Logic Glossary

Logic Glossary

Affirming the consequent

Like denying the antecedent, affirming the consequent is a formal fallacy. The fallacy lies solely in the form itself. It has the following pattern: if p then q, q, therefore p. Any argument that fits this pattern is invalid, that is, even if the premises are true, the conclusion that follows from these premises may not be true. Whereas, a valid form guarantees that, if the premises are true, the conclusion will be true. Indeed, if an argument has a valid form and true premises, then it is impossible for the conclusion to be false.
An argument is a piece of reasoning with one or more premises and a conclusion. Arguments are usually divided into two kinds, deductive and inductive. So defined, an argument is to be distinguished from a disagreement. One may use an argument, in the logician’s sense, in order to win an argument, in the everyday sense of a dispute. Clearly the logician’s “argument” is not as dramatic as a verbal fight. For an example of an inductive argument see argument from analogy; for an example of a deductive argument see hard determinism.
Argument from analogy
An argument from analogy is an argument that has the form:

All P are like Q
Q has such-and-such characteristic.
Thus P has such-and-such characteristic.

Thus, for example, a few years ago one Republican congressman, who had been a fighter pilot during the Vietnam War, argued in a caucus prior to the election of the Speaker of the House:

Not voting to re-elect Newt Gingrich would be like abandoning your wingman.
Abandoning your wingman is wrong.
So not voting to re-elect Newt would be wrong.

One evaluates such an argument by examining the analogy. It is a weak analogy, and thus fallacious, if there are not many similarities. For instance, in this example there is some similarity between the two situations. The Congressman no doubt felt that with Speaker Gingrich having been charged with ethics violations that he was under attack as a fighter pilot’s wingman could be. But there are also dissimilarities. Voting for Speaker of the House is not a life-or-death situation. Moreover, n combat, one neither gets to choose one’s wingman nor one’s mission. Yet it is the obligation of a congressman to vote for the officers of the House of Representatives as s/he sees fit.
Here’s a stronger analogy:

Premise: Learning logic is like learning a foreign language.
Premise: You can’t learn a language by cramming; you have to study it regularly.
Conclusion: You can’t learn logic by cramming; you have to study it regularly.

Notice the form is the same for a weak or a strong analogy. What makes a weak analogy fallacious is not the pattern of reasoning but a lack of compelling similarities to warrant the alleged one.
A conclusion is the supported claim that is being made. In an argument one expects that a claim will be supported with reasons or premises. Moreover, these premises will be true and will, in fact, lead to the conclusion. Hence arguments can be evaluated as to how well they do this: Are the premises true? Is the reasoning good?
A conditional statement is an if-then statement and consists of two parts, an antecedent and a consequent. The antecedent, or that which goes before, is preceded by the “if”; the consequent, or that which comes after, may be preceded by a “then”. English sentences sometimes reverse the order: John studies hard if he thinks that he will do well in a class. But the logic of this sentence is: If John thinks that he will do well in a class, then he studies hard. Here the antecedent is “John thinks that he will do well in a class” and the consequent is “he studies hard”.
Consistency is much prized in reasoning. Ideally, one would like for one’s beliefs to fit together without any contradictions. Consistency is the intuitive notion that is the basis for the understanding of validity: we expect true premises to lead to a true conclusion. When we find that we have true premises and a false conclusion we lack consistency between premises and conclusion and know that the argument form is invalid.
A contradiction occurs when one asserts two mutually exclusive propositions, such as, “Abortion is wrong and abortion is not wrong.” Since a claim and its contradictory cannot both be true, one of them must be false. Few people will assert an outright contradiction, but one may fall into an inconsistency.
A counterexample is an example that runs counter to (opposes) a generalization, thus falsifying it. A TV newscast that limited its coverage of “mayhem and misery” (in Bob Inman’s phrase) would falsify a claim that all local TV newscasts focused on crime and disasters. Consequently, careful thinkers avoid rash generalizations (see hasty generalization) by qualifying their generalizations. If there are local TV newscasts that do not focus on “mayhem and misery,” one could say, “Most local TV newscasts focus on “mayhem and misery.”
A deductive argument is one that derives the truth of the conclusion from the truth of the premises. If the argument form, or structure of the argument, is valid, then the conclusion will always follow from the premises. The hard determinism argument below is an example of a deductive argument that makes use of two modus ponens arguments in which the conclusion of the first serves as the premise of the second, or so it appears.
Denying the antecedent
Denying the antecedent, like affirming the consequent, is a formal fallacy. Denying the antecedent has the following form, or pattern: if p then q, not-p, therefore not-q, or

