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4.5: Commons Forms of Arguments

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  • 22 Commons Forms of Arguments38

    Disjunctive Syllogism (DS)

    The basic form disjunctive syllogism gets its name from the feature that one of the two premises is a disjunction. The disjunction tells us that at least one of its disjuncts must be true in order for the disjunction to be true. Now since the other premise asserts that one of the disjuncts is false (that is, its negation is true). It follows that the other disjunct must be true.

    p v q or p v q

    ~p ~q

    q p

    Notice that in Propositional Logic, the order of the premises does not matter. So the following two are treated as the same form.

    p v q ~p

    ~p = p v q

    q q

    We can prove that the form disjunctive syllogism is valid using the truth table.

    p q p v q ~ p q

    T T T T T F T T

    T F T T F F T F

    F T F T T T F T OK!

    F F F F F T F F

    After we know what the form looks like, the next step is to identify it from a written argument. Here is an example of disjunctive syllogism:

    Either interest rates go up or inflation gets worse. Since interest rates have not gone up, we can be sure that inflation is getting worse.

    After symbolizing the argument as

    U v W

    ~U

    W

    we can tell that it is an instance of disjunctive syllogism. In this way we can find out that it is valid without constructing its truth table.

    Recognizing Common Forms

    In learning the basic argument forms, we use “p”, “q”, “r” and “s” as variables. They serve as place holders in argument forms. If we replace each variable in a basic form with a capital letter, we would of course end up with an instance of the form. But we can also replace a variable with a compound sentence. The resulting argument would also be an instance of the form. Such substitutions give us more flexibility in constructing instances of the basic forms. It also helps us identify them. Take a look at the next argument.

    The current economic growth cannot be sustained unless inflation is under control. Inflation is not under control. Therefore, the current economic growth cannot be sustained.

    The argument can be symbolized as

    ~S v U

    ~U

    ~S

    We can then see that it is an instance of disjunctive syllogism by the following substitutions:

    p = ~S

    q = U

    Is the following a valid argument?

    p v q p Therefore, ~q.

    It is not! Think about this: You can have mustard or ketchup on your hot dog. You had mustard, therefore you didn't have ketchup. Why can't you have both? You can! Because we're using the INCLUSIVE or, remember?

    I. Decide whether each of the arguments is one of the common forms. If it is, identify the name of the form and decide its validity. If it is not a common form, label it as “No Form” and use a truth table to determine its validity.

    7. B v C

    ~C

    B

    9. G v ~N

    N

    G

    19. C v ~A

    ~A v N

    C v N

    More Practice exercises

    Determine whether each pair of sentences is logically equivalent.

    1. A, ~A

    2. A, A v A

    6. ~(A&B), ~A v ~B

    9. [(A v B) v C], [A v (B v C)]

    10. [(A v B)&C], [A v (B &C)]

    Determine whether each argument is valid or invalid.

    7. A v B, B v C, ~A, B &C

    8. A v B, B v C, ~B, A&C

    For the following sentences, let R mean `You will cut the red wire' and B mean `The bomb will explode.'

    21. If you cut the red wire, then the bomb will explode.

    22. The bomb will explode only if you cut the red wire.

    Sentence 21 can be translated partially as `If R, then B.' We will use the symbol `' to represent logical entailment. The sentence becomes R B. The connective is called a conditional. The sentence on the left-hand side of the conditional (R in this example) is called the antecedent. The sentence on the right-hand side (B) is called the consequent.

    Sentence 22 is also a conditional. Since the word `if' appears in the second half of the sentence, it might be tempting to symbolize this in the same way as sentence 21. That would be a mistake. The conditional R B says that if R were true, then B would also be true. It does not say that your cutting the red wire is the only way that the bomb could explode. Someone else might cut the wire, or the bomb might be on a timer. The sentence R B does not say anything about what to expect if R is false.

    Sentence 22 is different. It says that the only conditions under which the bomb will explode involve your having cut the red wire; i.e., if the bomb explodes, then you must have cut the wire. As such, sentence 22 should be symbolized as B R.

    It is important to remember that the connective `' says only that, if the antecedent is true, then the consequent is true. It says nothing about the causal connection between the two events. Translating sentence 22 as B R does not mean that the bomb exploding would somehow have caused your cutting the wire. Both sentence 21 and 22 suggest that, if you cut the red wire, your cutting the red wire would be the cause of the bomb exploding. They differ on the logical connection. If sentence 22 were true, then an explosion would tell us| those of us safely away from the bomb| that you had cut the red wire. Without an explosion, sentence 22 tells us nothing.

