Tag Archives: deductive reasoning

Boy/girl hair puzzle

A boy and a girl are chatting.

“I am a boy”, said the child with black hair.

“I am a girl”, said the child with white hair.

At least one of them lied. What colour hair does the boy have?

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Post Office Burglary solution

Solution: Derek was the culprit.

Looking at Brian’s statement if it was Charles, then Brian was lying in his first statement, which makes the second statement true. Which would mean that it was both Charles and Alan. So it can’t be Charles.

Which means Derek was lying in his first statement, which makes the second statement true. Therefore it can’t be Alan.

So Eric’s second statement must be false, meaning his first statement was true, therefore it was Derek.

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Post Office burglary puzzle

After a local Post Office burglary, five suspects were being interviewed.

Below is a summary of their statements.

Police know that each of them told the truth in one of the statements and lied in the other.

From this information can you tell who committed the crime?

Brian said:
It wasn’t Charles
It was Alan

Derek said:
It was Charles
It wasn’t Alan

Charles said:
It was Brian
It wasn’t Eric

Alan said:
It was Eric
It wasn’t Brian

Eric said:
It was Derek
It was Alan

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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’s main concern is with the validity or cogency of arguments. 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.

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 deductiveinductive  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. An example of an inductive argument is: 

Premise 1: Almost all people are taller than 26 inches
Premise 2: George is a person
Conclusion: Therefore, George is almost certainly taller than 26 inches

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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Argument from authority

by Tim Harding

The Argument from Authority is often misunderstood to be a fallacy in all cases, when this is not necessarily so. The argument becomes a fallacy only when used deductively, or where there is insufficient inductive strength to support the conclusion of the argument.

The most general form of the deductive fallacy is:

Premise 1: Source A says that statement p is true.
Premise 2: Source A is authoritative.
Conclusion: Therefore, statement p is true.

Even when the source is authoritative, this argument is still deductively invalid because the premises can be true, and the conclusion false (i.e. an authoritative claim can turn out to be false).[1] This fallacy is known as ‘Appeal to Authority’.

The fallacy is compounded when the source is not an authority on the relevant subject matter. This is known as Argument from false or misleading authority.

Although reliable authorities are correct in judgments related to their area of expertise more often than laypersons, they can occasionally come to the wrong judgments through error, bias or dishonesty. Thus, the argument from authority is at best a probabilistic inductive argument rather than a deductive  argument for establishing facts with certainty. Nevertheless, the probability sometimes can be very high – enough to qualify as a convincing cogent argument. For example, astrophysicists tell us that black holes exist. The rest of us are in no position to either verify or refute this claim. It is rational to accept the claim as being true, unless and until the claim is shown to be false by future astrophysicists (the first of whom would probably win a Nobel Prize for doing so). An alternative explanation that astrophysicists are engaged in a worldwide conspiracy to deceive us all would be implausible and irrational.

“…if an overwhelming majority of experts say something is true, then any sensible non-expert should assume that they are probably right.” [2]

Thus there is no fallacy entailed in arguing that the advice of an expert in his or her field should be accepted as true, at least for the time being, unless and until it is effectively refuted. A fallacy only arises when it is claimed or implied that the expert is infallible and that therefore his or her advice must be true as a deductive argument, rather than as a matter of probability.  Criticisms of cogent arguments from authority[3] can actually be a rejection of expertise, which is a fallacy of its own.

The Argument from Authority is sometimes mistakenly confused with the citation of references, when done to provide published evidence in support of the point the advocate is trying to make. In these cases, the advocate is not just appealing to the authority of the author, but providing the source of evidence so that readers can check the evidence themselves if they wish. Such citations of evidence are not only acceptable reasoning, but are necessary to avoid plagiarism.

Expert opinion can also constitute evidence and is often accepted as such by the courts.  For example, if you describe your symptoms to your doctor and he or she provides an opinion that you have a certain illness, that opinion is evidence that you have that illness. It is not necessary for your doctor to cite references when giving you his or her expert opinion, let alone convince you with a cogent argument. In some cases, expert opinion can carry sufficient inductive strength on its own.


[1] If the premises can be true, but the conclusion can be false, then the argument is logically invalid.

[2] Lynas, Mark (29 April 2013) Time to call out the anti-GMO conspiracy theory.

[3] An inductive argument based on strong evidence is said to be cogent.

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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.

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