if p then q

Both formal fallacies are easily confused with two valid argument forms: modus ponens and modus tollens. Here is an analysis of the four forms according to affirmation-denial and antecedent-consequent:

antecedent consequent
affirm (1)modus ponens (2)affirm the consequent
deny (3)deny the antecedent (4)modus tollens

(1) and (4) are valid argument forms; (2) and (3) are invalid.

In popular use a dilemma can be almost any sort of difficult choice, but in logic a dilemma is a choice in which there are only two options, attractive or not. One can refute a dilemma, that is, show that is not a real dilemma, by finding a third possibility.
Disjunctive Syllogism
If there are only two possibilities, one of which is true, and then, if one is eliminated, the remaining one is true. Hence the following argument form:

Either X or Y
Not X
Therefore Y

This form of argument is a disjunctive syllogism. It is a syllogism, that is, an argument with two premises, and one of the premises is a disjunction. Here is an ordinary language example:

Either you pass logic or you do not graduate.
You will not pass logic.
Therefore you will not graduate.

Fortunately, the first premise is not true. Hence the argument, while valid is not sound.
From a Greek word meaning “to experiment,” it is used by philosophers to mean that which has to do with sense experience.
Empirical generalization
Empirical (or inductive) generalizations are general statements based upon experience.

Most student desks in older classroom buildings at UNC Charlotte have gum stuck underneath the desk tops.

A good generalization will be developed from a large number of varied experiences. For instance, one could offer as a justification for the previous generalization:

I’ve looked underneath several desks in several classrooms.

Generalizations drawn from a small number of instances or from anecdotal evidence are said to be hasty generalizations.
An explanation identifies the cause of an event, thus answering the question why something is what it is or why it occurs. Historical explanations show how something came to be what it is. For instance, Old Shell Road in Mobile got its name because at one time the street was paved with shells dredged from Mobile Bay. A scientific explanation identifies the conditions that must be present for something to occur. For instance, an explanation of why matches light would identify, among other things, the presence of oxygen, a phospherous tip, a wooden stick and friction.The following example, contributed by Lee-Marie Davis, a student in one of my critical thinking classes, explains why a particular explanation is an explanation:

Explanations identify causal relationships. They tell why or how something happens. The following is an example of an explanation:My father was diagnosed with lung cancer two years ago. Of course, one of the very first questions out of his mouth was, “Why did this happen?” The doctor explained to my dad that he fit into three categories of risk factors that contribute to the onset of cancer in some patients. The first category that the doctor said my did fit into was that he had a history of cancer in his family. The second category was that my dad had smoked for almost 30 years, and the third category was that my dad had gone through a period of high stress.
This is an explanation because the doctor tells my dad why he had cancer. The doctor gives him three reasons that had put my dad at risk for lung cancer. He told him that he fit into the risk categories of family history, high stress and was a smoker. The explanation of the doctor helped my dad better understand why he had the cancer by telling him the cause of the cancer.

