A Brief Introduction to Logic:
The Science of Correct Reasoning

Viewed from a certain angle, philosophy and logic are about what, if anything, we should believe. Much philosophy, such as ethics or social and political philosophy, has to do with how we should act and how we should structure our society’s institutions, so one might say that viewing philosophy as all about beliefs ignores these aspects. But how we decide to act or how we decide to structure society ultimately comes down to what we believe about the nature of the world, our nature as human beings, the facts of the particular situation, what we value, and what is good and right. Thus, for present purposes, it is accurate enough to say that philosophy is all about what we should believe.

There are many sources for our beliefs - parents and friends, newspapers and television, government, churches, schools, textbooks, and our own observations - but not all these sources are equally good, nor is any of them good all the time. Most of us form most of our beliefs without consciously attending to the fact that we have done so. Many beliefs attained through casual observation and most of the teaching and socialization of childhood is simply absorbed without our realizing it. Despite our best efforts, this uncritical acceptance of claims and formation of beliefs continues through adulthood. But such uncritical thinking is a dangerous thing, for it enslaves you to the influence and manipulations of other individuals and institutions. Thus, since our beliefs - the claims we accept - constitute our view of the world, and importantly affect how we act, it'is important to examine more carefully every belief we hold. This applies both to large scale philosophical issues (Does a god exist, and if so, what sort? Do I have a soul? Am I free?), and to small scale mundane affairs (Should I be blunt with others, politically correct, or somwhere in between depending on the situation? Do I really need an SUV? Should I believe the judge?)

This is pretty much the focus of logic: examining fundamental philosophical beliefs about the world, ourselves, and our values. In this brief piece on logic, we will introduce a framework of terms to use when examining claims and the arguments or evidence offered in support of them. They will be immediately useful in assesing even the most mundane concerns.

When you accept a claim or form a belief on the basis of other claims or beliefs, you are making an inference. Not all inferences are created equal-some are clearly bad, some are clearly good, and for others it is more difficult to tell. In logic we study rules and methods for determining which inferences are good and which bad. To get a firm grasp on things we will focus on arguments. 'Argument’ does not mean a shouting match or angry disagreement (though sometimes these accompany the sort of argument we are interested in). An argument is a set of declarative sentences, one of which-the conclusion-is supposed to be supported by the others-the premises.

Declarative sentences—as opposed to questions, commands, exclamations—are sentences which attempt to state a fact. ‘Statement’, ‘sentence’, and ‘claim’ will appear interchangeably. Such sentences are either true or false (though we may not know which)-that is, they have a truth-value: Either the truth-value true or the truth-value false.

When we encounter an argument, the premises and conclusion may come in any order. Consider the following versions of a classic example:

• Socrates is mortal, for all humans are mortal, and Socrates is human
  
• Given that Socrates is human, Socrates is mortal; since all humans are mortal
  
• All humans are mortal; Socrates is human; therefore, Socrates is mortal

Three statements are involved (two premises and a conclusion) for each bullet point, and despite the fact that they appear in different order, all three examples express the same argument. For the sake of clarity we often transcribe arguments into what is called standard form: list the premises, draw a line, then write the conclusion. For the above argument:

Despite the variability of statement order, when not expressed in standard form, a good writer usually makes clear which sentences are premises and which is the conclusion. This is usually done through contextual clues, including the indicator words/phrases used above. Below are two brief lists of indicators:

It's usually a simple to task to put arguments into standard form. Of course, if the argument is long and complex, with sub-conclusions acting as premises for further conclusions, things can get messy. Later on we'll divide arguments, and, hence, the inferences they express, into three basic types, Deductive, Inductive, and Adductive. We'll have similar but different things to say about what counts as “good” for each, but let's first introduce some common errors of reasoning.

