The Rules of Inference: Forms of Valid Argument
AS DEBATE and correct reasoning require that the arguments we submit are valid, studying this topic is very essential as it could help in developing one’s sharpness in reasoning. This lecture teaches various argument forms that are necessarily valid.
The rules of inference (or transformation rules) stand for deductive argument forms which comprise statement variables and operators that are syntactically arranged in such a way that when the premise/s is/are true, the conclusion cannot be false. Rules as they are, the consistent replacement of their variables by statements automatically results in valid arguments. In other words, any argument having their forms is a valid argument, hence the importance of studying them.
The ten (10) rules of inference tackled in this lecture are specifically called rules of implication because they are basic argument forms whose premises imply their conclusions. You will understand the topic more as we study its various forms so we better start enumerating the different types.
Modus Ponens (MP)
Also named implication elimination, the rule of inference called modus ponens (“asserting mode”) consists of a conditional premise, a second premise that asserts the antecedent of the conditional premise, and a conclusion that asserts the consequent. It can be summarized as “P implies Q; P is asserted to be true, so therefore Q must be true.” Example:
If the car fails to start, then something is wrong with it.
The car fails to start.
Therefore, something is wrong with it.
Notice however that if you affirm the consequent instead of the antecedent in the second premise, and conclude by asserting the antecedent, the argument is invalid:
If the car fails to start, then something is wrong with it.
Something is wrong with the car.
Therefore, the car will fail to start.
We know that not all kinds of car damages prevent it from starting, hence the conclusion is not necessarily true. In fact, this is a case of a fallacy (invalid reasoning) called “affirming the consequent.” Any argument that has the form of affirming the consequent is an invalid argument.
Modus Tollens (MT)
Also known as modus tollendo tollens and“denying the consequent,” modus tollens (“denying mode”) consists of one conditional premise, a second premise that denies the consequent of the conditional premise, and a conclusion that denies the antecedent. Its form is closely associated with that of modus ponens:
This rule of inference has this form: “P implies Q; Q is false, so therefore P must also be false.”
Example:
If he is a voter, then he is of legal age.
He is not of legal age.
Therefore, he is not a voter.
As with modus ponens, there is an invalid argument form commonly mistaken for modus tollens. This is the fallacy of “denying the antecedent” which consists of a conditional premise, a second premise that denies the antecedent of the conditional, and a conclusion that denies the consequent. Taking this invalid form of argument, our previous example will appear this way:
If he is a voter, then he is of legal age.
He is not a voter.
Therefore, he is not of legal age.
Among other possibilities, a person can be of legal age even without being a voter—as in the case of those who failed to register in any polling precinct.
Hypothetical Syllogism (HS)
Also called “pure hypothetical syllogism,” “the chain argument,” “chain rule,” or “the principle of transitivity of implication,” this argument form consists of two premises and one conclusion, all of which are hypothetical (conditional) statements.
It takes the form: “P implies Q; Q implies R, so therefore P implies R.” Example:
If Jordan will play, then Pippen will play.
If Pippen will play, then Rodman will play.
Therefore, If Jordan will play, then Rodman will play.
Hypothetical syllogism is necessarily valid because its premises link together like a chain in such a way that the consequent of the first premise is identical to the antecedent of the second.
Disjunctive Syllogism (DS)
Historically known as modus tollendo ponens, disjunctive syllogism which is also named “disjunction elimination” and “or elimination” is that which has a disjunctive statement for one of its premises, a negation of either of the disjuncts for another premise, and an affirmation of the remaining disjunct for the conclusion.
Example:
Your crime is either a felony or misdemeanor.
(The judge says) It is not a felony.
Therefore, it is a misdemeanor.
Your crime is either a felony or misdemeanor.
(The judge says) It is not a misdemeanor.
Therefore, it is a felony.
Disjunctive syllogism is valid because if we are told that at least one of two statements is true; and also told that it is not the former that is true; we can infer that it has to be the latter that is true. Or, if the second premise tells that it is not the latter that is true, then we can conclude that it has to be the former that is true.
Do not mistake it for an argument form that embodies an invalid inference, the fallacy of affirming the alternative: “Either P or Q; It is P, so therefore it is not Q.”
The invalidity of this form is clear in the following example:
Mr. X is either rich or intelligent.
(Somebody confirms) He is rich.
Therefore, He is not intelligent.
The argument is flawed because it is possible for a person to be both rich and intelligent. Notice that the first premise is an inclusive disjunction, that is, it allows the possibility of both disjuncts being true.
Conjunction (Conj.)
Also called “conjunction introduction,” this rule of inference states that if the proposition P is true, and proposition Q is true, then the logical conjunction of the two propositions “P and Q” is true. That is, two propositions asserted separately may be conjoined in whatever order we choose (either “P & Q” or “Q & P”).
Example:
Kris is an actress.
Noynoy is a politician.
Therefore, Kris is an actress and Noynoy is a politician.
Or: Therefore, Noynoy is a politician and Kris is an actress.
Notice that this rule allows the derivation of a conjunction from the truth of both of its conjuncts.
Simplification (Simp.)
Equivalent to “conjunction elimination,”simplification permits us to infer the truth of any of the conjuncts of a conjunction. Remember that a true conjunction means that the two propositions composing it are both true. Thus, each of them is necessarily true separately.
This inference conveys that if the conjunction “P and Q” is true, then P is true, and Q is true.
