Propositional Logic and Its Role in AI

Atomic

Atomic propositions are the simplest kind: single statements that are either true or false. For instance:

• "2 + 2 is 4" is an atomic proposition — true by definition.
• "The Sun is cold" is also an atomic proposition — though demonstrably false.

Compound

Compound propositions, by contrast, combine atomic propositions using logical connectives such as AND ( ∧ ), OR ( ∨ ), NOT ( ¬ ), IMPLIES ( → ), and IF AND ONLY IF ( ↔ ). Given AI's need to model multifaceted scenarios, compound propositions are far more common in practice. An example:

• "Literal Labs is a company, and its offices are in Britain" is a compound proposition, joining two factual assertions.

How Propositional Logic is Used in AI

Propositional logic plays a pivotal role in knowledge representation, the process by which AI systems structure, manipulate, and reason over information. In Literal Labs’ AI models, including those powered by Tsetlin machines, this logical backbone is combined with Tsetlin automata to enable effective pattern recognition and decision-making.

1. Encoding Facts: At its core, propositional logic encodes facts as true or false statements. For example, "Patient has a fever" may be represented as proposition P. Whether P is true or false depends on real-world observation.
2. Modelling Relationships: Logical connectives help articulate relationships between facts. "Patient has a fever AND cough" becomes P ∧ C, where P and C are distinct propositions.
3. Inference and Reasoning: AI systems use propositional logic to draw conclusions from known facts. If "Patient has a fever AND cough" (P ∧ C), and "Fever implies flu" (P → F), the system may infer that "Patient likely has the flu" (F).
4. Decision-Making: Logical rules enable AI systems to evaluate propositions systematically and select appropriate actions. The clear true/false structure of propositional logic ensures that decisions remain both transparent and explainable.

How It Differs from Other Logical Approaches Within AI

While propositional logic provides a robust foundation, several other logical frameworks exist within AI, each adding layers of complexity or increasing the compute resources needed:

• First-Order Logic (FOL): An extension of propositional logic, First-Order Logic introduces variables and quantifiers (such as "for all" and "there exists"). This allows reasoning about objects, their properties, and their relationships, going beyond simple true/false statements.
• Fuzzy Logic: Departing from the binary clarity of classical logic, fuzzy logic accommodates degrees of truth — values between 0 and 1. While this allows for 'shades of grey,' it also tends to increase computational complexity and energy demands.
• Modal Logic: Adding another layer, modal logic introduces modalities such as necessity, possibility, belief, and time. It allows AI systems to consider different states of affairs, potential outcomes, or temporal conditions.
• Bayesian Logic: Rooted in Bayes’ theorem, Bayesian logic updates probabilities as new evidence becomes available. Instead of absolute truth, it works with likelihoods, helping AI models adjust conclusions dynamically based on incoming data.

In contrast to these more complex or probabilistic methods, propositional logic — particularly when paired with data binarisation and Tsetlin Machines — offers a uniquely efficient, interpretable, and resource-conscious approach to AI design and deployment.