ImpreciseLanguage

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LoGic -- MeaningAndDefinitions

Ordinary folk need imprecise language much more than the precise language used by philosophers and scientists. The following is an article which bears on this as it reflects on the Sorites argument. It leans heavily on what I learned from [Max Black]'s Margins of Precision.

The Sorites is a form of argument in which the concept of consensus can dispose of a long standing confusion.

The standard form of this disturbing argument goes something like this.

1. A million grains of sand makes a heap.
2. Removing a single grain of sand from a heap doesn't destroy the heap.
3. A single grain of sand makes a heap.

Many philosophers and logicians (better prepared than I) have confronted this argument and registered their analysis. Some, like [Bertrand Russell], simply deny that logic works with vague concepts. Others go so far as destruction of all arguments of this form, including mathematical induction (which is not really a Sorites argument in my opinion).

My attempt to clarify matters goes as follows:

Many of the examples of this argument use words which refer to members of a vaguely defined set with an underlying quantitative scale which can be used to make precise analogs. For example, We could define a p-heap which has at least p grains of sand. We would then have a precise analog for which the Sorites argument would clearly fail because statement 2) above could not be applied to all p-heaps. There would be a least p-heap to which the item could be applied.

Consider the height form of the argument.

1. A man whose height is seven feet is tall.
2. Reducing the height of a tall man by one inch leaves him still tall.
3. A man whose height is four feet is tall.

And consider this

1. A man whose height is seven feet is considered tall by everyone.
2. Reducing the height of a man considered tall by consensus may change the consensus or not. If the reduction is small, then the consensus may only change slightly.
3. A man whose height is four feet is considered tall by very few people.

I believe that the usefulness of language is the consensus we share on the definitions of terms. Precise terms have a mechanism by which we can persuade someone that a specific application of the term is valid. Vague terms have no such mechanism. If a person insists on calling a seven foot man short, we might suspect that he is using NBA centers as his reference set, but we would hardly accuse him of a logic error. Vague terms are useful to the extent that we have consensus. When we use them out of context, we confuse each other.

The Sorites merely illustrates that the growth and destruction of consensus is a required element in the logical analysis of how we use vague language. It is a fallacy to treat a vague term as if everyone agreed with its definition. We may agree in its application to some but not all members of the universe of discourse.

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