Semantic Tableaux in Predicate Logic

Semantic Tableaux, also known as Truth Trees, can be used as a proof technique in Predicate Logic (which includes FOPL) to determine the satisfiability or validity of a set of formulas. It is a systematic method that uses a tree-like structure to explore all possible interpretations of the formulas.

The process of constructing a semantic tableau involves the following steps:

1. Initialization: Write down the negation of the conclusion (if proving validity) or the set of formulas (if checking satisfiability) at the root of the tableau.
2. Branching: Look for a formula in the tableau that can be decomposed or expanded according to the rules of FOPL. These rules include quantifier rules, logical connective rules, and equality rules. Choose a formula and apply the corresponding rule to expand the tableau.

• Quantifier Rules: If a quantifier (∀ or ∃) appears in a formula, introduce a new branch for both the universal and existential cases. In the universal case, replace the quantified variable with a specific term and continue expanding. In the existential case, introduce a new individual constant or variable and continue expanding.
  • Logical Connective Rules: Apply the rules for logical connectives (∧, ∨, ¬, →, ↔) to decompose or simplify formulas.
  • Equality Rules: If an equality symbol (=) appears in a formula, apply the rules to decompose or simplify the formulas involving equality.

• Closing the Branches: At each step of branching, if a contradiction is derived within a branch (i.e., a formula and its negation appear in the same branch), close the branch. Closing a branch indicates that the formulas within that branch are unsatisfiable.
• Termination: Continue branching and closing branches until all branches are either closed or no further expansion is possible. If all branches are closed, the set of formulas is unsatisfiable, meaning there is no interpretation that satisfies all the formulas. If no branches are closed, the set of formulas is satisfiable.
• Final Decision: Based on the state of the branches, determine the final result. If all branches are closed, the set of formulas is unsatisfiable. If at least one branch remains open, the set of formulas is satisfiable.

Semantic Tableaux provide a systematic and visual approach to analyzing the truth values of formulas in Predicate Logic. By exploring all possible interpretations and identifying contradictions, it can help determine the satisfiability or validity of a set of formulas.

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