Preface
Aristotelian logic is fundamentally different from the mathematical logic taught in most textbooks. Three features set it apart:
1. It is a philosophical logic, not only a mathematical one. Its sentences are called judgments and aim to describe reality and formal structures.
2. It works with incomplete information. You do not need to know everything about the world to draw valid conclusions.
3. It is paraconsistent. Contradictory observations do not make everything explode into contradictions.
These three features are deeply interconnected, and they shape the entire system.

Philosophical vs. Mathematical Truth

In mathematics, a sentence is true if it can be derived from axioms using valid deduction rules. The axioms are true by definition, and deduction preserves truth. This gives mathematical logic an unusual clarity: contradictions cannot happen unless the system itself is broken. If you encounter a contradiction, you stop.
Aristotelian logic does not have that luxury. Its truth criterion is different:
A sentence is true if and only if it corresponds to a real situation.
This shifts everything. Deduction alone cannot guarantee truth. The world is independent of our reasoning about it. Information comes primarily from perception, and perception is partial, contextual, and sometimes conflicting.

An Example: The Bent Stick

A straight stick is half-submerged in water. Looking at it, you perceive it as bent at the water's surface. You pull it out. Now you perceive it as straight.
The two perceptions are inconsistent: the stick cannot be both bent and straight. Yet both perceptions are genuine. Each faithfully reports a real sensory situation.
In classical mathematical logic, this would be disastrous. Once you have a contradiction, the principle of explosion allows you to derive anything. The system becomes trivial.
But perception-based reasoning cannot afford explosion. The contradiction is not a failure of logic - it is a signal that your information is incomplete. The solution is not to reject one perception or embrace both as "simultaneously true." Instead, you add context: the stick is straight, but light refracts at the water's surface. The conflict dissolves through additional information, not through choosing sides.

Paraconsistency: Three Approaches

Modern logic has developed paraconsistent systems - formal logics that do not explode in the presence of contradictions. Two main approaches are well known (https://plato.stanford.edu/entries/logic-paraconsistent/).
(a) Da Costa's C-systems restrict the structural rules of inference so contradictions do not propagate.
(b) Priest's dialetheism accepts some contradictions as genuinely true.
Aristotelian logic follows a third path:
(c) Contradictions signal over-determination due to incomplete information. They are resolved not by restricting inference or embracing paradox, but by adding information and integrating partial observations into a coherent framework of scientific explanation .
This is precisely what the formal machinery in this tutorial is designed to handle: reasoning under incomplete and potentially conflicting information, guided by deductive structure.

Why "Aristotle's World"?

This tutorial is modeled in spirit after Barwise and Etchemendy’s Tarski’s World, the classic interactive introduction to first-order logic. But the starting point is fundamentally different.
Tarski's World assumes a fully known, mathematically precise universe. You see all objects, all their properties, all their relations. Truth is verification against a complete model.
Aristotle's World assumes the opposite: a partially known reality, accessed through perception and structured by learning. Individuals may be known only through some of their properties. Predicates may apply in some cases and remain unknown in others. Contradictions may arise and await resolution. Truth is correspondence to situations you do not fully control.
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