A fallacy is an attractive but unreliable piece of reasoning, or affirming the consequent and denying the antecedent. Informal fallacies include begging the question, composition, division, equivocation, false cause, false dichtomy, hasty generalization, personal attack, red herring, slippery slope, straw man, weak analogy. There are many other examples of bad reasoning that have been identified by logicians, but these are enough to illustrate the idea of a fallacy.
Arguments often exhibit one or more reasoning patterns. These patterns, such as modus ponens or an argument from analogy, are called forms and are to be distinguished from the content of the actual argument. Just as a coffee cup or mug has a distinctive shape and is distinguishable from what you put it (the coffee or content), so argument forms are identifiable and not to be confused with the actual premises and conclusions used.
Hard determinism
Determinism is the view that all events are caused. One form of determinism, one that pushes the notion of universal causation to unacceptable consequences, is hard determinism. Here, in summary form, is an argument for this extreme view. I offer it as an example of a flawed deductive argument:

1. All events are caused.2. If all events are caused, then there are no free actions.
3. There are no free actions (from 2 and 1 by modus ponens).
4. If there are no free actions, then there is no personal responsibility.
5. There is no personal responsibility (from 4 and 3, once again, by modus ponens).

There is nothing apparently wrong with the form of this argument, for modus ponens is a valid argument form. Unless one is prepared to accept the consequences that we lack both freedom and responsibility, then one must find some other error.
Hasty Generalization
A generalization based on too little or unrepresentative data. The relevant rule that it violates is: Generalizations should be based on a large number of various representative examples. Here is a note I once received from a student (the names have been changed):

Mr. Eldridge, As you notice, I was not in class Thursday, due to the flu. I gave my paper to Justin because he was going to class. On Sunday, I found out he did not attend class. Here is my revised paper. Let’s hope this will work! Now, I’ve learned not to trust other people. Laura Walker

Hasty generalizations should not be confused with the fallacy of composition. In a hasty generalization one infers a general statement on the basis of an atypical instance; whereas in the fallacy of composition you take something that is true of each of the parts and attribute to the whole. Composition, like division, confuses distribution and collectivity (whether something is considered individually or as a whole); hasty generalizations infer something to be true generally on the basis of a limited number of unrepresentative instances.
Inconsistency is to be avoided, for it indicates error. It is an implicit contradiction. An inconsistent set of statements will not be an outright contradiction but will lead to one. For example, if one declares:

All UNC Charlotte students are hardworking.
Jim Schwartz is a UNC Charlotte student, and
Jim Schwartz is lazy,

then s/he is being inconsistent. There is no contradiction here, such as,

Jim Schwartz is hardworking and Jim Schwartz is lazy,

but, clearly, there is an inconsistency. For if all UNC Charlotte students are hardworking, then it is impossible for Schwartz to be a UNC Charlotte student and not be hard-working. It is implicitly contradictory to say that Schwartz is UNC Charlotte student (and thus hard-working) and to claim that he is lazy, that is, not hard-working. See consistency.
Unlike deductive arguments, inductive ones promise only probability, not certainty. Thus, if one argues that having watched several different newscasts in several different cities on many different nights one may infer that newscasts emphasize, in Bob Inman’s phrase, “mayhem and misery”, then one is making an inductive argument. (In this case, an inductive (or empirical) generalization. Another kind of inductive argument is an argument from analogy. Inductive arguments are judged by their reliability, where one expects only a high degree of probability, not one hundred percent reliability as with deduction.
Logic is the study of correct reasoning. It both describes and evaluates the way in which we draw inferences. Inferences are formulated as arguments and then evaluated as to their validity and soundness. The aim is to find generally reliable (see inductive) or always reliable (see deductive) arguments. Although logicians describe our reasoning patterns, this task is more properly the work of psychologists. The logician’s primary concern is normative–how we should reason. The value of this ancient enterprise, which can be traced back to Aristotle and his predecessors, notably Zeno of Elea, is well expressed by the British philosopher, Patrick Shaw, in the preface to Logic and Its Limits (Oxford University Press, 1997):

