A Theory on Logic and Science

Imagine two people trying to solve a puzzle. One insists that all the pieces must fit a certain way, while the other keeps changing the rules about what "fit" means.

They quickly realize that before solving anything, they need to agree on some basic starting points: what counts as a rule, what counts as a piece, and what it even means for something to "fit."

Logic essentially works by beginning with assumptions that act as starting rules, and from there, we figure out what conclusions must follow. This article explores how those assumptions shape reasoning, how contradictions arise, and how conclusions are derived within a logical system.

It also looks at how we extend these ideas to the real world through science, and why our understanding of truth always depends on the starting points we are willing to accept.

Finally, it raises a broader question: if everything we know depends on assumptions, how certain can we ever be about what is true?

The Foundations of Logic

Logic is a human-constructed system for determining what follows from a set of assumptions. Every logical framework begins with assumptions.

These assumptions can be divided into two broad categories below.

• Assumptions that establish general rules or relations.
• Assumptions that establish the existence or identity of particular things.

For example:

• "All men are mortal" is a rule-type assumption.
• "Socrates is a man" is an existence-type assumption, because it specifies what Socrates is.

Not every word must be exhaustively defined. What matters is not the words themselves, but the meanings they represent within the constructed framework.

However, sufficient clarification is still necessary for the framework to function coherently. In the end, logic is not the words themselves, but their meanings.

In ordinary reasoning, the intended meanings matter more than the specific wording, although formal logic also studies symbolic structure independently of meaning. Philosophy of language is a separate topic, and I will not explore it further here.

Contradiction and Consistency

Imagine a system with no assumptions at all. In such a state, nothing is constrained.

There would be infinitely many possible assumptions, which could be added while maintaining internal consistency. Now introduce the assumption, "All men are mortal".

The set of assumptions that are now logically impossible also becomes infinite. Infinite assumptions that could be added while maintaining consistency still remain, but some are now excluded.

For example, the statement "All men are immortal" would now contradict the framework. "Some men are immortal" would also contradict it.

Next introduce "Socrates is a man". This further increases the number of impossible assumptions.

The statements "Socrates is not a man" and "Socrates is immortal" are now inconsistent with the assumptions already present. This is because the system would allow us to derive both 'Socrates is mortal' and 'Socrates is not mortal.'

This situation fits the standard definition of a contradiction. A contradiction occurs when it is possible to derive the statement X, and also derive the statement not X.

Derivation

A statement is derivable when there exists no logically consistent interpretation of the framework in which the assumptions are true and the statement is false. Given the assumptions:

• All men are mortal.
• Socrates is a man.

there exists no internally consistent possibility in which Socrates is not mortal. Therefore, "Socrates is mortal" is a derived statement.

However, infinitely many undetermined possibilities still remain. Statements such as:

• "Socrates is intelligent,"
• "Socrates likes pink,"
• "Socrates is an idiot,"

are neither required nor forbidden by the framework. There will always be such possibilities.

Let's back up now. Contradiction is what I originally called "mutual inconsistency."

My "rule assumptions" roughly correspond to axioms or universal premises, while my "existence assumptions" correspond more closely to postulates or existential premises.

From Logic to Science

Internal consistency alone does not guarantee correspondence with reality. Logic only establishes that if the assumptions are true, then the derived conclusions must also be true.

The application of logical systems to reality is the domain of science. It is useful to people, because it allows us to understand reality.

There is no absolute obligation to follow science. Science is made up of theories, and theories are beliefs.

Specifically, a belief that certain assumptions are true, based on our observations. If one values predictive capability, then science becomes an extremely useful method.

We have many types of theories, even unjustified ones, but when someone generally speaks about a scientific theory, they mean one that can explain phenomena, and predict that something will happen, which is then observed.

An extremely famous example of this process is how Albert Einstein attempted to explain irregularities in Mercury's orbit through general relativity. His theory predicted that the light from distant stars would be bent by gravity.

Einstein's prediction of gravitational light bending was first successfully observed in 1919, during a total solar eclipse, propelling the theory to acquire international acclaim.

The Limits of Certainty

Most, if not all assumptions that we make about our universe are a posteriori, defined as knowledge gained from experience. To be fully sure, we need to derive a logical conclusion based on our assumptions, known as a priori, which is knowledge derived independent of experience.

But we can never, in reality, be sure whether what we think are the assumptions are the founding rules of the universe. We cannot be sure of uniformity, causality, or anything.

We cannot even be sure that there is only one truth, for even that is an assumption.

But nevertheless we assume so, because it aligns with what we have seen, and observed, over our entire lives. We have never seen a light that was both on and off.

We have never seen someone float in the sky for no apparent reason. We have never seen someone both telling the truth and lying.

We have never seen something, that is, but is not. This is the reason why we adopt certain fundamental assumptions.

All our experiences lead us to not be able to conceive a world where those assumptions are false. That is how experience plays a crucial role in how we form assumptions about reality.

I would like all the readers to be critical of this article, and think before accepting what I propose. There are infinitely many possibilities that could be true, and what I have written here is just a few of them.

We can never truly know whether any theory fully captures reality.