6. Assess mentally: L→M, (L→M) → (A→B), A | B
L→M and (L→M)→(A→B) gives us A→B. And A→B and A gives us B.
Valid.
115
7. Write the symbolization and assess mentally: “If God exists then the world would be welldesigned. And the world is well designed. Therefore God exists.”
G→W, W |- G
From to go from G→W and W to G is to commit the Fallacy of Assuming the Consequent.
Invalid.
Written out, the “proof” is:
1. G→W
2. W
P
P – want G
Chapter 1 - Conditionals 62
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
3. G
116
1, 2 FAC!
8. Write the symbolization and assess mentally: “If the world is well-designed then God exists.
And the world is well designed. Therefore God exists.”
W→G, W |- G
From W→G and W, we can conclude G.
Valid.
Written out, the proof is:
1. W→G
2. W
3. G
117
P
P – want G
1, 2 →O
9. Write the symbolization and assess mentally: “People have an innate sense of right and
wrong. People can have an innate sense of right and wrong only if God exists. And provided
that God exists, evil-doers will be punished. Therefore, evil-doers will be punished.”
I, I→G, G→P |- P
From I and I→G we get G. From G and G→P we get P.
Valid
Written out, the proof is:
1. I
2. I→G
3. G→P
4. G
5. P
118
P
P
P – want P
1, 2 →O
3, 4 →O
10. Write the symbolization and assess mentally: “People have an innate sense of right and
wrong. People can have an innate sense of right and wrong if God exists. And provided that
God exists, evil-doers will be punished. Therefore, evil-doers will be punished.”
I, G→I, G→P |- P
To from I and G→I to G is to commit the Fallacy of Assuming the Consequent. From G and
G→P we can get P, but that doesn’t matter because we can’t validly get G in the first place.
Chapter 1 - Conditionals 63
Dona Warren, Department of Philosophy, The University of Wisconsin – Stevens Point
Invalid
Written out, the “proof” is:
1. I
2. G→I
3. G→P
4. G
5. P
119
P
P
P – want P
1, 2 FAC!
3, 4 →O
11. “If lying can be morally permissible provided that it sometimes serves to promote the
greatest happiness for the greatest number of people then utilitarianism is right. In fact,
utilitarianism is right. And lying does sometimes serve to promote the greatest happiness for
the greatest number of people. Therefore, lying can be morally permissible.”
(G → P) → U, U, G |- P
1. (G → P) → U
2. U
3. G
4. G → P
5. P
P
P
P – want P
1, 2 FAC!
3, 4 →O
Invalid
120
12. “If utilitarianism is right then lying can be morally permissible provided that it sometimes
serves to promote the greatest happiness for the greatest number of people. In fact,
utilitarianism is right. And lying does sometimes serve to promote the greatest happiness for
the greatest number of people. Therefore, lying can be morally permissible.”
U → (G → P), U, G |- P
1. U → (G → P)
2. U
3. G
4. G → P
5. P
Valid
P
P
P – want P
1, 2 →O
3, 4 →O