There exists First Order Logic (FOL) and Second Order Logic (SOL). There are other distinct logical systems which operate within their parameters. The construction of the FOL system and language could aptly be described as a paradigm shift. The reason for this is that it enabled the ‘mathematisation of logic’, or that it at least enabled a process that could be called the ‘mathematisation of logic’ to get underway. It was the development required to make modern mathematical discoveries expressible or tractable in the language of logic, it allowed the two disciplines to ‘speak to one another.’ It can be inferred from the prominent position FOL holds in the history of logic that this dialogue proved to be of interest.
First Order Logic (FOL) and Second Order Logic (SOL) both exist. Other logical systems with different names also exist and are related to FOL and SOL. There is a connection between FOL and mathematics. The particular nature of this connection is of significance to the history of logic.
There exists First Order Logic (FOL) and Second Order Logic (SOL). The share the same symbols and operations, but certain rules governing the course of deduction in FOL don’t apply in SOL. In SOL, for example, the ‘Law of Limited Dilation’ does not apply, this means that when exposed to certain forms of operational duress symbols in SOL are liable to balloon slightly. This effect is called ‘Proportional Drift.’
Other logical systems can work within the semantic and syntactic parameters of both SOL and FOL, in fact, it is only within the deductive environments furnished by both FOL and SOL that these systems can work at all. And yet these systems are distinct from FOL and SOL, operating within it like viruses or bacteria within the human body. Another analogy is more helpful at elucidating this dependent distinctiveness: imagine ‘Star Wars: A New Hope’ was remade. All the characters in the remake were the same as well as all the environments and the combination of each character and environment in each scene was the same and the plot was exactly the same – but the dialogue, although it resulted in the plot unfolding in exactly the way it does in the original and produced exactly the same relationships between characters as the original, was completely different. The proof is achieved, the deduction proceeds through the stages it has to in order to reach, produce, said proof, the symbols, the rules are the same – it’s just all done in a different way, different motivations and dynamics are sourced to produce the same results. This would undeniably be a different film.
The construction of the FOL system and language was especially significant, it was a paradigm shifting achievement. It made the ‘mathematisation of logic’ possible. Before the creation of FOL logical arguments were hampered by an inability to take quantitative differences into account, especially at higher levels of abstraction, whereas mathematical demonstrations were built on providing very detailed and powerful accounts of such differences. (This is not to say that there wasn’t a place for numbers in logical argument). This meant that logic had no handle on the objects dealt with by modern mathematics, they either slipped entirely through its fingers or were left horribly mangled by its cack-handed fondling. Any dialogue between the two disciplines was impeded. However, in FOL not only could a mathematical object receive adequate representation, but it could also be manipulated in exciting new ways, new avenues in and out of it could be discovered, new implications exposed. The new bridge it forged between mathematics and logic is the primary reason why FOL holds such a prestigious place in the history of logic.