Fundamental Logic Concepts

As we have already noted in the introduction, logic is a systematic process of reason, which enables us to ensure that ideas are consistent with each other, as well as to identify any ideas which are inconsistent.  Most people are vaguely familiar with logic, since they have to use some form of it just to survive from day to day.  However, for most the process may be rather haphazard and undisciplined, involving a high percentage of guesswork, and thereby frequently yielding disappointing results.  Here we'll consider how we might improve our thinking, and eliminate some bad habits or unwarranted preconceptions along the way.

Statement

To begin, let's consider the fundamental building block of logic: the statement.  A statement is just that; it is a declaration about something—anything—a declaration which can be evaluated as either true or false.  "I am reading this sentence" is a statement, and if indeed you have looked at it and comprehended its meaning, then it is safe to say that that statement can be evaluated as true.

However, not everything we hear or say, read or write, is a statement, since not every utterance declares something that is either true or false.

"I have climbed that tree."
This is a statement.  It declares something, and can be evaluated as true or false, depending upon whether or not I have actually climbed the tree.

"Climb that tree!"
This is not a statement, but a command.  It gives an instruction, but does not declare anything, and therefore cannot be evaluated as true or false.

"Have you climbed that tree?"
This is not a statement, but a question.  It asks for information, but does not declare anything, and therefore cannot be evaluated as true or false.

"Dwight Eisenhower was the 34th President of the United States."
This is a statement. It declares something (in this case a historical fact), and evaluates as true.

"Christopher Columbus discovered Hawaii."
This is a statement.  It declares something, and evaluates as false, since Columbus's explorations extended no further than the Atlantic Ocean, and Hawaii is in the Pacific.

"Rome was founded in the year 753 BCE."
This is a statement.  It declares something, and although we can't say with certainty that it evaluates as true, or that it evaluates as false, we can at least say with certainty that it evaluates as one or the other, and not as something else (such as "sweet" or "red" or "five").

"Patrick Henry said, 'Give me liberty or give me death!'"
This is a statement, taken as a whole.  Although the quote, "Give me liberty or give me death," is a demand and thus not a statement, the declaration that Patrick Henry said it is a statement, which (if our history books are not in error) evaluates as true.

In logic, the condition of whether a statement is true or false is called its truth value.  As we have seen, any logical statement has a truth value, either true or false, even though in some cases we might not happen to know which.

For the purposes of logic, a statement should be as unambiguous as possible.  For example, if we make a statement about "fish," we should make it clear whether or not we mean to include such things as sharks, shellfish, cuttlefish, or silverfish, if there is any room for question or doubt.  In some cases, a valid conclusion might reasonably be disputed on the basis of accidental misunderstanding; in others, an ambiguous statement might be deliberately introduced in order to support a specious conclusion.

This brings us to a question which must puzzle many people who haven't formally studied logic, and which we ought to address before we move on:

Why do logicians talk like robots?

Believe it or not, there's a very good reason for this.  (And it's not because logicians are robots!)  In the process of critiquing ideas and their relationships, logicians customarily state those ideas in a bare-bones format.  This helps ensure that nothing gets hidden or lost, that all essential points are clear, that the evaluation is as free as possible of irrelevancy and ambiguity, and that the function and relationship of each statement with respect to the whole is evident.  Thus evaluation and comparison of similar or competing ideas is as fair and objective as possible.

To convert a statement to a standard form that is easy to analyze, compare, and contrast, it is customary, first of all, to convert everything to present tense (provided, of course, that time relationships are not a relevant factor in the argument).  Thus, "I bought apples yesterday because they were on sale" becomes "I buy apples because they are on sale."

Next, we break complex statements down into units, and state them in a standard form.

• If apples are on sale, then I buy apples.

• Apples are on sale.

• I buy apples.

Obviously, this is not the way people normally talk; however, it shows the thought process in a simplified and easily analyzed form, without distorting any of the significant ideas (beyond the tense switch).  Note that the "because" clause in the original becomes an "if" clause in the analysis.  This is an example of converting to a standard form: In logic, an "if - then" statement is the accepted standard form of a conditional statement.  (Note that using some other form in normal discourse is perfectly all right, and does not in the least detract from the logic of what we say or write.  The "if - then" construction just makes it easier to analyze.)  "If - then" syntax can be logically substituted for nearly any conditional statement in ordinary language.  Note the logical equivalence of these examples:

• Whenever apples are on sale, I buy apples.

