# 4.1: Logic as a Discipline

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

### Arguments and Inference

#### HomeThe Discipline of Logic

Human life is full of decisions, including significant choices about what to believe. Although everyone prefers to believe what is true, we often disagree with each other about what that is in particular instances. It may be that some of our most fundamental convictions in life are acquired by haphazard means rather than by the use of reason, but we all recognize that our beliefs about ourselves and the world often hang together in important ways.

If I believe that whales are mammals and that all mammals are fish, then it would also make sense for me to believe that whales are fish. Even someone who (rightly!) disagreed with my understanding of biological taxonomy could appreciate the consistent, reasonable way in which I used my mistaken beliefs as the foundation upon which to establish a new one. On the other hand, if I decide to believe that Hamlet was Danish because I believe that Hamlet was a character in a play by Shaw and that some Danes are Shavian characters, then even someone who shares my belief in the result could point out that I haven’t actually provided good reasons for accepting its truth.

In general, we can respect the directness of a path even when we don’t accept the points at which it begins and ends. Thus, it is possible to distinguish correct reasoning from incorrect reasoning independently of our agreement on substantive matters. Logic is the discipline that studies this distinction—both by determining the conditions under which the truth of certain beliefs leads naturally to the truth of some other belief, and by drawing attention to the ways in which we may be led to believe something without respect for its truth. This provides no guarantee that we will always arrive at the truth, since the beliefs with which we begin are sometimes in error. But following the principles of correct reasoning does ensure that no additional mistakes creep in during the course of our progress.

In this review of elementary logic, we’ll undertake a broad survey of the major varieties of reasoning that have been examined by logicians of the Western philosophical tradition. We’ll see how certain patterns of thinking do invariably lead from truth to truth while other patterns do not, and we’ll develop the skills of using the former while avoiding the latter. It will be helpful to begin by defining some of the technical terms that describe human reasoning in general.

#### HomeThe Structure of Argument

Our fundamental unit of what may be asserted or denied is the proposition (or statement) that is typically expressed by a declarative sentence. Logicians of earlier centuries often identified propositions with the mental acts of affirming them, often called judgments, but we can evade some interesting but thorny philosophical issues by avoiding this locution.

Propositions are distinct from the sentences that convey them. “Smith loves Jones” expresses exactly the same proposition as “Jones is loved by Smith,” while the sentence “Today is my birthday” can be used to convey many different propositions, depending upon who happens to utter it, and on what day. But each proposition is either true or false. Sometimes, of course, we don’t know which of these truth-values a particular proposition has (“There is life on the third moon of Jupiter” is presently an example), but we can be sure that it has one or the other.

The chief concern of logic is how the truth of some propositions is connected with the truth of another. Thus, we will usually consider a group of related propositions. An argument is a set of two or more propositions related to each other in such a way that all but one of them (the premises) are supposed to provide support for the remaining one (the conclusion). The transition or movement from premises to conclusion, the logical connection between them, is the inference upon which the argument relies.

Notice that “premise” and “conclusion” are here defined only as they occur in relation to each other within a particular argument. One and the same proposition can (and often does) appear as the conclusion of one line of reasoning but also as one of the premises of another. A number of words and phrases are commonly used in ordinary language to indicate the premises and conclusion of an argument, although their use is never strictly required, since the context can make clear the direction of movement. What distinguishes an argument from a mere collection of propositions is the inference that is supposed to hold between them.

Thus, for example, “The moon is made of green cheese, and strawberries are red. My dog has fleas.” is just a collection of unrelated propositions; the truth or falsity of each has no bearing on that of the others. But “Helen is a physician. So Helen went to medical school, since all physicians have gone to medical school.” is an argument; the truth of its conclusion, “Helen went to medical school,” is inferentially derived from its premises, “Helen is a physician.” and “All physicians have gone to medical school.”

#### HomeRecognizing Arguments

It’s important to be able to identify which proposition is the conclusion of each argument, since that’s a necessary step in our evaluation of the inference that is supposed to lead to it. We might even employ a simple diagram to represent the structure of an argument, numbering each of the propositions it comprises and drawing an arrow to indicate the inference that leads from its premise(s) to its conclusion.

Don’t worry if this procedure seems rather tentative and uncertain at first. We’ll be studying the structural features of logical arguments in much greater detail as we proceed, and you’ll soon find it easy to spot instances of the particular patterns we encounter most often. For now, it is enough to tell the difference between an argument and a mere collection of propositions and to identify the intended conclusion of each argument.

