This article is about Hegel meeting Boole meeting Wittgenstein. The World meets Logic. The issue is that the world is not logical. People are not logical. Science, in spite of everything it says, is not logical. Logic, in itself, is bookkeeping for observations. Observations are “tick-ticks” and “boom-boom-boom”. We make boxes, and we count the number of ticks.
G. Boole is bold in titling his book as “An Investigation of the Laws of Thought” (Boole, n.d.). The operation of “thought” belongs to the human ego mind’s perception and experience of the world. But as we argue, the world in itself is not subject to logic. But logic is our coherence bookkeeping for observations based on the assumptions of:
- \(P \lor \lnot P = \text{True}\) (Law of Excluded Middle)
- \(P \land \lnot P = \text{False}\) (Law of Noncontradiction)
Thus we do not expect Logic to do the real work of discovering the contingent truths of the world. Contingent truths are propositions which are true, but which are not necessary a priori. Contingent truths are truths which don’t have to be truths, but which are in the world. Hegel’s perspective (Zizek 2015) is that the contingent truths are the most transcendental truths, being the boundaries of what is tautological and what is contradictory. For us, these contingent truths are the true subject of physics and natural philosophy properly so-called. “How wondrous are thy works O Lord, in wisdom thou hast made them all.”
We read G. Boole (Boole, n.d.) and ask ourselves “Are there any nontrivial syllogisms or inferences in Boolean logic?” The Aristotelian syllogisms are almost “word play” where everything is immediately clear from the form of the propositions, this logical form being independant of the content supposed to be represented by the Boolean symbols. This idea can be found in (Wittgenstein, n.d.).
With time we have learned to appreciate the differences between strict logical statements and deductions, and contingent inductive statements which depend on observation. Mathematics and physics often confuse these ideas, believing that the world can be known strictly by logic and formalism. However it’s not until one actually tries discovering original mathematics, that one realizes two things:
Logic by design only produces trivial statements and conclusions.
Creative mathematics is not known a priori, but is discovered by experiment.
For example, there are many functions \(f: X\to Y\) to study in mathematics, but almost all properties of \(f\) cannot be deduced from the definition of \(f\). Instead these properties are only learned by computing \(f(x)\) at various inputs and observing the result. But the evaluation of \(y=f(x)\) is not necessarily included in the definition of \(f\).
In fact given a boolean proposition \(P\), the nontrivial problem is deciding \(\texttt{isTautology}(P)\)$ and \(\texttt{isContradiction}(P)\). If \(P\) strictly contained Boolean variables, then these decision problems are a strictly Boolean property. However if \(P\) is an interesting nontrivial statement about the world, then \(P\) contains contingent variables (i.e. variables which depend on the state of the world).
There is an idea that logical deductions are a type of topological closure operation. But this closure is not taken in any absolute sense, but only in the limited Boolean logical topology. And indeed, constructing the closure \(\overline{X}\) of a collection of propositions \(X\) strictly delimits \(X\) and its consequences. Our claim is that this “closure operation” is formal and not representative of inductive logic and actual scientific discovery. Inductive statements are beyond this trivial logical boundary.
What’s the point? In a sense our discussion is rehashing the Frankfurt logical positivist view represented by early Wittgenstein (Wittgenstein, n.d.). But we are mindful that good mathematics does not necessarily consist of tautological statements, or what Emmanuel Kant might call “analytic” statements.
Thus we argue that interesting statements and claims are not formal, i.e. are non tautological. Therefore in debates and real discourse, whatever the consensus is going to be, there’s no proper Boolean logic unless the issue in question is tautological and trivial.
But this begs the question of What is reason? Or jurisprudence and the art of persuasion? This is the art of approximation, of the ability to persuade others that certain complex settings \(P\) can be reduced to more obvious statements \(Q,R,S,\ldots\) which are not necessarily contained within the definition of \(P\) itself. One has to argue that \(P\) is approximately tautological conditioning on certain auxiliary hypotheses \(Q, R, S, \ldots\).
But boolean logic, in strict terms, assumes that people are logical and consistent, and frequently in debates there is no apparent law of noncontradiction. People can be explicitly “yes” but implicitly “no” on the same question. This uncertainty seems inherent to natural real world phenonomenon.
A keystone of mathematical reasoning is method of proof by Reductio ad absurdum (Reduction to the absurd). These arguments seek to prove that \(\lnot P \to 0\), and therefore obtain the syllogism \[(P \lor \lnot P = 1), (\lnot P \to 0) \vdash P.\]
In practice, most mathematicians reason by the method of Reductio ad obvious. This method looks to reduce propositions \(P\) to the simplest possible form, e.g. saying that \(P\) implies \(1=1\) or \(2=1\). Thus \(P\) either reduces to a tautology or contradiction.
There is another methodology which we call Implicit Bayesian. This is a weaker version of reductio ad absurdum. The argument typically proceeds with Alice declaring a proposition \(Q\). But Bob says that “\(Q\) can’t be true, because \(Q\) would imply \(R, S, T\), \(\ldots\), but this contradicts my (Bob’s) subjective belief that \(\lnot R\), \(\lnot S\), \(\lnot T, \ldots\), are true.
In practice Bob does not know whether \(R\) or \(\lnot R\) are true, but Bob has a strong subjective belief in \(\lnot R\), say. And this a priori belief is an obstruction to Bob’s accepting \(Q\).
In debates Bob’s assumptions are usually implicit and not stated prior to the debate but only emerge as the argument develops. These assumptions are drawn out by Alice’s proposal \(Q\), and these assumptions reveal the internal rearrangements which are required for Bob’s logical consistency. However an individual’s internal logic is not necessarily consistent. What is revealed in actual debates, if one is tracking the parameters correctly, is that people form their opinions based on their own implicit background assumptions. And these assumptions are not the subject of debate, but are held as fixed constants in the person’s mind. And in general, person’s are not open to questioning these a priori assumptions.
Here we highlight how the human ego enters into logic, because the observations requires some mechanism (“tick-tick” or “boom boom”) by which we count events. This is the physical aspect of observation. However we emphasize that logical proofs are also a type of observation. Indeed the algebraic proof of a Boolean deduction also requires an observation, typically of the form “LHS equals RHS” \(A=B\).
So what are we saying? That we have to be clear on what logic is and what logic is not. Everything is look and see, “see to believe”. One either sees a boolean proof which is always trivial and tautological, or one sees basic physical events and makes statements about these events. But one cannot really prove statements of these events. This means confronting Hegel against Boole, which is the contingent nature of reality and the world versus the tautological logical propositions of Boolean logic.
And this is how we understand the sense of Wittgenstein that “all facts are independant of one another.” There really is no contradiction inherent in \(Q\) or \(\lnot Q\). The world could be in either case. And this is decided only inductively by observation.