First-Order Logic Natural Deduction Solver Calculator

Solve symbolic arguments with guided deduction support. Compare premises, conclusions, quantifiers, and derived proof hints. Download results, inspect charts, and practice formal reasoning confidently.

Calculator Input

Premises

Use symbols like ∀, ∃, !, &, |, ->, and <->.

Conclusion

Example: Mortal(Socrates)

Sample domain size

This controls guidance only. It does not force full first-order model enumeration.

Example Data Table

Case	Premises	Conclusion	Expected Insight
1	∀x (Human(x) -> Mortal(x)); Human(Socrates)	Mortal(Socrates)	Universal instantiation then modus ponens.
2	P -> Q; P	Q	Direct propositional support.
3	R \| S; !R	S	Disjunctive syllogism suggestion.
4	∀x (Bird(x) -> Animal(x)); Bird(Tweety)	Animal(Tweety)	Predicate reasoning with a named constant.

Formula Used

This page combines structural counting, truth-functional entailment, and natural deduction rule hints. The propositional core tests whether every checked model that satisfies all premises also satisfies the conclusion.

Semantic entailment is approximated with: Premises ⊨ Conclusion when no checked truth assignment makes all premises true and the conclusion false.

Natural deduction guidance applies common rules: modus ponens, conjunction elimination, disjunctive syllogism, double-negation elimination, biconditional elimination, and basic universal or existential instantiation.

Complexity is estimated with: 2 × premises + connectives + 2 × quantifiers + atomic symbols. This gives a quick workload score for the argument.

How to Use This Calculator

Enter one premise per line in the premises box.
Write the target statement in the conclusion field.
Use symbols like ∀, ∃, !, &, |, ->, and <->.
Choose a sample domain size for quantified rule guidance.
Click Solve Logic Argument.
Read the summary, metrics, derived proof suggestions, and chart.
Export the result block as CSV or PDF when needed.
Use the example table to test known argument patterns quickly.

Frequently Asked Questions

1. What does this solver check first?

It first normalizes your symbols, counts logical structure, and scans the propositional core for support or counterexamples.

2. Does it fully solve every first-order proof?

No. It is a practical hybrid assistant. It gives reliable structural guidance, common derivation hints, and an approximate entailment scan.

3. Why are quantified statements mapped to placeholders?

The truth-table engine treats complex quantified segments as structured atoms. This keeps the page fast while still preserving proof guidance.

4. Which connectives can I enter?

You can use negation, conjunction, disjunction, implication, biconditional, universal quantifiers, and existential quantifiers with standard symbolic syntax.

5. What does the complexity score mean?

It is a quick workload estimate based on premises, connectives, quantifiers, and atomic symbols. Higher scores usually mean more involved reasoning.

6. What does a counterexample mean?

A counterexample is a checked assignment where every premise holds but the conclusion fails. That blocks propositional support.

7. Can I export my result?

Yes. The page includes CSV export for tabular review and PDF export for notes, assignments, or sharing.

8. Is this useful for teaching and revision?

Yes. It works well for classroom drills, self-study, and proof pattern review because it shows structure, rule hints, and examples together.