(1) Fallacies

Generally speaking, a fallacy is any mistake in reasoning, but some fallacies are so common that they have earned names. (Try to avoid the term logical fallacy in conversation. Friends will call you a nerd...and rightly so.) The term logical fallacy refers to a thinking error independent of the truth of the premises and the conclusion in an argument. It is a flaw in the structure of an argument as opposed to an error in its premises or the conclusion.

Circularity is one very common type of fallacy. For instance, "I believe in Allah because the Qur'an tells me He exists. I know I can trust the Qur'an because it is the revealed word of Allah." See also the list of other common errors.

(2) Deductive Arguments

Deductive arguments are those in which the premises are supposed to (if true) guarantee the truth of the conclusion. There are two questions we want to ask when assessing a deductive argument. The question of validity has to do with the structure of the argument, and the question of soundness has to do with the truth of the premises.

Validity: An argument is valid if and only if it's impossible for ALL the premises to be true AND the conclusion false. I.e., an argument is valid if and only if the assumed truth of the premises would guarantee the truth of the conclusion. Mull it over and you’ll see that these say basically the same thing. Validity is a question of truth preservation, and this is a question of form, so the actual truth-values of the premises and conclusion are themselves irrelevant. In this sense, a valid deductive argument is a little like algebra: a² + b² = c² no matter what real numbers you plug in for a and b.

Soundness: An argument is sound if and only if it is valid AND all its premises are true.

Consider the following two groups of arguments:

Upon reflection, you should be able to see that all three arguments of Form 1 are valid, and that this has nothing to do with the actual truth-values of the component claims. That is, despite the actual truth-value of the component sentences, there is no way (it is NOT possible) that the premises could ALL be true AND the conclusion false. Take 1D, for example, despite the fact that all the claims are false, it has to be the case that IF the premises were true, then the conclusion would be true as well-just imagine the premises are true; could the conclusion be false? NO, so it’s valid.

As you will have noticed, 1A, 1B, and 1D are all of the same form. Indeed, it is in virtue of having this form that each of the three instances are valid. Any argument in which we consistently substitute noun phrases for the place-holders F, G, and H will be a valid argument. Thus, Form 1 is a valid argument form and 1A, 1B, and 1D are valid arguments because they are instances of a valid form. Validity is a question of form. That is why the actual truth-values of the component claims is irrelevant. What is relevant is whether the form is such that it is not possible to have all premises true and the conclusion false. Valid arguments are truth preserving. A good metaphor for this is plumbing: if you hook up the pipes correctly (if your argument has a valid form) you know that if you put water in at the top (true premises), you’ll get water out at the bottom (true conclusion)-but it doesn’t actually matter whether you do, indeed, put any water in.

What, then, of 1C? There can be no such instance of Form 1. It is impossible, the argument is valid. Compare the arguments in Form 2, 2C in particular. Note that of the three arguments in group 1, only 1A is also sound. This is because, in addition to being valid, it has premises which are actually true.

Now look at the four arguments of Form 2. First, note that they are all invalid. That is, despite the actual truth-values of the component sentences, in each instance it IS possible for all the premises to be true and the conclusion false. 2A, 2B, and 2D, will require some imagination, but you will see that you can consistently imagine all the premises true and the conclusion false in each case (by the way, an alternate definition of validity would be: an argument is valid if and only if assuming the truth of the premises is inconsistent with assuming the falsehood of the conclusion). 2C takes no imagination at all. Here we have an instance of the form in which the premises are all actually true and the conclusion is actually false. Obviously this particular argument is invalid, since the premises ARE all true and the conclusion IS false, it is POSSIBLE for all the premises to be true and the conclusion false. Since validity is a matter of form, once we have an instance like 2C before our eyes, we know that the form is bad, and so is any instance (such as 2A, 2B, or 2D). Notice that it was just the C-type instance (all Ps are T, C is F) that the valid Form 1 lacked. When we find such a C-type argument we are said to have found a counter-example to the argument form. Valid arguments do not have counter examples.

Lastly, because an argument must be valid to be sound, and none of the instances of Form 2 is valid, none of them is sound-in fact, once we determine that an argument is invalid, we don’t bother with the question of soundness.

