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Encyclopedia > Problem of multiple generality

The problem of multiple generality names a failure in Aristotelian logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if: Aristotelian logic, also known as syllogistic logic, is the particular type of logic created by Aristotle, primarily in his works Prior Analytics and De Interpretatione. ...

Some cat is feared by every mouse

then it follows logically that:

All mice are afraid of at least one cat

However, it is possible to express this inference in Aristotle's system, with such expressions as the following:

There exists cat such that for all mouse, cat is feared by mouse.
For all mouse, there exists cat such that mouse fears cat.

When medieval logicians recognised this problem, they saw that it was possible to add further, more complex syllogisms to the theory to allow such inferences. They developed the "for all" and "there exists" statements to account for this.

The first logical calculus capable of dealing with such inferences was the above "regular" logic. Logic, from Classical Greek Î»ÏŒÎ³Î¿Ï‚ (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...

Modern research on term logic have shown how the problem may be solved in a syllogistic theory, as well. Traditional logic, also known as term logic, is a loose term for the logical tradition that originated with Aristotle and survived broadly unchanged until the advent of modern predicate logic in the late nineteenth century. ...

Results from FactBites:

 Problem of multiple generality - Wikipedia, the free encyclopedia (192 words) The problem of multiple generality names a failure in Aristotelian logic to describe certain intuitively valid inferences. When medieval logicians recognised this problem, they saw that it was possible to add further, more complex syllogisms to the theory to allow such inferences. Modern research on term logic have shown how the problem may be solved in a syllogistic theory, as well.
 General Linear Models (GLM) (13045 words) The general purpose of multiple regression (the term was first used by Pearson, 1908) is to quantify the relationship between several independent or predictor variables and a dependent or criterion variable. With regard to the generality of the multiple regression model, its only notable limitations are that (1) it can be used to analyze only a single dependent variable, (2) it cannot provide a solution for the regression coefficients when the X variables are not linearly independent and the inverse of X'X therefore does not exist. The general implication of the theory of estimability of linear functions is that hypotheses which cannot be expressed as linear combinations of the rows of X (i.e., the combinations of observed levels of the categorical predictor variables) are not estimable, and therefore cannot be tested.
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