Logic is a branch of philosophy. It is the study of the correct rules (or laws) of reasoning. Logic can be divided into two main branches. The first, deductive logic, is the study of those rules (or laws) of reasoning that guarantee a preservation of truth. It concerns the structure of arguments in which true statements always lead to true statements. The second branch, inductive logic, is the study of those rules of reasoning that are likely, but not guaranteed, to preserve truth. It concerns the structure of arguments in which true statements have a high probability of leading to true statements.
Introduction. Logic is about arguments. An argument is a series of statements in which reasons are offered for a conclusion. Philosophy uses logic as one of its tools, and what logic helps us do is evaluate arguments. Logic is not about establishing truth. It’s aim is simply to determine the strength of an argument. Whether or not the reasons really do constitute true claims is a matter for the appropriate field. So, for instance, with logic we can evaluate medical arguments, though the question of what medical claims are true is to be determined by medical research, not by logic. Similarly, with logic we can evaluate arguments in law, business, chemistry, art, history, and everything else. Logic is powerful tool, but it is not alone enough. Without it, however, we easily fall into false reasoning. In other words, logic is a necessary but not a sufficient tool for discovering truth.
The 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.
The 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.”
An argument is a series of declarative statements, one of which (the conclusion), is intended to be supported by the others (the premises). A declarative statement is a sentence that is either true or false. The only thing logic is concerned with is whether arguments are good or bad, correct or incorrect as 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.
Our 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 logic studies deductive arguments. A deductive argument is a series of statements in which some of them (the premises) are intended to provide absolute support for another of them (the conclusion). Another word for “argument” in this context is “inference.”
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 absolute and complete support for its conclusion, so that it would be utterly inconsistent to suppose that the premises are true but the conclusion false.
Notice that every 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.
Inductive logic studies inductive argument. An inductive argument is a series of statements in which some of them (the premises) are intended to prove some support for another of them (the conclusion). With inductive arguments the aim is for the conclusion to be probable given the truth of the premises. Again, another word for “argument” in this context is “inference.”
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.
Truth and Validity
Since deductive reasoning requires such a strong relationship between premises and conclusion, it is important to spend time studying various patterns of deductive inference. Doing so will provide conceptual clarity, something very important to have when you approach the murky waters of induction. 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:
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 sound argument; 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.
So, logic is solely concerned with the form (or structure) of arguments. What, then, is a deductive argument exactly? And what is an inductive argument exactly?
When defining “Deductive Argument” any of the following will do, at least to begin with. The last one is probably best.
- an argument in which the truth of the premises is intended to guarantee the truth of the conclusion.
- an argument in which the truth of the premises is intended to provide absolute support for the truth of the conclusion.
- an argument in which the certainty of the conclusion is held to follow from the certainty of the premises.
- an argument which has a form that is supposed to guarantee the preservation of truth.
When defining “Inductive Argument” any of the following will do, at least to begin with. The last one is probably best.
- an argument in which the truth of the premises is intended to provide strong support for the truth of the conclusion.
- an argument that is not deductive, but the conclusion is probable given the probable truth of the premises.
- an argument in which the truth of the premises is provides some support for the truth of the conclusion.
When evaluating an argument the first thing to do is to identify the conclusion. The second thing to do is to identify the reasons for the conclusion (i.e., the premises). Below are lists of common conclusion indicators and premise indicators. However, these lists are not exhaustive. Further, not all arguments provide indicators. You must learn to identify the conclusion of an argument and its premises by paying attention to what the argument is trying to accomplish.
Take the following as an example:
Alice will do well in this course. This is because she pays attention in class, studies hard, and asks relevant questions when she does not understand. Since this is what a student needs to do to do well in this course, she will do well.
Next, the argument can be put in standard form. An argument is in standard form when it is laid out as a series of steps, in which the supporting reasons (premises) occur first and the defended statement (conclusion) occurs last. So, the standard form of the argument above is as follows:
P1) To do well in this course a student needs to pay attention in class, study hard, and ask relevant questions when she does not understand
P2) Alice pays attention in class, studies hard, and asks relevant questions when she does not understand.
C) Alice will do well in this course.
In putting the argument in standard form, notice that I have had to restate some claims to make them clearer. In addition, sometimes you will need to decide whether a statement is really a part of the argument, or merely some superfluous statement. This argument is inductive, since the truth of the premises do not guarantee the truth of the conclusion. Rather, they make the conclusion more likely.
Put each of the following arguments in standard form.
- If you believe in fighting evil, then you support the current war. I see that you support the current war, so you must also believe in fighting evil.
- Flanders has used his influence to persecute his opponents. Angstrom Flanders is the worst governor the state has ever had. He has embezzled money from the state’s treasury. He has ruined the state economy. I am horrified by the things he has done, and his obvious ineptitude.
- All men are mortal. Hence, Socrates is mortal, since Socrates is a man.
- Spontaneous combustion is a real phenomenon because no scientist has ever been able to disprove it.
- No one knows what evil lies in the hearts of men, but if anyone might know it will be Stanley; everything he does is so nasty.
Once you are able to put arguments in standard form, the next task is to evaluate the arguments. The specific tools used for evaluating inductive arguments are different from those used for evaluating deductive arguments. So, we need to decide whether an argument is inductive or deductive before we begin an evaluation. Which of the above arguments are deductive?
There are two main branches of deductive logic. The first is called Aristotelian logic, the second is called modern logic. Aristotelian logic focuses on classes (all, some, none), and negation (not). Modern logic focuses on a set of logical operators (and, or, if_then, if and only if) and negation (not).
To evaluate a deductive argument we need to put each of the statements in logical form. To put a statement in logical form is to identify its logical operators. The logical operators are: not, and, or, if-then, if-and-only-if
To evaluate a deductive argument we need to put each of the statements in logical form. To put a statement in logical form is to identify its logical operators.