What follows are some points to remember about valid arguments, and brief list of some valid forms (there are an infinite number of valid argument form, so we can’t list them all. Aristotle (384-322 BC) was the first to systematically catalogue deductively valid forms. He dealt solely with categorical syllogisms (Forms 1 and 2 are examples), of which there are only finitely many. The limitations of Aristotle’s logic were not genuinely overcome until the end of the 19th and beginning of the 20th centuries, with the work of Peirce, Frege, Cantor, Russell, and others in the development of predicate logic, quantification theory, and set theory. Again...

• Validity is a question of truth preservation, and this is a question of form, so the actual truth-values of the premises and conclusion are irrelevant. All true premises and true conclusion, DO NOT make a valid argument! Look at 1B, 1D, and (especially) 2A.
• Soundness does have to do with the actual truth-value of the premises. If an argument is valid and its premises are true, then it is sound.
• We can see that a particular argument, and all arguments of the same form, are invalid either by consistently imagining that all the premises are true and the conclusion false, or by finding a counterexample—an instance which actually does have all true premises and a false conclusion.

Some Valid Forms:

Tempting But Invalid:

                         Evaluating Deductive Arguments
                      
       Is it possible for ALL the premises to be true AND the conclusion false?
                     /                                       \
            No, not possible                    Yes, possible
                    |                                         |
            Argument is VALID                 Argument is INVALID
                    |                                         |
      Are the premises actually true?            STOP
          /                        \
        YES                      NO
         |                          |
     Argument is             Argument is
     SOUND                   UNSOUND

(3) Inductive Arguments

There are a number of ways of marking the distinction between inductive and deductive arguments. Traditionally, deductive arguments are said to be those that proceed from the general to the particular, while inductive are said to move from the particular to the general. But this is incorrect. Many deductive arguments do not invoke general premises and do arrive at specific conclusions. The converse is true of many inductive arguments. We'll distinguish the two in the following way:

A deductive argument is one in which the truth of the premises is supposed to guarantee the truth of the conclusion. As discussed above, they are supposed to be truth-preserving - though sometimes they fail. In contrast,

Inductive arguments are those in which the truth of the premises is supposed to lend a degree of plausibility or probability to the truth of the conclusion. As such, inductive arguments are not valid, there is no (and generally no claim to) guarantee of truth preservation. We need a different measure of the “goodness” of inductive arguments.

Strength: An argument has inductive strength to the degree to which the premises (if true) provide evidence to make the truth of the conclusion plausible or probable.

Consider the following examples:

In neither of these cases does the assumed truth of the premises guarantee the truth of the conclusion. So how good are these arguments? Well, for the first one we have a pretty good idea-it is quite strong, barring unforeseen happenings, we would rate the probability of the conclusion at 80%. But for the second one it is unclear. It seems pretty strong, but that is a vague assessment. And this is often the case with non-statistical inductive arguments. Except where the argument is clearly very weak, often we can only give a vague assessment of its strength.

This is a point to remember about deductive versus inductive arguments: validity is like an on/off switch, an argument is either valid or invalid (and not both); but inductive strength is a matter of degree. Moreover, unlike validity of deductive arguments, the strength of inductive arguments is not simply a matter of form (though form is often relevant). (More on this below.) Two common and simple forms of induction are:

With enumeration, generally speaking, the larger the sample, the stronger the argument. As the number of observed examples exhibiting the relevant property increases, so does the likelihood of the conclusion (unless the population is infinite). Moreover, the narrower (or more conservative) the conclusion, the stronger the argument. For example, it is a narrower conclusion, and so a safer bet, that the next fish will be infected, than that all fish are (perhaps there are a very small number of resistant fish). With arguments by analogy, strength tends to vary with the number of shared properties-the more the two objects (or groups) have in common, the more likely the object in question will also have the further property.

