2.6: Deductive Reasoning and Argument Forms
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Deduction is the process by which we determine new truths from old. It is sometimes claimed that nothing truly new can come from deduction, the truth of a statement that is arrived at by deductive processes was lying (perhaps hidden somewhat) within the hypotheses. This claim is something of a canard, as any Sherlock Holmes aficionado can tell you, the statements that can sometimes be deduced from others can be remarkably surprising. A better argument against deduction is that it is a relatively ineffective way for most human beings to discover new truths – for that purpose inductive processes are superior for the majority of us. Nevertheless, if a chain of deductive reasoning leading from known hypotheses to a particular conclusion can be exhibited, the truth of the conclusion is unassailable. For this reason, mathematicians have latched on to deductive reasoning as the tool for, if not discovering our theorems, communicating them to others.
The word “argument” has a negative connotation for many people because it seems to have to do with disagreement. Arguments within mathematics (as well as many other scholarly areas), while they may be impassioned, should not involve discord. A mathematical argument is a sequence of logically connected statements designed to produce agreement as to the validity of a proposition. This “design” generally follows one of two possibilities, inductive reasoning or deductive reasoning. In an inductive argument a long list of premises is presented whose truths are considered to be apparent to all, each of which provides evidence that the desired conclusion is true. So an inductive argument represents a kind of statistical thing, you have all these statements that are true each of which indicates that the conclusion is most likely true. . . A strong inductive argument amounts to what attorneys call a “preponderance of the evidence.” Occasionally a person who has been convicted of a crime based on a preponderance of the evidence is later found to be innocent. This usually happens when new evidence is discovered that incontrovertibly proves (i.e. shows through deductive means) that he or she cannot be guilty. In a nutshell: inductive arguments can be wrong.
In contrast, a deductive argument can only turn out to be wrong under certain well-understood circumstances.
Like an inductive argument, a deductive argument is essentially just a long sequence of statements; but there is some additional structure. The last statement in the list is the conclusion – the statement to be proved – those occurring before it are known as premises. Premises may be further subdivided into (at least) five sorts: axioms, definitions, previously proved theorems, hypotheses and deductions. Axioms and definitions are often glossed over, indeed, they often go completely unmentioned (but rarely unused) in a proof. In the interest of brevity this is quite appropriate, but conceptually, you should think of an argument as being based off of the axioms for the particular area you are working in, and its standard definitions. A rote knowledge of all the other theorems proved up to the one you are working with would generally be considered excessive, but completely memorizing the axioms and standard definitions of a field is essential. Hypotheses are a funny class of premises – they are things which can be assumed true for the sake of the current argument. For example, if the statement you are trying to prove is a conditional, then the antecedent may be assumed true (if the antecedent is false, then the conditional is automatically true!). You should always be careful to list all hypotheses explicitly, and at the end of your proof make sure that each and every hypothesis got used somewhere along the way. If a hypothesis really isn’t necessary then you have proved a more general statement (that’s a good thing).
Finally, deductions – I should note that the conclusion is also a deduction – obey a very strict rule: every deduction follows from the premises that have already been written down (this includes axioms and definitions that probably won’t actually have been written, hypotheses and all the deductions made up to this point) by one of the so-called rules of inference.
Each of the rules of inference actually amounts to a logical tautology that has been re-expressed as a sort of re-writing rule. Each rule of inference will be expressed as a list of logical sentences that are assumed to be among the premises of the argument, a horizontal bar, followed by the symbol \(∴\) (which is usually voiced as the word “therefore”) and then a new statement that can be placed among the deductions.
For example, one (very obvious) rule of inference is
\(\begin{array} &&\underline{A ∧ B} \\ ∴ &B \end{array} \)
This rule is known as conjunctive simplification, and is equivalent to the tautology \((A ∧ B) \implies B\).
The modus ponens rule1 is one of the most useful.
\(\begin{array} & &A \\ &\underline{A \implies B} \\ ∴ &B \end{array} \)
Modus ponens is related to the tautology \((A ∧ (A \implies B)) \implies B\).
Modus tollens is the rule of inference we get if we put modus ponens through the “contrapositive” wringer.
\(\begin{array} & &¬B\\ &\underline{A \implies B} \\ ∴ &¬A \end{array}\)
Modus tollens is related to the tautology \((¬B ∧ (A \implies B)) \implies ¬A\).
Modus ponens and modus tollens are also known as syllogisms. A syllogism is an argument form wherein a deduction follows from two premises. There are two other common syllogisms, hypothetical syllogism and disjunctive syllogism.
Hypothetical syllogism basically asserts a transitivity property for implications.
\(\begin{array} & &A \implies B\\ &\underline{B \implies C} \\ ∴ &A \implies C \end{array}\)
Disjunctive syllogism can be thought of as a statement about alternatives, but be careful to remember that in Logic, the disjunction always has the inclusive sense.
\(\begin{array} & &A ∨ B\\ &\underline{¬B\;\;\;\;\;\;\;} \\ ∴ &A \end{array}\)
Convert the \(A ∨ B\) that appears in the premises of the disjunctive syllogism rule into an equivalent conditional. How is the new argument form related to modus ponens and/or modus tollens?
