Logic, Quantifiers and Generalizations

An examination
by Chris Hibbard

Logic, Quantifiers and Generalizations

Generalizations are broad statements or beliefs based on a limited number of facts, examples or statistics. They require inductive reasoning, taking specific or detailed facts and using them to extrapolate to form far more general principles.

Most generalizations require quantifiers to support their claim. Quantifiers are words in natural language that describe an amount. All, Some, Many, A Lot, Few, None, and One are all examples of quantifiers. These quantifiers can be countered by falsification conditions – counter-quantifiers or counter-examples that invoke ‘more’ or ‘less’ or the item which the quantifier is claiming,

For instance, a generalization that “all lions are cats”, is universal in scope. It would be an incorrect of faulty generalization were it to be reversed, now claiming that “all cats are lions”. This particular example is universal in scope through its use of a very narrow quantifier in the word “all”. Generalizations such as this (all, none, every, each) require absolutely no exceptions to the rule. If one were to come up with a single counter-example, such as locating a lion that was not a member of the feline family, then the generalization above would be automatically incorrect. Universal generalizations like this require little in the way of falsification conditions and are, in other words, quite easy to refute.

A decent rule to follow in language and logic is this: the broader the scope of the claim, the easier it is to refute. Claims that use quantifiers such as some, many, or most, are much more difficult to dispute than others such as every, all, or none. In claims that use the former it must be shown that there is tremendous evidence against such a claim, whereas in the latter, only one or two examples need be found to cancel the claim completely.

Quasi-quantifiers such as ceteris paribus and ceteris normalibus dictate that a claim may be true or false under standard or ‘normal’ conditions, with all things being equal. These often accompany prima facie generalizations; those which are accepted and permitted at face value – unless they are explicitly provable otherwise. Without these ‘normalized’ quasi-quantifiers, arguments and debate can easily reduce into ‘apples vs. oranges’ discussions.

While dealing with evaluations, (i.e. whether something is true or false), one must remember that evaluation presupposes classification, while classification presupposes personal standards. True or false can be described as corresponding to reality or failing to, much as right or wrong and good or bad can be seen as corresponding to ethics, standards, and societal conventions. But since evaluative properties such as these are neither empirical nor sensory; thereby lacking in ‘natural’ properties that can be scientifically observed and measured, then there must be non-natural properties involved, as well as some set of standards against which the claim is being measured or weighed. Since we can not observe such properties as are involved in notions of good and bad, we can only conclude that we know them some other way, be it intuition, education or indoctrination.

Consequently, generalizations such as ‘marijuana should be decriminalized’, ‘capital punishment is wrong’, and ‘abortion should be illegal’, are all quite broad in scope. Perhaps marijuana should be decriminalized in some countries but not in others. Perhaps in some circumstances, capital punishment truly does fit the crime. Perhaps abortion should be illegal, unless in extreme instances of rape or incest. These particular generalizations, all quite common issues today, do not conform to any empirical or sensory knowledge as such, nor do they take into account quasi-quantifiers such as ceteris paribus that would substantially alter the claim to allow for exceptions.

~ by Chris Hibbard on November 24, 2008.

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