But, as Hume teaches us, inductive arguments are not so simple. Consider the following, which are of exactly the same forms as the above arguments:

Note that neither of these arguments is particularly strong. Despite the rather large sample size in the first one, the conclusion that I will live forever is extremely unlikely, and while the conclusion that I will live through the next day seems to be stronger (because the conclusion is narrower), it is not particularly comforting… The argument involving the elephant is obviously ludicrous, mainly because the similarities cited are largely irrelevant to the question of my having a trunk. Analogical arguments are stronger when the similarities cited are relevant to the target property, and when there are few relevant dissimilarities.

Determining the relevance of similarities and dissimilarities (as well as the question of the strength of enumerations) depends to a large degree on background knowledge-knowledge which is often left unstated, but which, when made explicit, or inserted as a new premise, may strengthen or weaken the argument. In some cases (say when hypothesizing about the effect on humans of a drug tested on mice) we may not be entirely sure of how relevant the similarities and dissimilarities are, and this affects our assessment of the strength of the argument. This is part of why we say that strength is not a question of form.

In accord with the breadth of our definition, there are many ways to argue inductively and many ways to critique inductive arguments, but we won't try to survey them here. Let's leave this topic with one final point: If an argument is valid, this status cannot be changed by the introduction of additional evidence in the form of further premises.* In inductive arguments, however, the introduction of further evidence in the form of further premises can increase or decrease the strength of the argument. (Imagine pointing out relevant dissimilarities between me and Bingo, or, in the argument about the mail carrier, adding the evidence that the roads are flooded.)

*Even if we introduce a new premise which contradicts one of the old premises! The reason is that, if the premises contradict one another, then it is not possible for them all to be true. So, it is not possible for all premises to be true AND the conclusion false. So, the argument is valid. Indeed, every argument with contradictory or inconsistent premises is valid. Note, however, that in such “degenerate” cases of validity the argument can never be sound.

(4) Adductive Reasoning (Inference to the Best Explanation)

Some authors keep adductive reasoning strictly separate from inductive reasoning. Usually this is because, while other inductive inferences often aim at making a prediction or generalization on the basis of the premises, adductive inference is explicitly aimed at explaining the truth of the premises. That is, instead of finding a conclusion which is probable when we assume the truth of the premises, we try to find a conclusion (the explanation) which, if true, would explain the truth of the premises, (in part) by showing the premises to be probable. This may seem backwards, but at least one way good explanations achieve their explanatory power is by showing why, on the basis of more general considerations (the explanatory conclusion), the premises are to be expected. Precisely distinguishing adductive from inductive is not important here, so let's just continue with the discussion.

As with inductive arguments, adductive reasoning is not truth-preserving. We can only make our conclusion plausible or probable with adductive reasoning, there is no guarantee of truth. Adductive arguments are distinguished by their attempt to provide an explanatory hypothesis for some set of data. The try to answer the question of why something is the way it is. Thus, adductive reasoning is often described as inference to the best explanation. In fact, the example involving the mail carrier is a miniature adductive argument. The conclusion that it is a holiday, would, if true, explain the absence of mail at 5pm on a Monday in a manner consistent with the carrier’s past punctuality. It is not, however, the only hypothesis which would explain it, flooded roads might well explain the absence of mail. Which is the better, or best explanation? That depends on how much evidence we have and how much more we can gather. Some loose criteria follow:

• The more known data an explanation can account for in a consistent and coherent manner, the better the explanation.
• The better an explanation coheres with established theory, the better it is (e.g., hypothesizing that the mail is there, but is invisible, does not (among other problems) cohere well with established physical theory).
• Moreover, if an explanation successfully predicts further data not originally observed (I check my calendar and find that it is July 4th; or I turn on the news and learn that a water main has burst), then that explanation is better than one which cannot do so.
• There are no precise guidelines for adductive reasoning. Especially if we focus on the notion of the ‘best’ explanation. Much depends on plausibility relative to our background knowledge and the quality and quantity of our evidence. Almost always there is more evidence to be had (in principle), and new evidence (as with inductive, but not valid deductive arguments) can radically alter our assessment of the quality of the inference.