    The paraphrased sentence `A only if B' is logically equivalent to `If A, then B.' `If A then B' means that if A is true then so is B. So we know that if the antecedent A is true but the consequent B is false, then the conditional `If A then B' is false. What is the truth value of `If A then B' under other circumstances? Suppose, for instance, that the antecedent A happened to be false. `If A then B' would then not tell us anything about the actual truth value of the consequent B, and it is unclear what the truth value of `If A then B' would be.

    In English, the truth of conditionals often depends on what would be the case if the antecedent were true| even if, as a matter of fact, the antecedent is false. This poses a problem for translating conditionals into SL. Considered as sentences of SL, R and B in the above examples have nothing intrinsic to do with each other. In order to consider what the world would be like if R were true, we would need to analyze what R says about the world. Since R is an atomic symbol of SL, however, there is no further structure to be analyzed. When we replace a sentence with a sentence letter, we consider it merely as some atomic sentence that might be true or false.

    In order to translate conditionals into SL, we will not try to capture all the subtleties of the English language `If…then…' Instead, the symbol `' will be a material conditional. This means that when A is false, the conditional AB is automatically true, regardless of the truth value of B. If both A and B are true, then the conditional AB is true. In short, AB is false if and only if A is true and B is false. We can summarize this with a characteristic truth table for the conditional.

    A B A B

    T T T

    T F F

    F T T

    F F T

    The conditional is asymmetrical. You cannot swap the antecedent and consequent without changing the meaning of the sentence, because AB and BA are not logically equivalent.

    Interestingly, the following is logically true: If God exists, then there is evil in the world. This is because the consequent, “There is evil in the world” is True, and whenever the consequent is True, the whole conditional is True, regardless of the truth of the antecedent!

    Similarly, if the antecedent is False, then the conditional is always True: If gravity makes things go up, then I am a billionaire. Since the antecedent is False, this sentence is always true! Interesting, right?

    Not all sentences of the form `If…then…' are conditionals. Consider this sentence:

    23. If anyone wants to see me, then I will be on the porch.

    If I say this, it means that I will be on the porch, regardless of whether anyone wants to see me or not| but if someone did want to see me, then they should look for me there. If we let P mean `I will be on the porch,' then sentence 23 can be translated simply as P.

    Biconditional

    Consider these sentences:

    24. The figure on the board is a triangle only if it has exactly three sides.

    25. The figure on the board is a triangle if it has exactly three sides.

    26. The figure on the board is a triangle if and only if it has exactly three sides.

    Let T mean `The figure is a triangle' and S mean `The figure has three sides.'

    Sentence 24, for reasons discussed above, can be translated as T S.

    Sentence 25 is importantly different. It can be paraphrased as, `If the figure has three sides, then it is a triangle.' So it can be translated as S T.

    Sentence 26 says that T is true if and only if S is true; we can infer S from T, and we can infer T from S. This is called a biconditional, because it entails the two conditionals S T and T S. We will use `' to represent the biconditional; sentence 26 can be translated as S T. We could abide without a new symbol for the biconditional. Since sentence 26 means `T S and S T,' we could translate it as (T S)&(S T). We would need parentheses to indicate that (T S) and (S T) are separate conjuncts; the expression T S &S T would be ambiguous.

    Because we could always write (A B)&(B A) instead of A ↔ B, we do not strictly speaking need to introduce a new symbol for the biconditional. Nevertheless, logical

    languages usually have such a symbol. SL will have one, which makes it easier to translate phrases like `if and only if.'

    A↔B is true if and only if A and B have the same truth value. This is the characteristic truth table for the biconditional:

    A B A↔B

    T T T

    T F F

    F T F

    F F T

    Other symbolization

    We have now introduced all of the connectives of SL. We can use them together to translate many kinds of sentences. Consider these examples of sentences that use the English-language connective `unless'~

    27. Unless you wear a jacket, you will catch cold.

    28. You will catch cold unless you wear a jacket.

    Let J mean `You will wear a jacket' and let D mean `You will catch a cold.' We can paraphrase sentence 27 as `Unless J, D.' This means that if you do not wear a jacket, then you will catch cold; with this in mind, we might translate it as ~J D. It also means that if you do not catch a cold, then you must have worn a jacket; with this in mind, we might translate it as ~D J. Which of these is the correct translation of sentence 27? Both translations are correct, because the two translations are logically equivalent in SL.