Examples:
There’s a typhoon and it’s raining.
Therefore, there’s a typhoon.
There’s a typhoon and it’s raining.
Therefore it’s raining.
Addition (Add.)
Addition or “disjunction introduction” states that if P is true, then “P or Q” must be true. It entails that when a statement is asserted, it may be joined disjunctively with any proposition.
Examples:
A. Oprah is a talk-show host.
Therefore, Oprah is a talk-show host or an actress.
B. Oprah is a talk-show host.
Therefore, Oprah is a talk-show host or a UFC fighter.
Both arguments are valid. Argument B, no matter how funny it may appear, is also valid because the conclusion is a disjunction and so all it needs to be true is at least one true disjunct. Notice that the conclusion uses the connective “or” and not “and”—so it’s like offering alternatives in which one of the options is confirmed to be true (in our examples, Oprah’s being a talk-show host).
This notion thus reminds us that in addition, the new statement must always be joined disjunctively and never conjunctively to the given proposition. When one affirms that “Oprah is a talk-show host,” we are never justified to conclude, “Therefore, Oprah is a talk-show host and a UFC fighter”.
Absorption (Abs.)
This inference states that if it is true that “P implies Q,” then we are warranted to conclude that “P implies both P and Q.” Evidently, the rule is called absorption because the term Q is “absorbed” by the term P in the consequent.
Example:
If Bryant will play in the Olympics, then James will play too.
Therefore, if Bryant will play in the Olympics, then both Bryant and James will play.
Our example allows us to easily grasp the correctness of the law of absorption. If we are assured that James will play once Bryant decides to play, then we are certain, as expressed in the conclusion, that if Bryant plays, then two players are sure to play—Bryant and James.
Constructive Dilemma (CD)
This rule of inference consists of a conjunctive premise made up of two conditional statements, a disjunctive premise that affirms the antecedents in the conjunctive premise, and a disjunctive conclusion asserting the consequents of the conjunctive premise. Constructive dilemma takes this form: “Both P implies Q and R implies S; It is P or R; Therefore, it is Q or S.”
Example:
If I drink pineapple juice, I become healthy; but if I drink milk instead, I become stronger.
(When I get home) I’ll drink either pineapple juice or milk.
Therefore, either I’ll become healthy or stronger.
Constructive dilemmais the disjunctive version of modus ponens as it involves two modus ponens steps. In our example, the first premise states that if I drink pineapple juice, I become healthy; and if I drink milk, I become stronger. And since, by the second premise, I’ll drink either pineapple juice or milk, it follows by modus ponens that either I’ll become healthy or stronger.
Destructive Dilemma (DD)
Like the constructive dilemma, destructive dilemma includes a conjunctive premise made up of two conditional statements and a disjunctive premise. But instead of affirming the antecedents of the conditionals, its disjunctive premise denies the consequents of the conditionals. Using the principle of modus tollens, the conclusion then logically denies the antecedents of the conditionals: “Both P implies Q and R implies S; Either not Q or not S; Therefore, either not P or not R.”
Example:
If Pao is sick, then she will stay home; and if Senna is awake, then she will want to eat.
But, Pao will not stay home or Senna will not want to eat.
Therefore, Pao is not sick or Senna is not awake.
Application
The applicability and practicability of studying the rules of inference rest on the fact that any argument possessing a form that is identical to any of the rules of inference is certainly valid. In determining whether or not an argument exhibits a valid form, you may try to symbolize it and see afterwards whether the symbolized argument fits the pattern of any of the rules of inference.
The ten (10) basic rules of inference discussed are also the common argument forms we use everyday. Even the seemingly complicated inferences like the constructive dilemma and destructive dilemma are functional as we use them, knowingly or otherwise, especially as we engage in discussions, establishing our points, and debates.
For instance, someone who is pessimistic about the country’s economic condition may comment in this manner:
If we encourage competition, we will have no peace, and if we do not encourage competition, we will make no progress. Since we must either encourage competition or not encourage it, we will either have no peace or make no progress.
In countering this argument, someone who is rather optimistic about government’s decision-making may alternatively respond this way:
If we encourage competition, we will make progress, and if we do not encourage competition, we will have peace. Since we must either encourage competition or not encourage it, we will either make progress or have peace.
Notice that both arguments have the form of the constructive dilemma. Indeed handy, constructive dilemma, just like the other rules of inference, can be used in proving either side of a particular proposition. (© 2014 by Jensen DG. Mañebog/MyInfoBasket.com)
Also Check Out:
Reasoning and Debate: A Handbook and a Textbook by Jensen DG. Mañebog
Also Check Out: From Socrates to Mill: An Analysis of Prominent Ethical Theories, also by author Jensen DG. Mañebog
INTERACTIVE ONLINE ACTIVITY
Go online to www.OurHappySchool.com. Through its search engine (upper right section), look for the article “Who should be ‘the’ Philippine National Hero?” Click the ‘share’ button below the article and accomplish sharing by writing your stand and reason concerning the issue using your Facebook account. Ask at least four friends (not from your school) to leave a comment on your post. Print your post together with your friends’ comments. Submit the print out to your professor.
SUPPLEMENTARY ONLINE READING
Look for the article “Is the theory of Evolution authentically scientific? (II)” through the search engine (upper right section) of www.OurHappySchool.com. Pay attention to the various forms of inference used in the article.