• Because apples are on sale, I buy apples.

• I buy apples unless apples are not on sale.

• I buy apples except when apples are not on sale.

Each of these is logically equivalent to: "If apples are on sale, then I buy apples."  While the point of this might not be immediately obvious, the value of using a standard form becomes readily apparent, when we consider that the logical principles governing virtually all conditional statements are the same, regardless of how the statement is phrased.  So once we have learned the few basic rules applying to "if - then" statements, we don't have to learn similar sets of rules for "whenever" statements, "because" statements, "unless - not" statements, "except - not" statements, and so on.  So to boil all this down into logical terms:

• A common cause of problems is confusion.

• A common cause of confusion is needless complexity.

• One set of rules for analyzing conditional statements is less complex than many sets of rules.

• Translating comparable statements into a single standard syntax lets us use one set of rules.

• So, using one standard syntax for logical analysis reduces problems.

Granted, this form of expression sounds extremely "dry."  This is why we use more colorful and varied phrasing when we want to persuade others rather than put them to sleep.  But when we try to cut to the core of a complex argument in order to find out whether it really makes sense, such simplification makes the task much easier, and much less subject to guesswork and error.

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Terms:

• statement

• truth value

Argument and Syllogism

In colloquial terms, we often think of an "argument" as a contentious disagreement, ranging from an exchange of insults to open warfare.  In logic, however, the term "argument" has nothing to do with heated or violent confrontation, but applies instead to a collection of statements systematically arranged to support or refute a particular idea.  Indeed, a logical argument might not involve any sort of disagreement whatever.  However, if and when it does, the organized presentation of the opposing arguments—the points of view and their respective supporting material—is called a "debate," not an argument.

A simple but very useful form of logical argument is the syllogism.  Syllogism is a very old tool of logic, dating back at least to the Classical period of Greece.  A syllogism is a formal grouping of three statements, such that two of them, called the major and minor premises, lead to the third, called the conclusion.  Here's an example:

• Major premise: "All trees are plants."

• Minor premise: "All maples are trees."

• Conclusion: "All maples are plants."

Note how the conclusion inevitably derives from the two premises.  As long as both of the premises are true, then a conclusion based on both of them should also be true.  Also note that the statements could be in any sequence.  The minor premise could be stated before the major premise, and the conclusion could be stated at the outset, or between the two premises, instead of at the end.

• Conclusion: "All maples are plants."

• Minor premise: "All maples are trees."

• Major premise: "All trees are plants."

What determines which statement is which is not their sequence, but their individual functions, and their relationship to each other.  The major premise makes a general statement, while the minor premise makes a particular statement, and the conclusion derives from the two premises considered together.

However, also note that the truth value of each statement is affected by precisely how it is stated. While "All trees are plants" evaluates as true, "All plants are trees" is clearly false.  Even so, it is entirely possible for false premises to yield a true conclusion.

• Minor premise: "All maples lay eggs." [False.]

• Major premise: "All egg-laying things are plants." [False.]

• Conclusion: "All maples are plants. [True.]

In this example, the absurdity is obvious, but in actual practice it is usually subtle.  In some cases, the error might be accidental, the result of honest mistake; but in others it might be intentional, constituting a deliberate attempt to mislead.  (Indeed, it is not unheard of for arguers to mislead themselves through such errors!)

Another thing of which to be wary is a so-called "conclusion" which does not actually derive from the premises.

• All trees are plants. [True.]

• All maples are trees. [True.]

• God made all plants. [Irrelevant to premises.]

Regardless of whether a statement is true or false, if it is not logically supported by the premises, then it cannot be considered a valid conclusion, and so the whole does not hold together as a syllogism. (This logical error, or fallacy, is specifically known by the Latin term non sequitur, which means "not following."  We'll look at several common types of fallacy in another section.)