         Most of the time, the ordinary person does think straight. In countless ways social life depends on doing so. Balancing the housekeeping money, locating a fault in a wiring system, planning a day out–all involve, tacitly or otherwise, working out what is compatible with what. I cannot spend this pound and save it; if the bulb works in another socket then the fault does not lie in the bulb; either we catch the five o’clock train or we will not be able to get to the concert. These are the kind of commonplaces that underpin any sort of planned, purposive behaviour. They are largely taken for granted, and any mistakes in reasoning quickly run up against the harsh corrective of experience.         Problems arise when the test of experience is neither so immediate nor overwhelming. People speculate on what the facts might be when the facts are not obvious; and they disagree in their speculations. Also people pronounce upon, and disagree about, what ought to be the case, or whether one thing is better than another. They are not disagreeing about what is the case, so they cannot appeal straightforwardly to experience.
         When these kinds of disagreement occur, when the competing claims cannot be easily and obviously tested, attention is bound to turn to the route by which a cotnroversial conclusion was reached. We are forced to become self-conscious about the reasoning process. How far reasoning will take us remains to be seen, but so far as it leads we must be sure that it is sound.

Modus ponens
valid argument form, not to be confused with affirming the consequent, modus ponens consists of a conditional statement and one other premise. The second premise affirms the antecedent of the conditional, yielding the consequent as the conclusion:

if p then q

Modus Tollens
valid argument form, modus tollens is not to be confused with denying the antecedent. Modus tollens consists of a conditional statement and one other premise. The second premise denies the consequent of the conditional, yielding the denied antecedent as the conclusion:

If p then q

Necessary and sufficient conditions
If event A must occur for event B to occur, then we say that A is necessary for B. If event A may cause B but there could be some other cause as well, then we say that A is sufficient to cause B.
Statements offered as reasons to support a conclusion are premises. Logicians generally pay more attention to the reasoning, that is, the relationship between premises and conclusion. They rely on scientists to determine the accuracy of the premises.
Salva Veritate
A Latin phrase which literally mean “saving truth”; salva veritate is used by logicians to express the concept of truth preservation, which is the test of a valid deductive argument. If a deductive argument does not preserve the truth of the premises (assuming they are in fact true), then it has an invalid argument form. Salva veritate is the necessary and sufficient condition for a valid argument form.
deductive argument is said to be sound if it meets two conditions: valid argument form and true premises. (Notice that validity and true premises constitute necessary and sufficient conditions for soundness.)
Every proposition is either true or false. This status is called “truth-value”.
Unstated premises
Not every argument is fully expressed. Sometimes premises or even conclusions are left unexpressed. If one argues that Rover is smart because all dogs are smart, he is leaving unstated that Rover is a dog. Here the unstated premise is no problem; indeed it would probably be obvious in context. But sometimes unstated premises are problematic, particularly if two parties in a discussion are making differing assumptions. If one person thinks violence depicted in the media encourages violent behavior and another does not, then an argument that proceeds as follows will be evaluated differently by the two parties:

There’s too much violence on TV.
No wonder we have so much violence among kids these days.

What will appear obvious to the person making these statements will not be so clear to the person who may be wondering what is the connection between the premise–there’s too much violence on TV–and the conclusion–no wonder we have so much violence among kids these days. Hence the need for critical awareness. One function of critical thinking is to make the reasoning under discussion explicit.
Validity is a characteristic of good deductive argument forms, those patterns which are one hundred percent reliable. It is impossible for a valid deductive argument with true premises to have a false conclusion. See soundness.
Venn diagrams
Diagrams developed by John Venn, an English logician, in 1881 to represent categorical propositions and categorical syllogisms. They consist of two (for propositions) or three (for syllogisms) overlapping circles and are commonly used in introductory logic courses to represent and test the validity of categorical syllogisms.
Weak Analogy
An argument that infers that because two objects or situations are alike, then what is true of the one is true of the other, yet fails to notice a telling difference between the two objects or situations.

No one objects to a physician looking up a difficult case in medical books. So no one should object to nursing students, when taking a logic exam, being permitted to use their reference materials.

A weak analogy is anargument from analogy; it is just not a very good one.