Here is another example:

I hear scratching in the walls,
I hear the scurrying-clicking sound of little paws at night
My cereal and rice boxes have holes chewed in them      
I have mice

My conclusion, together with other relevant information about the behavior of mice, would well explain the data expressed in the premises. Assuming the presence of mice, the observations are to be expected. Hence, we have a pretty good explanation. Of course, I may have an eccentric neighbor who enjoys practical joking. But until I gather some evidence that she is at work (cheese disappears from the unsprung mousetraps; nary a mouse is to be seen; the sounds of mice do not occur on nights when I lock up the house) the mice hypothesis seems to be the best.

(5) The Three Laws of Thought: Excluded Middle, Contradiction and Identity

These are ancient, though there is uncertainty about their universal applicability, e.g. in metaphysics. They should be quite satisfactory for mundane affairs, however.

• Excluded Middle - Every proposition is either true or not true. Note that "not true" and "false" are not equivalent. Excluded middle exists in stronger form as bivalence: every proposition is either true or false. A limitation to excluded middle is exposed with the clause, "this statement is not true" and then asking whether the proposition in quotes is true or not true.
• Contradiction - No proposition can be both true and not true. There are two other forms, as well: that nothing can both have and not have a given property simultaneously; and that nothing can be both the case and not the case at the same time.
• Identity - Every thing is what it is and not another thing. Conversely, every thing is not what it is not and only the thing it is. For example, a circle is a circle and never, ever a square. Circles and squares (a type of rectangle, really) have mutually exclusive mathematical definitions. The Law of Identity has been neglected by logicians if not the judges who discovered that corporations and living persons are the same.

(6) Formal Logic

There are seven basic symbols: ~, V, &, ® , «, $, ".
Respectively: Not, Or, And, Implies (if...then), If and Only If (iff),There Exists, For All.

Applying the basic symbols to a few of the earlier examples we have:

This is just to give you a taste of formal logic, also called symbolic logic, in which there are many more symbols than what you see above. Search the web or find an introductory logic text if you are interested in learning the grammar logicians and mathematicians have developed.

(7) Conclusion

One last word about critical thinking. People are often suspicious of logic, seeing it as a sort of “double-speak”. But this is usually because they do not know any logic. Moreover, all of us are to some degree resistant to change in our worldviews-and there are some good reasons for this-but in part this is because we are lazy and don’t want to go through the discomfort of having our beliefs shaken up and having to go to the trouble of sorting things out again and possibly (gasp!) changing our minds. In addition to this laziness (which is not a good thing), many people often center their identity-their sense of self-in a certain fixed set of beliefs, or in a certain authoritative institution (such as a church, government, etc.). When you view yourself in this way, any attempt by others (or yourself) to examine or criticize those beliefs or that institution becomes extremely threatening-for it threatens the core of your self-identity. A more intelligent, and healthier, alternative is to learn some logic and reasoning, question your own beliefs a bit, and center your self identity not in some set of beliefs or some institution, but in your ability to consider alternatives, to think carefully and critically, and to change your mind if the situation calls for it. This is the difference between being an unreflective believer and a critical thinker. The latter is more flexible and less subject to manipulation by others, hence the critical thinker is, in an important sense, more free.*

*Lest we be accused of purveying a false dilemma, we do recognize that these are not the only two ways one can view oneself. In addition, the unreflective believer and the flexible critical thinker are really two poles of a continuum along which our position varies at different times in our lives, even different times of the day, and with regard to different subject matter. It's also wise to bear in mind that

" Mere intellect makes one insane. "
- Mohandas Ghandi

[Note: Much of this essay is the work of a Professor Gregory (Colgate U.) The original was obtained on-line in October 2001 and edited for length, organization, and content, including the addition of sections 5 and 6]