    Sentence 28, in English, is logically equivalent to sentence 27. It can be translated as either ~J D or ~D J. When symbolizing sentences like sentence 27 and sentence 28, it is easy to get turned around. Since the conditional is not symmetric, it would be wrong to translate either sentence as J ~D. Fortunately, there are other logically equivalent expressions. Both sentences mean that you will wear a jacket or|if you do not wear a jacket| then you will catch a cold. So we can translate them as J v D. (You might worry that the `or' here should be an exclusive or. However, the sentences do not exclude the possibility that you might both wear a jacket and catch a cold; jackets do not protect you from all the possible ways that you might catch a cold.)

    If a sentence can be paraphrased as `Unless A, B,' then it can be symbolized as A v B.

    The sentence `Apples are red, or berries are blue' is a sentence of English, and the sentence `(AvB)' is a sentence of SL. Although we can identify sentences of English when we encounter them, we do not have a formal definition of `sentence of English'. In SL, it is possible to formally define what counts as a sentence. This is one respect in which a formal language like SL is more precise than a natural language like English.

    If we add the conditional and the biconditional to our truth table of connectives, you can see how they function with relation to the rest of the connectives.

    A B A &B AvB AB AB

    T T T T T T

    T F F T F F

    F T F T T F

    F F F F T T

    Common Argument Forms40

    NOTE: ∙ is & in the text below. There are different notations and in some of them they use ∙ to represent conjunctions. Additionally, ≡ is ↔ for the same reasons.

    In the previous section we learned how to use truth tables to determine whether deductive arguments are valid. As arguments get longer, their truth tables would have more rows. Using truth tables to determine their validity can become quite time-consuming. For example, the truth table of the following argument has 16 rows, and can take quite a bit of time to construct.

    If interest rates are raised, the stock market will be hurt. If the stock market is hurt, the economy will slow down. But if interest rates are not raised, inflation will get worse. If inflation gets worse, the economy will slow down. So the economy will slow down.

    So using truth tables to determine validity can be tedious, and there is an incentive to find a more efficient way.

    Arguments such as this are built by combining small and basic valid argument forms. This means that if we can recognize the small forms and see how they are put together to form longer arguments, then we can determine validity without constructing truth tables.

    Basic Valid Forms

    There are six basic forms that are commonly used:

    1. Disjunctive Syllogism (DS) (covered in the last chapter)

    2. Hypothetical Syllogism (HS)

    3. Modus Ponens (MP)

    4. Modus Tollens (MT)

    5. Constructive Dilemma (CD)

    6. Destructive Dilemma (DD)

    We are going to study them and learn how to recognize them.

    Hypothetical Syllogism (HS)

    A hypothetical syllogism has a distinct feature that helps us recognize it. The argument consists of three conditionals. The first conditional says that p is a sufficient condition for q. The second one says that q in turn is a sufficient condition for r. It would then follow that p is a sufficient condition for r.

    p q

    q r

    p r

    Example:

    If more prisons are built, public education will get worse due to lack of funding. If public education gets worse due to lack of funding, there will be more criminals. As a result, if more prisons are built, there will be more criminals.

    has the form

    B W

    W C

    B C

    and thus is an instance of hypothetical syllogism.

    Modus Ponens (MP)

    “Modus Ponens” is the Latin term for “Affirmative Mode.” We can also call it “Affirming the

    Antecedent” because one of its premises affirms that the antecedent of the conditional is true. It is a valid form based on the concept of sufficient condition. If p is a sufficient condition of q, and p is true, then q must be true.

    p q

    p

    q

    Modus Ponens is one of the most commonly used valid forms. Here is an example:

    If Republicans favor free market economy, then they should oppose farm subsidies. Republicans favor free market economy. So they should oppose farm subsidies.

    The argument is symbolized as

    F O

    F

    O

    We can see that its form is Modus Ponens and thus is valid.