Even that isn’t always easy, since arguments embedded in ordinary language can take on a bewildering variety of forms. Again, don’t worry too much about this; as we acquire more sophisticated techniques for representing logical arguments, we will deliberately limit ourselves to a very restricted number of distinct patterns and develop standard methods for expressing their structure. Just remember the basic definition of an argument: it includes more than one proposition, and it infers a conclusion from one or more premises. So “If John has already left, then either Jane has arrived or Gail is on the way.” can’t be an argument, since it is just one big (compound) proposition. But “John has already left, since Jane has arrived.” is an argument that proposes an inference from the fact of Jane’s arrival to the conclusion, “John has already left.” If you find it helpful to draw a diagram, please make good use of that method to your advantage.

HomeOur primary concern is to evaluate the reliability of inferences, the patterns of reasoning that lead from premises to conclusion in a logical argument. We’ll devote a lot of attention to what works and what does not. It is vital from the outset to distinguish two kinds of inference, each of which has its own distinctive structure and standard of correctness.

#### Deductive Inferences

When an argument claims that the truth of its premises guarantees the truth of its conclusion, it is said to involve a deductive inference. Deductive reasoning holds to a very high standard of correctness. A deductive inference succeeds only if its premises provide such absolute and complete support for its conclusion that it would be utterly inconsistent to suppose that the premises are true but the conclusion false.

Notice that each argument either meets this standard or else it does not; there is no middle ground. Some deductive arguments are perfect, and if their premises are in fact true, then it follows that their conclusions must also be true, no matter what else may happen to be the case. All other deductive arguments are no good at all—their conclusions may be false even if their premises are true, and no amount of additional information can help them in the least.

#### HomeInductive Inferences

When an argument claims merely that the truth of its premises make it likely or probable that its conclusion is also true, it is said to involve an inductive inference. The standard of correctness for inductive reasoning is much more flexible than that for deduction. An inductive argument succeeds whenever its premises provide some legitimate evidence or support for the truth of its conclusion. Although it is therefore reasonable to accept the truth of that conclusion on these grounds, it would not be completely inconsistent to withhold judgment or even to deny it outright.

Inductive arguments, then, may meet their standard to a greater or to a lesser degree, depending upon the amount of support they supply. No inductive argument is either absolutely perfect or entirely useless, although one may be said to be relatively better or worse than another in the sense that it recommends its conclusion with a higher or lower degree of probability. In such cases, relevant additional information often affects the reliability of an inductive argument by providing other evidence that changes our estimation of the likelihood of the conclusion.

It should be possible to differentiate arguments of these two sorts with some accuracy already. Remember that deductive arguments claim to guarantee their conclusions, while inductive arguments merely recommend theirs. Or ask yourself whether the introduction of any additional information—short of changing or denying any of the premises—could make the conclusion seem more or less likely; if so, the pattern of reasoning is inductive.

#### HomeTruth and Validity

Since deductive reasoning requires such a strong relationship between premises and conclusion, we will spend the majority of this survey studying various patterns of deductive inference. It is therefore worthwhile to consider the standard of correctness for deductive arguments in some detail.

A deductive argument is said to be valid when the inference from premises to conclusion is perfect. Here are two equivalent ways of stating that standard:

• If the premises of a valid argument are true, then its conclusion must also be true.
• It is impossible for the conclusion of a valid argument to be false while its premises are true.

(Considering the premises as a set of propositions, we will say that the premises are true only on those occasions when each and every one of those propositions is true.) Any deductive argument that is not valid is invalid: it is possible for its conclusion to be false while its premises are true, so even if the premises are true, the conclusion may turn out to be either true or false.

Notice that the validity of the inference of a deductive argument is independent of the truth of its premises; both conditions must be met in order to be sure of the truth of the conclusion. Of the eight distinct possible combinations of truth and validity, only one is ruled out completely:

Premises Inference Conclusion
True Valid True
XXXX
Invalid True
False
False Valid True
False
Invalid True
False

The only thing that cannot happen is for a deductive argument to have true premises and a valid inference but a false conclusion.

Some logicians designate the combination of true premises and a valid inference as a soundargument; it is a piece of reasoning whose conclusion must be true. The trouble with every other case is that it gets us nowhere, since either at least one of the premises is false, or the inference is invalid, or both. The conclusions of such arguments may be either true or false, so they are entirely useless in any effort to gain new information.