The Implications of Adding Contradictions to Formal Systems: A Look at 12
There is an anecdote about a lecture delivered by the renowned philosopher and mathematician Bertrand Russell, where he discussed material implication in the context of formal logic. The story goes that during his lecture, a student raised a question related to the implication of a false proposition. Russell's response, as described, provides a deep insight into the foundational aspects of logical systems and their consistency.
The Story of 10 and 12
The tale begins when Russell mentioned that in the sense of material implication, a false proposition can imply any proposition. A student interrupted the lecture, posing the question: 'What if 10?' To which Russell replied, 'In general, adding a contradiction such as 12 given that we already know 1≠2 or more generally A and ?A, to a formal system makes all statements trivially provable because a contradiction implies everything. This makes inconsistent formal systems rather uninteresting and completely useless.'
The Nature of Inconsistent Systems
When a contradiction is introduced into a formal system, it can lead to a situation where all statements become trivially provable. This is because anything can be inferred from a contradiction. For instance, if we add the axiom 12 but also allow statements like 112, we create an inconsistent axiomatic system. In such a system, one can prove any proposition P and its negation ?P. This is a fundamental property of inconsistent systems.
Creating a Consistent and Valuable System
The challenge lies in creating a consistent axiomatic system that is both interesting and practically useful. Simply adding a contradiction like 12 does not produce a valuable system. What Russell is emphasizing is that to create something interesting, one must go beyond trivial statements and introduce meaningful axioms. For example, stating 22 is not the only axiom one can use; a system based on this axiom alone would be limited.
Russell further stresses that creating a valuable system requires more than just one dogma. One must develop a mathematical system that is not only consistent but also meaningful and well-structured. While it is possible to create an inconsistent system, this would not provide insightful or practical value. Instead, a system that is equivalent to a part of existing mathematics but uses an awkward code or notation might also be less effective.
Garbage in, Garbage Out
The idea that adding contradictions can render a formal system useless is akin to the principle 'Garbage in, Garbage out.' Just as inputting meaningless or contradictory data into a computer program will yield nonsensical output, adding contradictions into a logical system can lead to trivial or nonsensical conclusions.
The importance of consistency in formal systems cannot be overstated. Without it, the system becomes uninteresting and devoid of practical or theoretical value. In essence, the effort to create a valuable logical system requires careful and deliberate reasoning, not the introduction of contradictions or meaningless statements.
Understanding the implications of contradictions in formal systems is crucial for anyone working in logic, mathematics, and related fields. It underscores the need for rigorous and consistent axioms, as well as the importance of avoiding contradictions that can lead to meaningless or trivial conclusions.