Calculator
Example Data Table
| Premises | Conclusion | Derivation | Expected Result |
|---|---|---|---|
| P -> Q P |
Q | P -> Q | premise P | premise Q | 1,2 | MP |
Valid |
| P -> Q !Q |
!P | P -> Q | premise !Q | premise !P | 1,2 | MT |
Valid |
| P | Q P -> R |
R | Optional | Invalid |
Formula Used
The calculator tests validity with the semantic condition: premises entail the conclusion when no valuation makes every premise true and the conclusion false.
Implication is evaluated as false only when the antecedent is true and the consequent is false. Biconditional is true when both sides share the same truth value. Conjunction needs both sides true. Disjunction needs at least one side true. Negation reverses the truth value.
For derivation lines, the checker compares a selected rule pattern against referenced formulas. It supports premise, assumption, reiteration, MP, MT, conjunction introduction, conjunction elimination, disjunction introduction, and double negation.
How to Use This Calculator
Enter each premise on its own line. Enter one conclusion. Add optional derivation lines with vertical bars. A common format is statement | references | rule. For premises, use statement | premise. Press Calculate. The result appears above the form. Use the export buttons to save a report.
Article: Sentential Logic Derivations
Purpose
A sentential logic derivations calculator helps students test formal arguments with less guesswork. It accepts premises, a conclusion, and optional proof lines. Then it builds valuations for every detected variable. Each valuation is a possible truth assignment. The tool checks whether any assignment makes all premises true while making the conclusion false. Such a row is a counterexample. If no counterexample exists, the argument is valid.
Why Truth Tables Matter
Truth tables give a complete semantic view. They do not rely on intuition. They show the behavior of each connective. Negation flips a value. Conjunction requires two true parts. Disjunction accepts at least one true part. Implication fails only in the true to false case. Biconditional requires equal truth values. These rules make validity testable in a clear way.
Proof Line Checking
Derivations add a syntactic layer. A proof line should follow from earlier lines by an accepted rule. This calculator checks common classroom rules. It can review modus ponens, modus tollens, conjunction rules, disjunction introduction, reiteration, assumptions, and double negation. The report is not a replacement for a full proof tutor. It is a practical checker for many standard exercises.
Helpful Workflow
Start with simple symbols such as P, Q, and R. Enter each premise separately. Add the intended conclusion. Run the truth table first. If the argument is invalid, study the first counterexample. It shows exactly where the conclusion fails. If the argument is valid, add derivation lines. Use references carefully. A wrong reference often causes a good formula to fail its rule check.
Study Benefits
The calculator supports careful practice. It separates semantic validity from proof construction. That distinction is important. An argument may be valid even when a typed derivation has errors. A derivation may contain correct early lines but still miss the conclusion. CSV and PDF exports help learners save work, compare attempts, and discuss results with instructors or classmates. Use it to check homework drafts, prepare quizzes, or explain why an argument succeeds. Clear symbols and numbered proof lines make every review easier. Advanced users can also test alternate conclusions. They can compare competing derivations quickly. This makes review sessions more focused. It also reveals hidden assumptions, weak rule choices, and notation mistakes before final submission. Regular practice builds fluency with formal reasoning over time.
FAQs
What is sentential logic?
Sentential logic studies whole statements and connectives. It uses symbols like P and Q. It checks how truth values combine under not, and, or, implication, and biconditional.
What does a valid argument mean?
An argument is valid when no truth assignment makes all premises true and the conclusion false. Validity concerns structure, not real-world content.
What is a counterexample?
A counterexample is a truth row where every premise is true, but the conclusion is false. One counterexample proves invalidity.
Which proof rules are checked?
The checker supports premise, assumption, reiteration, MP, MT, conjunction introduction, conjunction elimination, disjunction introduction, and double negation.
How should I type implication?
Use P -> Q for implication. You may also use symbols like ! for negation, & for conjunction, and | for disjunction.
Why limit the number of variables?
Truth tables double with each added variable. Eight variables already require 256 rows. Limits keep the page responsive and readable.
Can this replace a proof instructor?
No. It checks common patterns and semantic validity. Complex natural deduction systems may need additional rules and instructor review.
What do the export buttons include?
The CSV export includes truth rows. The PDF export gives a compact report with the main result, variable list, and table preview.