Denying the antecedent

Denying the antecedent is a fallacy in formal logic where in a standard if/then premise, the antecedent (what comes after the ‘if’) is made not true, then it is invalidly concluded that the consequent (what comes after the ‘then’) is not true.  The fallacy confuses the directionality of logical relationships.

The structure of the fallacy takes the following form:

If P, then Q.

Not P.

Therefore, not Q.

Because this structure does not state that Q is exclusively a condition of P, it is invalid to deduce that Q is not true if P is not true. In most cases, there are other reasons that Q could be false.  One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example:

If it is raining, then the grass is wet.

It is not raining.

Therefore, the grass is not wet.

The conclusion is invalid because there are other reasons why the grass could be wet at the time (someone could have watered it).  To give another example:

Any person who is hopping on one foot, must be alive.

A sleeping person is not hopping on one foot.

Therefore, all sleeping people are dead.

An implication for skepticism is that quacks and conspiracy theorists sometimes use this fallacy in their propaganda, for example:

If pharmaceutical companies were always honest, then we could trust drug X.

Pharmaceutical companies are not always honest,

Therefore, we cannot trust drug X.

Recognising and exposing such a denial of the antecedent could be helpful to skeptics in our debates against our opponents.

Non sequitur

Non sequitur (Latin for “it does not follow”), in formal logic, is an argument in which its conclusion does not follow from its premises. In a non sequitur, the conclusion could be either true or false, but the argument is fallacious because there is a disconnection between the premises and the conclusion.

All invalid arguments are special cases of non sequitur.  Many types of known non sequitur argument forms have been classified into various logical fallacies, such as Affirming the Consequent, Begging the question and  Fallacies of Composition and Division.

The term has special applicability in law, having a formal legal definition.

 

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What is logic?

The word ‘logic‘ is not easy to define, because it has slightly different meanings in various applications ranging from philosophy, to mathematics to computer science. In philosophy, logic determines the principles of correct reasoning. It’s a systematic method of evaluating arguments and reasoning, aiming to distinguish good (valid and sound) reasoning from bad (invalid or unsound) reasoning.

The essential difference between informal logic and formal logic is that informal logic uses natural language, whereas formal logic (also known as symbolic logic) is more complex and uses mathematical symbols to overcome the frequent ambiguity or imprecision of natural language. Reason is the application of logic to actual premises, with a view to drawing valid or sound conclusions. Logic is the rules to be followed, independently of particular premises, or in other words using abstract premises designated by letters such as P and Q.

So what is an argument? In everyday life, we use the word ‘argument’ to mean a verbal dispute or disagreement (which is actually a clash between two or more arguments put forward by different people). This is not the way this word is usually used in philosophical logic, where arguments are those statements a person makes in the attempt to convince someone of something, or present reasons for accepting a given conclusion. In this sense, an argument consist of statements or propositions, called its premises, from which a conclusion is claimed to follow (in the case of a deductive argument) or be inferred (in the case of an inductive argument). Deductive conclusions usually begin with a word like ‘therefore’, ‘thus’, ‘so’ or ‘it follows that’.

A good argument is one that has two virtues: good form and all true premises. Arguments can be either deductive, inductive  or abductive. A deductive argument with valid form and true premises is said to be sound. An inductive argument based on strong evidence is said to be cogent. The term ‘good argument’ covers all three of these types of arguments.

Deductive arguments

A valid argument is a deductive argument where the conclusion necessarily follows from the premises, because of the logical structure of the argument. That is, if the premises are true, then the conclusion must also be true. Conversely, an invalid argument is one where the conclusion does not logically follow from the premises. However, the validity or invalidity of arguments must be clearly distinguished from the truth or falsity of its premises. It is possible for the conclusion of a valid argument to be true, even though one or more of its premises are false. For example, consider the following argument:

Premise 1: Napoleon was German
Premise 2: All Germans are Europeans
Conclusion: Therefore, Napoleon was European

The conclusion that Napoleon was European is true, even though Premise 1 is false. This argument is valid because of its logical structure, not because its premises and conclusion are all true (which they are not). Even if the premises and conclusion were all true, it wouldn’t necessarily mean that the argument was valid. If an argument has true premises and its form is valid, then its conclusion must be true.

