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Logic Proof Calculator
Check whether a propositional logic argument is valid using a full truth table. Enter your premises and a conclusion, and see exactly why the argument holds, or a counterexample if it doesn't.
Check your argument
One premise per line. Symbols: ¬ ∧ ∨ → ↔, or use ~ & | -> <->.
No row makes every premise true while the conclusion is false.
Full truth table
Quick examples
How to use this logic proof calculator
Enter your premises
One statement per line, using letters as your variables.
Enter your conclusion
The statement you're trying to show is guaranteed to follow.
Check validity
Every possible true/false combination gets tested automatically.
Review the table
See exactly which row would break the argument, if any.
What is a logic proof calculator?
A logic proof calculator checks whether a conclusion is guaranteed to follow from a set of premises in propositional logic. Rather than walking through a formal step by step natural deduction proof, this tool uses the truth table method: it tests every possible combination of true and false values for your variables and confirms whether the conclusion holds whenever all the premises do.
This makes it a reliable way to check homework problems, verify an argument you've constructed, or spot exactly why an argument that looks convincing on the surface actually fails.
How the truth table method works
For every variable appearing anywhere in your premises or conclusion, the calculator generates every possible combination of true and false values. For each combination, it evaluates all of your premises and your conclusion.
If even one row exists where every premise comes out true but the conclusion comes out false, that row is a counterexample, and the argument is invalid. If no such row exists anywhere in the table, the argument is valid, guaranteed by exhaustive enumeration rather than by matching a known pattern.
Common valid argument forms
Certain valid patterns come up so often in logic that they have their own names. Modus Ponens says that from P → Q and P, you can conclude Q. Modus Tollens says that from P → Q and ¬Q, you can conclude ¬P. Hypothetical Syllogism chains two implications together, and Disjunctive Syllogism eliminates one side of an OR statement once you know it's false.
This calculator doesn't need to recognize these named forms to check them, since the truth table method verifies any argument's validity directly, but recognizing these patterns can help you construct valid arguments faster by hand.
Common invalid patterns to avoid
Affirming the Consequent is a classic fallacy: from P → Q and Q, it's tempting to conclude P, but this doesn't follow, since Q could be true for reasons that have nothing to do with P. Denying the Antecedent makes a similar mistake, concluding ¬Q from P → Q and ¬P.
Both patterns look structurally similar to genuinely valid forms like Modus Ponens and Modus Tollens, which is exactly why they're such common mistakes, and exactly the kind of error a truth table check catches immediately.
A full worked example
Take the premises P → Q and ¬Q, with the conclusion ¬P, the Modus Tollens pattern. With two variables, P and Q, there are four possible rows: TT, TF, FT, and FF.
In row TT, P → Q is true and ¬Q is false, so this row doesn't have all premises true and is skipped. In row TF, P → Q is false, also skipped. In row FT, P → Q is true but ¬Q is false, skipped again. In row FF, P → Q is true and ¬Q is true, so both premises hold, and ¬P is also true in this row. Since this is the only row where all premises hold, and the conclusion holds there too, the argument is valid.
Now compare that to Denying the Antecedent: premises P → Q and ¬P, conclusion ¬Q. In row FT, P is false and Q is true, so P → Q is true and ¬P is true, meaning both premises hold. But ¬Q is false in this row, since Q is true. That row is a counterexample, proving the argument invalid, even though it looks superficially similar to the valid Modus Tollens pattern above.
Frequently asked questions
What makes a logical argument valid?
An argument is valid if there's no possible way for all of its premises to be true while its conclusion is false. This calculator checks every possible true or false combination for your variables and confirms an argument is valid only if the conclusion holds in every single row where all the premises hold.
What do the symbols ∧, ∨, ¬, →, and ↔ mean?
¬ means NOT, ∧ means AND, ∨ means OR, → means IMPLIES (if...then), and ↔ means IFF, or if and only if. You can type these directly, or use the ASCII shortcuts this calculator accepts: ~ or ! for NOT, & for AND, | for OR, -> for IMPLIES, and <-> for IFF.
What is a counterexample, and why does one prove an argument invalid?
A counterexample is a specific true or false assignment to your variables that makes every premise true while the conclusion comes out false. Finding even one such row is enough to prove an argument invalid, since a valid argument can never allow that combination to occur.
What's the difference between a valid argument and a sound argument?
Validity is purely about logical structure, whether the conclusion follows from the premises regardless of whether those premises are actually true in reality. A sound argument is a valid argument whose premises also happen to be true. This calculator only checks validity, since truth of real-world premises isn't something a truth table can determine.
Why does this calculator use a truth table instead of listing inference rules?
A truth table method is exhaustive and mathematically guaranteed to catch every possible case, which makes it reliable for any propositional argument you enter, not just ones matching a handful of named patterns. Common inference rules like Modus Ponens are included as reference material below, but the validity check itself doesn't depend on recognizing a specific named form.
Can an argument with false premises still be valid?
Yes. Validity only cares about the relationship between premises and conclusion, not whether the premises are true in the real world. An argument built entirely on false premises can still be perfectly valid if its conclusion would necessarily follow whenever those premises happened to be true.
What happens if my argument has too many variables?
This calculator supports up to 8 distinct variables, which generates a truth table of up to 256 rows. Beyond that, the table becomes impractical to display and check visually, so try breaking a very large argument into smaller sub-arguments if you hit that limit.
What's the difference between validity and the truth of the premises?
These are two separate questions. Whether P is actually true is a factual question about the world, while whether an argument using P is valid is a purely structural question about whether the conclusion is guaranteed to follow whenever the premises hold. This calculator only answers the structural question.
This tool is for educational purposes only. Always verify important results with a qualified professional.