Syllogisms, by the way, can be separated into different groups. The kind we have been considering so far makes its point by observing whether things belong in the same or different categories, and is appropriately named categorical syllogism.  Two other kinds of syllogism we are likely to encounter fairly frequently are constructive and disjunctive syllogisms.

Constructive syllogism cascades two related conditional statements, such that the result of the second is seen to be the necessary outcome of the first as well.  Notice that the conclusion of this kind of syllogism is also a conditional statement.

• Major premise: If Spasky and I play chess, I will lose.

• Minor premise: If I lose, I will be disappointed.

• Conclusion: If Spasky and I play chess, I will be disappointed.

Whenever the effect of the one condition becomes the cause of another, then it becomes logically possible to bypass the "middle-man" element.  That's not to say we can actually eliminate it, since it is an integral part of the process.  But as long as we know that two conditional statements always interact in this fashion, such that the cause of the first always produces the effect of the second, then we can derive a third conditional statement that expresses this overall relationship.

Disjunctive syllogism can be viewed as a process of elimination.  It considers a choice between two statements, at least one of which must be true.  If one of those statements is determined to be false, then the other must be true.

• Premise: Either we must ride the bus or we must ride the subway.

• Premise: We must not ride the subway.

• Conclusion: We must ride the bus.

Obviously, the major premise must be true for the rest of the argument to follow.  If for some reason we must not ride either the subway or the bus, then the first premise is false, and the conclusion is no longer adequately supported.

Not all groups of three statements constitute syllogisms, of course.

• I came.

• I saw.

• I conquered.

These statements are not designed or arranged so that two of them provide logical support for the third.  They do not conclude something on the basis of premises, but merely relate a sequence of events.  Such a grouping is not an argument, but a narrative.  A long, complex argument might include many related syllogisms, and these might be interspersed with statements that function neither as premises nor as conclusions: background, clarification, illustration, and testimony, for example.

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Terms:

• argument

• conclusion

• condition

• premise (major, minor)

• syllogism (categorical, constructive, disjunctive)

Validity and Soundness

There are good arguments and bad arguments. However, this is not judged subjectively, but is determined by specific criteria. One of these is validity, which indicates whether an argument is consistent within itself.

An argument is considered valid if, whenever all of its premises are true, its conclusion is also true.

• Premise: Plants are not animals. [True]

• Premise: Trees are plants. [True]

• Conclusion: Trees are not animals. [True]

If all of an argument's premises are true but its conclusion is false, then the argument is invalid.*

• Premise: Some insects sting. [True]

• Premise: Some insects are fireflies. [True]

• Conclusion: Fireflies sting. [False]

*[NOTE: In logic, the word invalid is an adjective, pronounced "in-VA-lid," with the second syllable stressed. It should not be confused with the noun, pronounced "IN-va-lid."]

Before we go further, it's appropriate to make a point about conditional statements, a point that is important, not only to the current discussion, but throughout the study of logic.  We have seen that arguments contain premises that in reality might be either true or false, depending on circumstances.  While we can accept "All fathers are male" as a universally true statement, "I am your father" might be true or false, depending on to which specific individuals the pronouns "I" and "you" happen to refer.  But some arguments have premises that are clearly not true, and perhaps could never be true in reality, for example, "If a horse has a horn, then the animal is called a unicorn."  Although (as far as we know) no horses have horns, evaluating an argument containing such a premise requires that we momentarily assume—"for the sake of argument," as they say—that some horses do have horns.  So when we consider "IF all premises are true," we must ignore for the moment that a premise is inherently false, and hypothetically consider what would be the effect on the argument if somehow the premise in question were true.  This is a recurring theme in logic, particularly useful when an arguer intends ultimately to show that an idea is false.

Now, on with validity, and beyond!  In the following example, one of the premises is false, and leads to a false conclusion.