    Modus Tollens (MT)

    “Modus Tollens” means “Denying Mode” in Latin. Its English name is “Denying the Consequent”

    because one of its premises denies that the consequent of the conditional is true. The validity of Modus Tollens can be easily explained using the concept of necessary condition. If q is a necessary condition of p, and q is false, then p must be false.

    p q

    q

    ∴∼p

    The next argument is an example of Modus Tollens:

    We should be against big corporations only if we are against their stock holders. We are not against the stock holders. So we should not be against big corporations.

    B S

    S

    ∴∼B

    Constructive Dilemma (CD)

    Constructive dilemma, like Modus Ponens, is built upon the concept of sufficient condition. The

    two conditionals p q and r s can be joined together as a conjunction or stated separately as two premises. They assert that p is a sufficient condition for q and r is a sufficient condition for s.

    Consequently, if at least one of the sufficient conditions is true, then at least one of the consequents must also be true.

    (p q) (r s) or p q

    p r r s

    q s p r

    q s

    After symbolizing the argument

    If consumers increase spending, then inflation will get worse. If consumers cut back on spending, then there will be a recession. Consumers either increase or cut back on spending. It follows that the inflation will get worse or there will be a recession.

    as

    (I W) (C R)

    I C

    W R

    We can see that it is a constructive dilemma, and thus is valid.

    p as one of the premises.

    (p q) (p s)

    p ∨∼p

    q s

    p is a tautology, the argument is sound if the premise (p q) (p s) is true. Here is an example of this commonly seen version of constructive dilemma:

    With protectionism, prices for consumer goods would become higher. Without protectionism, jobs would be lost. Since we either adopt protectionism or reject it, prices for consumer goods would become higher or jobs would be lost.

    (P H) (P J)

    P ∨∼P

    H J

    Destructive Dilemma (DD)

    Destructive dilemma is another common form based on the concept of necessary condition. The two conditionals assert that q is a necessary condition for p and s is a necessary condition for r. So if q is false or s is false, then it must be the case that p is false or r is false.

    (p q) (r s) or p q

    s r s

    p ∨∼r q ∨∼s

    r

    Here is an example of destructive dilemma:

    Global warming can be slowed down only if we switch to cleaner energy sources. But the current level of industrial production can be sustained only if we continue to use fossil fuels. We won’t switch to cleaner energy sources or we won’t continue to use fossil fuels. As a result, global warming cannot be slowed down or the current level of industrial production cannot be sustained.

    (G S) (P F)

    S ∨∼F

    ∴∼G ∨∼P

    It is also common to see a destructive dilemma with a tautologous disjunction as one of its premises:

    q)

    q ∨∼q

    r

    q). The following argument is an instance of such a form:

    GM can be competitive only if it increases outsourcing. UAW workers can have job security only if GM does not increase outsourcing. GM either increases or does not increase outsourcing. Therefore, either GM cannot be more competitive or UAW workers cannot have job security.

    I) I

    ∴∼C ∨∼J

    The Problem of Evil — a Destructive Dilemma

    There is a well-known philosophical problem called the problem of evil in western philosophy and religions. It argues that God is either not all-powerful or not all-good. The argument can be constructed as a destructive dilemma:

    1. If God is all-powerful (P), He would be able to eliminate evil (A).

    2. If God is all-good (G), He would want to eliminate evil (W).

    3. Evil exists. (This means that God is not able to eliminate evil, or God does not want to eliminate evil.)

    4. Therefore, God is either not all-powerful or not all-good.

    After symbolization,

    (P A) (G W)

    A ∨∼W

    G

    we can see that the argument indeed is an instance of destructive dilemma.

    Recognizing Common Forms

    In learning the basic argument forms, we use “p”, “q”, “r” and “s” as variables. They serve as place holders in argument forms. If we replace each variable in a basic form with a capital letter, we would of course end up with an instance of the form. But we can also replace a variable with a compound sentence. The resulting argument would also be an instance of the form. Such substitutions give us more flexibility in constructing instances of the basic forms. It also helps us identify them.

    In the next example, we should recognize A ∙ K as the antecedent p and D as the consequent q. As a result, it is a Modus Ponens.

    (A ∙ K) ~D

    (A ∙ K)

    ∴∼D

    To properly identify the next form as an instance of disjunctive syllogism, we need to apply the rule of double negation, which says that p is logically equivalent to ∼∼p. Afterwards, we substitute p for G M and q for D.