The word “dilemma” usually refers to a situation in which an individual is faced with an impossible choice. A cute example known as the Crocodile’s dilemma is as follows:
A crocodile captures a little boy who has strayed too near the river. The child’s father appears and the crocodile tells him “Don’t worry, I shall either release your son or I shall eat him. If you can say, in advance, which I will do, then I shall release him.” The father responds, “You will eat my son.” What should the crocodile do?
In logical arguments the word dilemma is used in another sense having to do with certain rules of inference. Constructive dilemma is a rule of inference having to do with the conclusion that one of two possibilities must hold.
\(\begin{array} & &A \implies B\\ & C \implies D\\ &\underline{A ∨ C\;\;\;\;\;\;\;} \\ ∴ &B ∨ D \end{array} \)
Destructive dilemma is often not listed among the rules of inference because it can easily be obtained by using the constructive dilemma and replacing the implications with their contrapositives.
\(\begin{array} & &A \implies B\\ & C \implies D\\ &\underline{¬B ∨ ¬D\;\;\;} \\ ∴ &¬A ∨ ¬C \end{array}\)
In Table 2.6.1, the ten most common rules of inference are listed. Note that all of these are equivalent to tautologies that involve conditionals (as opposed to biconditionals), every one of the basic logical equivalences that we established in Section 2.3 is really a tautology involving a biconditional, collectively these are known as the “rules of replacement.” In an argument, any statement allows us to infer a logically equivalent statement. Or, put differently, we could replace any premise with a different, but logically equivalent, premise. You might enjoy trying to determine a minimal set of rules of inference, that together with the rules of replacement would allow one to form all of the same arguments as the ten rules in Table 2.6.1.
| Table 2.6.1: The Rules of Inference | |
|---|---|
| Name | Form |
| Modus ponens | \(\begin{array} & &A \\ &\underline{A \implies B} \\ ∴ &B \end{array} \) |
| Modus tollens | \(\begin{array} & &¬B\\ &\underline{A \implies B} \\ ∴ &¬A \end{array}\) |
| Hypothetical Syllogism | \(\begin{array} & &A \implies B\\ &\underline{B \implies C} \\ ∴ &A \implies C \end{array}\) |
| Disjunctive Syllogism | \(\begin{array} & &A ∨ B\\ &\underline{¬B\;\;\;\;\;\;\;} \\ ∴ &A \end{array}\) |
| Constructive Dilemma | \(\begin{array} & &A \implies B\\ & C \implies D\\ &\underline{A ∨ C\;\;\;\;\;\;\;} \\ ∴ &B ∨ D \end{array} \) |
| Destructive Dilemma | \(\begin{array} & &A \implies B\\ & C \implies D\\ &\underline{¬B ∨ ¬D\;\;\;} \\ ∴ &¬A ∨ ¬C \end{array}\) |
| Conjunctive Simplification | \(\begin{array} & &\underline{A∧B}\\ ∴ & A\end{array}\) |
| Conjunctive Addition | \(\begin{array} & &A \\ & \underline{B\;\;\;\;\;\;\;\;\;\;}\\ ∴ & A∧B\end{array}\) |
| Disjunctive Addition | \(\begin{array} & &\underline{A\;\;\;\;\;\;\;}\\ ∴ & A∨B\end{array}\) |
| Absorption | \(\begin{array} & &\underline{A \implies B\;\;\;\;\;\;\;\;}\\ ∴ & A\implies(A∧B)\end{array}\) |
Exercises:
In the movie “Monty Python and the Holy Grail” we encounter a medieval villager who (with a bit of prompting) makes the following argument.
If she weighs the same as a duck, then she’s made of wood.
If she’s made of wood then she’s a witch.
Therefore, if she weighs the same as a duck, she’s a witch.
Which rule of inference is he using?
In constructive dilemma, the antecedent of the conditional sentences are usually chosen to represent opposite alternatives. This allows us to introduce their disjunction as a tautology. Consider the following proof that there is never any reason to worry (found on the walls of an Irish pub).
Either you are sick or you are well.
If you are well there’s nothing to worry about.
If you are sick there are just two possibilities:
Either you will get better or you will die.
If you are going to get better there’s nothing to worry about.
If you are going to die there are just two possibilities:
Either you will go to Heaven or to Hell.
If you go to Heaven there is nothing to worry about. If you go
to Hell, you’ll be so busy shaking hands with all your friends
there won’t be time to worry . . .
Identify the three tautologies that are introduced in this “proof.”
For each of the following arguments, write it in symbolic form and determine which rules of inference are used.
- You are either with us, or you’re against us. And you don’t appear to be with us. So, that means you’re against us!
- All those who had cars escaped the flooding. Sandra had a car – therefore, Sandra escaped the flooding.
- When Johnny goes to the casino, he always gambles ’til he goes broke. Today, Johnny has money, so Johnny hasn’t been to the casino recently.
- (A non-constructive proof that there are irrational numbers \(a\) and \(b\) such that \(a^b\) is rational.) Either \(\sqrt{2}^{\sqrt{2}}\) is rational or it is irrational. If \(\sqrt{2}^{\sqrt{2}}\) is rational, we let \(a = b = \sqrt{2}\). Otherwise, we let \(a = \sqrt{2}^{\sqrt{2}}\) and \(b = \sqrt{2}\). (Since \(\sqrt{2}^{\sqrt{2}^{\sqrt{2}}} = 2\), which is rational.) It follows that in either case, there are irrational numbers \(a\) and \(b\) such that \(a^b\) is rational.