Me Talk Good One Day When Language and Logic Fail to Coincide

Content and Logic Classic propositional logic has rules expressed in symbols – content is irrelevant. For example, this is a truth table for the conditional, which is of the format “if A then B”.  B is trueB is false A is true A  B is trueA  B is false A is false A  B is true

Content and Logic Classic propositional logic also has rules of deduction. A  B  (A & C)  B For example, if it is true that “If I am happy then I am smiling”, then it is also true that “If I am happy and grass is green then I am smiling”.

Content and Logic This format doesn’t always hold in natural language, because the content of the symbols might cause the deduction to fail. An example: A match is struck  The match will light  A match is struck and the match is wet  The match will light

Presupposition It turns out that this is not a failure of logic in language, but rather an instance of presupposition. This can be thought of as a set of implied assumptions. Natural language lets A  B essentially say the same thing as (A & 1 & 2 &…)  B, and everyone makes the same assumptions about 1, 2, and so on.

Presupposition Therefore, the example of the match deduction was faulty, and would be equivalent to saying (A & C)  B  (A & ~C)  B which is simply not true. The addition of water would contradict one of the unstated assumptions. The rules of logic hold.

Mathematical Induction While it is known that a conclusion reached through ordinary inductive reasoning is not certain, mathematical induction is different. An example: 0 is a natural number, and it is divisible by 1. If natural number n is divisible by 1, so is n+1.  Every natural number is divisible by 1.

The Sorites Paradox The Sorites Paradox is an example of the failure of mathematical induction in natural language. Here is an example: A person with no hairs on his/her head is bald. If a bald person grows a hair on his/her head, he/she is still bald.  No matter how many hairs a person has on his/her head, he/she is bald, so everyone is bald. Obviously, this is false, but the reasoning seems correct.

The Sorites Paradox The reason that the Sorites Paradox occurs in natural language is because of the existence of vague predicates. Words such as bald and poor are only partially defined. (This is not restricted to adjectives. Other vague words are quickly and neighbor.)

Vagueness

Vague predicates also create issues with some of the fundamental rules of classical propositional logic. Here’s Gollum. He’s not really bald, and he’s not really non-bald either.

Vagueness Because bald is only partially defined, it would be wrong to say that Gollum is either bald or non-bald, so ~(b v ~b). People would also agree that it would be correct to say that Gollum is not both bald and non-bald, so ~(b & ~b). But, classical propositional logic says that ~(b & ~b)  b v ~b. This is not the case with Gollum! So, it can be seen that predicates such as bald to not conform to the usual logical rules.

Why Do We Care? This is relevant if one were to try to write a program to analyze the validity of arguments (i.e., in a court case). This is also important for a program to pass the Turing Test.