    (G M) v ~D

    D

    G M

    Combining Basic Forms

    We can combine basic argument forms to construct longer and more complex arguments. The argument

    If interest rates are raised, the stock market will be hurt. If the stock market is hurt, the economy will slow down. But if interest rates are not raised, inflation will get worse. If inflation gets worse, the economy will slow down. So the economy will slow down.

    is built by combining two hypothetical syllogisms with a constructive dilemma. It is easier to see the logical structure from the symbolization.

    R S

    S E

    R W

    W E

    (R ∨∼R)

    E

    R as the last premise. The tautology goes unstated in the original argument because it is trivial that it is true. Now from the first two premises, we can derive R E as a conclusion based on hypothetical syllogism.

    R S

    S E

    R E

    Using hypothetical syllogism one more time, we can draw the conclusion R E from the third and the fourth premises.

    R W

    W E

    ∴∼R E

    R, we arrive at the conclusion E according to constructive dilemma.

    R E

    R E

    R ∨∼R

    E

    Using Basic Forms to Determine Validity

    As we saw above, the argument is constructed by combining three basic forms. Since each of them is valid, we can determine that it is valid without constructing a long truth table with 16 rows. Since long arguments are often constructed out of basic forms, we can determine their validity by identifying the basic forms. If all of the basic forms are valid, then the long argument is valid. But if one of the basic forms is invalid, then the long argument is invalid. This is not a rigorous formal process like the truth table method. But it does enable us to determine validity without constructing truth tables. When we try to break up a long argument into basic forms, we need to, when possible, identify valid forms first. To decide the validity of the argument,

    A new game console is in high demand only if there are a lot of exciting games available for it. However, many game developing companies won’t design new games for a new console unless it is in high demand. Since there are not a lot of exciting games available for a new console, it follows that many game developing companies won’t design new games for it.

    we first symbolize it as

    H E

    D H

    E

    ∴∼D

    Then we break it apart into the following two forms:

    H E and D H

    E H

    ∴∼H ∴∼D

    Modus Tollens Disjunctive Syllogism

    In the next example,

    If we continue the acceleration of production and consumption, we will deplete natural resources within one hundred years. If natural resources are depleted within one hundred years, life on earth will not be sustainable. We continue to accelerate production and consumption. As a result, life on earth will not be sustainable.

    the argument is symbolized as

    C D

    S

    C

    ∴∼S

    We can see that it is constructed from the following two valid forms, and thus is valid.

    S

    D ⊃∼S C

    S ∴∼S

    Hypothetical Syllogism Modus Ponens

    The next argument

    If one is a fiscal conservative, then one would be against big government spending. But if one is against big government spending, then one would support budget cut on military spending. President Bush supports budget cut on military spending. Therefore, he is a fiscal conservative.

    F A

    A S

    S

    F

    is invalid because after we separate it into two forms:

    F A and F S

    A S S

    F S F

    Hypothetical Syllogism Affirming the Consequent

    we find that the second form is invalid.

    Exercises

    I. Decide whether each of the arguments is one of the common forms. If it is, identify the name of the form and decide its validity. If it is not a common form, label it as “No Form”.

    1. A C

    A

    C

    2. H K

    H

    ∴∼K

    3. E G

    G

    ∴∼E

    4. S M

    M

    S

    D

    R

    ∴∼D

    P

    P

    F

    L

    L

    A

    10. K M

    M E

    K E

    11. D E

    P ⊃∼D

    P E

    V) (A U)

    V vU

    ∴∼S A

    F

    F

    C

    14. (R L) (R E)

    R ∨∼R

    L E

    (H N)

    H ≡ N

    ∴∼G

    16. (A ∙ B) K

    B)

    ∴∼K

    D)

    J ∨∼P

    ∴∼M D

    18. F (D E)

    (D E)

    ∴∼F

    S

    S T

    ∴∼(R C) T

    II. Use the common argument forms to derive the conclusion from the premises.

    1. A E

    E

    2. D H

    D

    3. C R

    R

    4. K B

    B R

    N

    G

    6. (K ∙ A) M

    M

    L

    L

    8. B R

    S

    B D

    D

    D

    10. A (C U)

    (C U)

    H) N

    P ⊃∼H

    R)

    A R

    E

    E ⊃∼P

    14. (J D) Q

    J D

    N)

    K

    16. B F

    L)

    F ∨∼(N ∨∼L)

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