Deductive logic is essentially about consistency. The rules of logic are not arbitrary, like the rules for a game of chess. They exist to avoid internal contradictions within an argument. For example, if we have an argument with the following premises:

Premise 1: Napoleon was either German or French
Premise 2: Napoleon was not German

The conclusion cannot logically be “Therefore, Napoleon was German” because that would directly contradict Premise 2. So the logical conclusion can only be: “Therefore, Napoleon was French”, not because we know that it happens to be true, but because it is the only possible conclusion if both the premises are true. This is admittedly a simple and self-evident example, but similar reasoning applies to more complex arguments where the rules of logic are not so self-evident. In summary, the rules of logic exist because breaking the rules would entail internal contradictions within the argument.

Inductive arguments

An inductive argument is one where the premises seek to supply strong evidence for (not absolute proof of) the truth of the conclusion. While the conclusion of a sound deductive argument is supposed to be certain, the conclusion of a cogent inductive argument is supposed to be probable, based upon the evidence given. Here’s a classic example of an inductive argument:

1. Premise: Every time you’ve eaten peanuts, you’ve had an allergic reaction.
2. Conclusion: You are likely allergic to peanuts.

In this example, the specific observations are instances of eating peanuts and having allergic reactions. From these observations, you generalize that you are probably allergic to peanuts. The conclusion is not certain, but if the premise is true (i.e., every time you’ve eaten peanuts, you’ve had an allergic reaction), then the conclusion is likely to be true as well.

Whilst an inductive argument based on strong evidence can be cogent, there is some dispute amongst philosophers as to the reliability of induction as a scientific method. For example, by the problem of induction, no number of confirming observations can verify a universal generalization, such as ‘All swans are white’, yet it is logically possible to falsify it by observing a single black swan.

Abductive arguments

Abduction may be described as an “inference to the best explanation”, and whilst not as reliable as deduction or induction, it can still be a useful form of reasoning. For example, a typical abductive reasoning process used by doctors in diagnosis might be: “this set of symptoms could be caused by illnesses X, Y or Z. If I ask some more questions or conduct some tests I can rule out X and Y, so it must be Z.

Incidentally, the doctor is the one who is doing the abduction here, not the patient. By accepting the doctor’s diagnosis, the patient is using inductive reasoning that the doctor has a sufficiently high probability of being right that it is rational to accept the diagnosis. This is actually an acceptable form of the Argument from Authority (only the deductive form is fallacious).

References:

Hodges, W. (1977) Logic – an introduction to elementary logic (2nd ed. 2001) Penguin, London.
Lemmon, E.J. (1987) Beginning Logic. Hackett Publishing Company, Indianapolis.

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Reasoning

Rationality may be defined as as the quality of being consistent with or using reason, which is further defined as the mental ability to draw inferences or conclusions from premises (the ‘if – then’ connection). The application of reason is known as reasoning; the main categories of which are deductive and inductive reasoning. A deductive argument with valid form and true premises is said to be sound. An inductive argument based on strong evidence is said to be cogent. It is rational to accept the conclusions of arguments that are sound or cogent, unless and until they are effectively refuted.

A fallacy is an error of reasoning resulting in a misconception or false conclusion. A fallacious argument can be deductively invalid or one that has insufficient inductive strength. A deductively invalid argument is one where the conclusion does not logically follow from the premises. That is , the conclusion can be false even if the premises are true. An example of an inductively invalid argument is a conclusion that smoking does not cause cancer based on the anecdotal evidence of only one healthy smoker.

By accident or design, fallacies may exploit emotional triggers in the listener (e.g. appeal to emotion), or take advantage of social relationships between people (e.g. argument from authority). By definition, a belief arising from a logical fallacy is contrary to reason and is therefore irrational, even though a small number of such beliefs might possibly be true by coincidence.