Aristotelian and mathematical logic

20.03.2015 |

Episode #5 of the course “Philosophical ideas that everyone should know”

Lasting through the 19th century, the science of logic took the path Aristotle had set it on more than 2000 years before. Syllogism became popular as a way to reason. Syllogism consists of two premises and a conclusion. The conclusion is directly derived from the premises. For example: “All men are mortal; humans are men; therefore, humans are mortal.”

The example given is a deductive argument. Here the conclusion is the logical result of the premises, and the argument seems “valid.” When the premises of logical arguments are true, then the conclusion is also true. The end of a deductive argument cannot go beyond the facts in the premises. As such, one cannot accept the premises and deny the conclusion.

Induction is another way to reach the conclusion from the premises. In inductive arguments, a general principle is taken from observing the world. For example, observing animals giving birth may lead to the assumption that all mammals give birth to their young. This argument is not deductively valid because the premises could be true and the conclusion false (for example, some mammals lay eggs). So, inductive reasoning goes beyond its premises in a way that deductive arguments do not. They are generalizations used to predict an outcome.

The philosopher David Hume claims that there are no valid grounds to use induction. This reasoning, he suggests, relies on assuming that the future will mimic the past if conditions are similar. However, nothing exists to support this. Similarly, people try to prove induction through past successes. But the assumption that something will always work relies only on past success, so the argument ultimately fails. In Hume’s view, humans cannot prevent inductive reasoning, but he insists that this thinking is not rational. Humes alleges that the “problem of induction” mainly affects science, but it applies to generalizations across many areas of thought.


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