5 - The nature of logic

Summary

Analyticity as logical truth

Some define an analytic truth as one the denial of which leads to a contradiction. Kant, for instance, supplemented his well-known treatment of analyticity in terms of the predicate concept being already included in the subject concept in this way. The problem in such a definition is the phrase ‘leads to’. The intent is that logical principles applied to the denial of a sentence will suffice for deriving a contradiction. Thus interpreted, the above definition is equivalent to a more affirmative statement: a sentence is analytically true precisely when it follows from the principles of logic alone. But since what follows here are the theorems or laws of logic, then analytic truth in this sense is the same as logical truth. We must turn here to examine Quine's thoughts on analyticity as logical truth. To begin with, we will present a distinctively Quinian definition of logical truth. This will lead us to consider the bounds of logic, that is, where does logic end and mathematics begin? We will take note of the way Quine expresses the principles of logic, and we will then consider some criticisms of the attempts to ground logic and mathematics non-empirically.

The definition of logical truth

Consider the logical truth

Brutus killed Caesar or Brutus did not kill Caesar.

The schema for this sentence is:

p or not p

Such truths are distinguished by the fact that they remain true no matter what expressions we, taking care to be grammatical, put in the place of the non-logical parts. In the above schema, the nonlogical parts are indicated by p.