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From Nordan Symposia
Jump to navigationJump to search→An example: the classic syllogism
[[Greek philosophy|Greek philosophers]] defined a number of [[syllogism]]s, correct three-part inferences, that can be used as building blocks for more complex reasoning. We'll begin with the most famous of them all:
[[Greek philosophy|Greek philosophers]] defined a number of [[syllogism]]s, correct three-part inferences, that can be used as building blocks for more complex reasoning. We'll begin with the most famous of them all:
All men are mortal
Socrates is a man
------------------
Therefore Socrates is mortal.
The reader can check that the premises and conclusion are true. The validity of the inference may not be true. The validity of the inference depends on the form of the inference. That is, a valid inference does not depend on the truth of the premises and conclusion, but on the formal rules of inference being used. In [[term logic|traditional logic]], the form of the syllogism is:
The reader can check that the premises and conclusion are true. The validity of the inference may not be true. The validity of the inference depends on the form of the inference. That is, a valid inference does not depend on the truth of the premises and conclusion, but on the formal rules of inference being used. In [[term logic|traditional logic]], the form of the syllogism is:
All A is B
All C is A
----------
All C is B
Since the syllogism fits this form, then the inference is valid. And if the premises are true, then the conclusion is necessarily true.
Since the syllogism fits this form, then the inference is valid. And if the premises are true, then the conclusion is necessarily true.
In predicate logic (a simple but useful formalization of [[Aristotle|Aristotelician logic]]), this syllogism can be stated as follows:
In predicate logic (a simple but useful formalization of [[Aristotle|Aristotelician logic]]), this syllogism can be stated as follows:
∀ X, man(X) → mortal(X)
man(Socrates)
-------------------------------
∴mortal(Socrates)
Or in its general form:
Or in its general form:
∀ X, A(X) → B(X)
A(x)
------------------------
∴B(x)
∀, the [[universal quantifier]], is pronounced "for all". It allows us to state a general property. Here it is used to say that "if any X is a man, X is also mortal".
∀, the [[universal quantifier]], is pronounced "for all". It allows us to state a general property. Here it is used to say that "if any X is a man, X is also mortal".
Consider the following:
Consider the following:
All fat people are musicians
John Lennon was fat
-------------------
Therefore John Lennon was a musician
In this case we have two false premises that implies a true conclusion. The inference is valid because it follows the form of a correct inference.
In this case we have two false premises that implies a true conclusion. The inference is valid because it follows the form of a correct inference.
An incorrect inference is known as a [[fallacy]]. Philosophers who study [[informal logic]] have compiled large lists of them, and cognitive psychologists have documented many [[cognitive bias|biases in human reasoning]] that favor incorrect reasoning.
An incorrect inference is known as a [[fallacy]]. Philosophers who study [[informal logic]] have compiled large lists of them, and cognitive psychologists have documented many [[cognitive bias|biases in human reasoning]] that favor incorrect reasoning.
[[Category: General Reference]]
==Automatic logical inference==
==Automatic logical inference==