• Premise: All weasels have fur. {True]

• Premise: Some weasels are birds [False]

• Conclusion: Some birds have fur. [False]

Believe it or not, this argument is valid!  Remember that the question of validity rests not on whether all premises are in fact true, but rather on what would be the result if they were hypothetically true.  As to the issue at hand, if we suppose that if we lived in an imaginary universe where some weasels really were birds, then both of the argument's premises would be true.  And considering the logical consequences of that, the conclusion in our example would also be true in that imaginary universe of weasel-birds.  Thus this argument satisfies the one criterion for validity: For an argument to be valid, it is necessary only that its logic be self-consistent, not that it make only true statements or conclude something true about the real world.  But, as we might suppose, though validity is an important factor in logic, it is not by any means the only one.

To ensure that an argument reaches a true conclusion, it must satisfy not only the test of validity, but also a further condition called soundness.  An argument is said to be sound if all of its premises are true, and if the argument as a whole is valid.

• Premise: All squirrels are mammals. [True]

• Premise: Some squirrels are albinos. [True]

• Conclusion: Some mammals are albinos. [True]

This argument is valid (the conclusion is true if all premises are true), and both of its premises are indeed true; so it is sound.

Because an argument can be sound only if those two conditions are true, then if either is not true (that is, if either the argument is invalid, or any of its premises are false), then it is unsound.  An argument can be unsound even if its conclusion happens to be true.

• Premise: No men can become pregnant. [True]

• Premise: All women can become pregnant. [False]

• Conclusion: No women are men. [True]

The foregoing argument is valid, even though the second premise is false, because the conclusion happens to be true, and would still be true (and furthermore well supported) if the faulty premise were true.  However, because the argument in fact contains a false premise, the argument as a whole is unsound.

• Premise: All birds are warm-blooded and have wings. [True]

• Premise: All bats are warm-blooded and have wings. [True]

• Conclusion: All bats are birds. [False]

In this case, both premises are true, but the conclusion is false, so the argument is invalid, and therefore unsound.  As we see, an argument can be valid without being sound, but any sound argument must also be valid.

Because logic is such an abstract subject, presenting it in a visual format can help to clarify relationships and reinforce concepts.  We can set up a table showing the necessary conditions for validity and soundness, with a YES for any condition that must be true, a NO for any condition that must not be true, and a blank cell for any condition that doesn't matter. The table illustrates exactly what we've discussed so far, and allows us to summarize briefly.

As we see, validity depends upon only one condition: that if all of the premises are true then the conclusion must inevitably be true as well—in other words, whether the entire argument is consistent within itself.  Whether or not the premises actually are true, or even if they cannot be true, has no bearing at all on an argument's validity.

Soundness is a very different matter, for it depends upon two conditions: that the argument is valid, and that all of the premises are true.  Since both conditions must be met in order for for the argument to be sound; if either condition is not met (regardless of the other condition), then the argument is unsound.

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Terms:

• soundness / sound / unsound

• validity / valid / invalid

Strength and Cogency

At this point we might guess that validity and soundness are imperative in creating a good argument.  But that's not necessarily true.  So far we have dealt with arguments whose premises have no qualifying stipulations.  But that isn't generally representative of real life.  Real life is full of possibilities and probabilities, but comes up rather short on absolutes and universals.  Premises can have conditions that might make a conclusion true some or even most of the time, but not all of the time.

• Premise: Nearly all cars manufactured since 1965 have heaters.

• Premise: Sophia's car was manufactured since 1965.

• Conclusion: Sophia's car has a heater.

Note that the first premise has the qualifier "nearly," and this makes the argument somewhat inconclusive.  There is the possibility that any randomly chosen car manufactured since 1965 might not have a heater.  If Sophia's car happens to be one of those few without, then the conclusion is false, and the possibility of a false conclusion renders the argument unsound.  However, the chance that Sophia's happens to be one of those few cars built without heaters is vanishingly slim, so despite its unsoundness the argument is strong.  It's a safe bet that Sophia's car has a heater; still, we must stop short of saying it's a sure thing.

This argument is not sound, since in some cases its conclusion could turn out to be false even if all the premises are true. Nevertheless, it is a strong argument, since all true premises should result in a true conclusion in "nearly" every case.  The strength of an argument is relative to the probability that the conclusion is true if all the premises are true.  If that probability is greater than 50 percent but less than 100 percent, the argument is said to be strong (in a range of slightly strong to very strong).  But if it is less than 50 percent, then the argument is weak.  Since strength varies with probability, a probability of 95 percent or more might be quite convincing, but a probability of 51 to 55 percent, while still technically on the "strong" side of the borderline, generates hardly enough strength to make the argument worth making.

