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Generate a truth table for a symbolic logic statement.
Confused? Try ~{[(A & G) v ~(U > ~S)] v I}
Compare two statements to determine if they are equivalent (always have the same outcome), contradictory (never have the same outcome), or neither.
Determine if an argument constructed of premises and a conclusion is valid (which means that if all of the premises are true, the conclusion is true), then construct a proof for the argument.
Premise 1
Conclusion
Logical operator quick reference
The first step in making a truth table is to translate your argument into the language of symbolic logic. Here's a logic translation cheat sheet to help out.
| Symbol | Alternative | Name | Kind | English connectives |
| ~ | ! | Tilde | Negation (NOT) | it is not the case that A; not A; it is false that A |
| & | &&, /\ | Ampersand | Conjunction (AND) | A and B; A but B; A yet B; A while B; A moreover B; A however B; A nonetheless B; A still B; A nevertheless B; A also B; A although B; both A and B; A additionally B; A furthermore B |
| v | ||, \/ | Wedge | Disjunction (OR) | A or B; A unless B |
| ⊃ | > | Horseshoe | Conditional (IF) | if A, then B; B only if A; A is a necessary condition for B; B is a necessary condition for A; B given that A; B provided that A; A implies that B; B on condition that A; B in case that A |
| ≡ | ==, = | Triple bar | Biconditional (IFF) | A if and only if B; A just in case that B; A is a necessary and sufficient condition for B; A is equivalent to B |
What is a truth table?
A truth table is a table that you can use to work with logic statements. Wikipedia says that:
What's a symbolic logic statement?
A symbolic logic statement is a way to represent a logical argument with symbols. So, you could write something like this:Now that we have a symbolic logic statement, we can generate a truth table for it. To do that, type it into the calculator. You should see an output like this:
At the very top, there's a thumbs up emoji indicating the statement is well-formed. If you added an error into the statement (like changing "&" to "&&") that would go away and you'd see an error message. There's also a "Prettify" button to replace operators like > or = with nicer-looking versions. Next, you'll see indicators showing that the statement is a contingency. This simply means that the statement could either be true or false, depending on whether the inputs (the identifiers) are true or false. Statements can also be tautologies (the statement is always true, no matter what the inputs are) or contradictions (always false). For example, A v ~A is a tautology because if you translate it to English, it's saying that "A is true or it's false", which of course is always true regardless of what A is. A & ~A is a contradiction because it's saying that "A is true and it's false", and it's impossible for something to be both true and false at the same time. Our example is a contingency because it's possible for me to be neither a human or a robot, or I could be a robot but not have a CPU for a brain.
Finally, we have a truth table for our statement. Each component of our statement is broken out into a column of the table. The main operator's column is highlighted; this is the "output" of the statement. Let's try using it! First, we have to determine what the value of each identifier is. I'm a human, I'm not a robot, and I don't have a CPU for a brain, so H is true, R is false, and C is also false. On the table, we can find the row where each of our identifiers has the value we determined:
And we can see that in this row, the main operator is true, meaning that the statement is true!
How to use a truth table?
To summarize, to make and use a truth table:
- Translate your statement into symbolic logic
- Paste your statement into the calculator to generate the truth table
- Find the row on the table where the values of the identifiers match your input values
- Read the value in the main operator column (highlighted) to see the result!
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