Deduction and Induction. Validity


Deductive reasoning – spelling out whatever conclusion follows logically from your premises, without references to any external information.

Deductive proof – demonstrating that a particular conclusion logically follows from certain premises, and that this conclusion must be true if these premises are true.

Valid reasoning – correctly applying deductive reasoning in drawing out the logical conclusion of your premises.

Truth-preserving – a quality of a valid deductive argument stating that if used correctly, deductive reasoning is guaranteed to preserve the truth of its premises in its conclusion (if they are true in the first place).

Valid deductive reasoning should follow one of these structures:

Affirming the antecedent:
If A, then B.
It is A.
Therefore, B.
For instance:
All sailors drink heavily.
He is a sailor.
Therefore, he drinks heavily.
Denying the consequent:
If A, then B.
It is not B.
Therefore, not A.
All politicians lie.
She doesn’t lie.
She is not a politician.

Invalid reasoning – incorrectly applying deductive reasoning so that your conclusion does not logically follow from your premises.

Denying the antecedent:
If A, then B.
Not A.
Therefore, not B.
All sailors drink heavily.
He is not a sailor.
Therefore, he does not drink heavily.
Affirming the consequent:
If A, then B.
It is B.
Therefore, A.
All politicians lie.
She lies.
Therefore, she is a politician.

Unwarranted conclusion – a conclusion that is not supported by the argument.

Sound argument – a deductive argument that is both valid and has true premises, meaning its conclusion must also be true.

Unsound argument – an argument that does not meet the standard of soundness, either because it is invalid or because one or more of its premises is untrue, or both.

Valid VS True

Argument valid, conclusion not true:
All poets are English. Mayakovsky is a poet. Conclusion: Mayakovsky is English.

Argument invalid, conclusion true:
All poets are English. Mayakovsky is English. Conclusion: Mayakovsky is a poet.

Validity + truth: a sound argument.


Inductive reasoning – a form of reasoning in which premises strongly support a conclusion, but where we can never be absolutely certain that it is true.

Example: Every summer for 20 years my mum has gone somewhere on a vacation. I guess she will do the same this year.

Ranking inductive arguments – determining which arguments are
more or less convincing relative to one another.

Inductive strength or inductive force – a measure of how likely we believe an inductive argument is to be true.

Cogent – an inductive argument that has a good structure, but whose conclusion we should not necessarily accept as true (similarly to a valid unsound deductive argument).

Example: two years in a row, it was heavily raining on my birthday in July. I guess it will happen again this year.

Inductively forceful – an inductive argument that has both a good structure and true premises, and whose conclusion we thus have good reason to accept as true (similarly to a sound deductive argument, although without its certainty).

Example: there has never been a female US president. The next president is not likely to be a female either.

The problem of induction – no matter how likely we believe something to be, an inductive argument can never actually prove it to be true.

Counter-example – an example whose discovery makes it necessary to rethink a particular position, because it directly contradicts a generalization previously believed to be true.

Black swan – an event that defies both previous experience and expectations based on that experience, making it almost impossible to predict.

Rational expectation – whatever it would be most reasonable to expect in a particular situation; this can be quite different to what somebody personally expects.

Sample – the particular cases you are using to stand for the entire category about which you wish to make an inductive generalization.
Any inductive argument based on a single instance is likely to be very weak.

Representative samples closely resemble the larger group about which claims are being made, while unrepresentative samples fail to do so.

Sampling bias – biases introduced by imperfect methods of selecting a sample.

Observational error – errors due to the accuracy of your measuring system, usually reported as ±X, where X is the potential difference between measured and actual values.

Margin of error – an expression of the degree to which results based on a
sample are likely to differ from those of the overall population.

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