There is yet another quality of arguments called cogency.  If an argument is strong (even if it is invalid for some reason), and if all of its premises are true, then it is said to be cogent.  Now, if we review the previous argument about Sophia's car, we see that it is indeed cogent, since it is strong and both of its premises are true.  If either condition for cogency is lacking (i.e., either the argument is weak, or it contains at least one false premise, or both), then it is said to be uncogent.

At this point we might have noticed a parallel between cogency and soundness, in that each depends upon some other quality plus all true premises.  Recall that a sound argument must be valid with all true premises.  In the same way, a cogent argument must be strong with all true premises.  While we are comparing, we might also notice a contrast.  On one hand, validity and soundness are either "on" or "off;" there is no in-between.  On the other, strength is a relative quality; there is a continuum of values between very strong and very weak.  (This comparison isn't particularly important from a logic standpoint, but it might make the concepts a little easier to learn.)

Now we can add strength and cogency to our previous table, for a single-glance summary of argument qualities and their respective requirements.  Each quality is color-coded with its opposite. Again, YES indicates a condition that must be true, NO indicates a condition that must not be true, and a blank cell indicates that a condition is irrelevant. If more than one condition must be met, then the value for each condition is shown in the same column; if one condition alone tips the balance, it is noted in a column by itself.

While the above table clarifies which conditions must or must not be true in each instance, the overall relationships are perhaps better appreciated in a different format (with the same color-coding, for consistency's sake).

Note that cogency trumps strength (since strength is a requisite of cogency), and that soundness trumps all other considerations, in the final analysis of how much confidence we may have that an argument's conclusion is true if all of its premises are true.  This is because strength is based on degrees of probability less than 100 percent.  In contrast, soundness either is there or it is not; and when it is, the argument is conclusive.

Generally, we want to stick to sound and cogent reasoning, and to avoid unsound and uncogent lines of thinking as much as possible.  If someone frequently uses unsound and uncogent argumentation, it might simply be that his or her reasoning skills need some tuning up.  However, if, having strengthened these skills, an arguer still finds that he or she must fall back upon poor logic to defend a position, then that's probably a clue that the position itself is in serious need of rethinking.

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Terms:

• cogency / cogent / uncogent

• strength / strong / weak

Deductive versus Inductive Reasoning

It is often said that deductive reasoning proceeds from the general to the particular, while inductive reasoning proceeds from the particular to the general.  While this is not entirely true, it does serve to highlight some of the overall effective differences between the two.

Deductive reasoning uses premises to derive conclusions.  If the premises are true, and if the argument is valid, and if the reasoning is sound, then the conclusion is necessarily true, and is said to be proved.  (Obviously, if some error or oversight creeps into the process unnoticed, then what is "proved" might well turn out to be false.  However, meticulous and unbiased scrutiny can usually keep the chances for error within defined limits.)

Inductive reasoning interpolates or extrapolates specific information in an attempt to formulate general principles.  As part of the process, it considers the relative strength and cogency of arguments.  By its nature, induction implies rather than proves.  Typically, its results must be verified empirically to determine whether or not they are indeed true.

Both deductive and inductive reasoning employ logic, but they use it in different ways.

• Deduction applies logic to accepted evidence and premises, in order to derive conclusions deserving a level of confidence equal to that deserved by the weakest of the premises.

• Induction applies logic to a combination of speculative and accepted premises, in order to develop hypotheses, which can then be independently tested to determine whether or not they deserve our confidence.

Inductive reasoning is the sort employed most often by inventors and theorists, in an effort to develop and test new ideas using existing knowledge as a base.  Deductive reasoning is the very pure sort, as exemplified by the abstractions of statement logic and predicate logic.  But with just a little practice and discipline, both can also be extremely useful to ordinary people in their day-to-day problem-solving and decision-making.

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Terms:

• deduction / deductive

• induction / inductive