by Paul Curzon, Queen Mary University of London
We have seen how to represent truth tables in lego. Truth tables are a way of giving precise meaning to logical operations like AND, OR and NOT. They are also give a way to do logical reasoning following a simple algorithm.
That’s Not Not True
You may have been pulled up in English and told you just said the opposite of what you meant, after saying something like “There ain’t no way I’m doing that”. This is a “double negative” as the “n’t” in “ain’t” is really “not” so followed by “no way” you are actually saying “not not way” or overall: “I am doing that”. Perhaps the most famous double negative is in the Rolling Stones song “(I can’t get no) satisfaction”. English is very flexible though and double negatives like this don’t cancel out but just become a different way of saying the negative version. In logic two negations do cancel out, though. Let’s take a purer version to work with: the statement “I am not not happy”. What does this mean? In logic the basic proposition here is “I am not happy”. The logical statement is “NOT (NOT (I am happy))”.
We can prove what this means using truth tables. We can do more than just prove what this single statement means. We can prove what all double negatives mean, more generally. We do this by replacing the proposition “I am happy” with a variable P. It now becomes NOT (NOT P) or in our lego version where we use a yellow brick to mean a proposition, P:
This is just syntax, just a sequence of symbols. It doesn’t give us any meaning on its own. We can build truth tables in Lego for that. We start from the variables that are at the inside of the logical expression which here is just the variable P. We list in a table column the possible values it can take (true or false).
This shows P (yellow) can be either be TRUE (blue) or FALSE (red). Now we build up the logical expression of interest a column at a time. NOT is applied to P, so we add a new column for NOT P and use the truth table for the operator, NOT, to tell us what lego brick to put in each row based on the lego brick already there. The NOT truth table is at the top of the page. It says if you have a blue brick in a row, place a red brick there. If you have a red brick, put a blue brick there. This gives us a new filled out column for (NOT P) which is just a copy of the NOT truth table (but bare with us that was just a simple case). We get:
Moving outwards in the expression NOT (NOT P)), we now look at the operator applied to (NOT P). It is NOT again. We add a new column to our truth table and again use the NOT truth table to work out the new values, but this time applied to the column before (the NOT P column). The NOT truth table says put a blue brick for a red brick, and a red brick for a blue brick in the column it is being applied to (the NOT P column). This gives:
The result is a truth table with coloured bricks identical to that of the original column for P. Switching back from lego bricks to what the columns mean, we have shown that the NOT(NOT P) column is the same as the P column, or in other words that NOT(NOT P) EQUALS P (whatever value P has).
We can actually go a step further though, because equivalence is just a logical operation with its own truth table. It gives true if the two operands have the same value and false otherwise (or in lego terms if the bricks are the same colour the answer is a blue brick and if they are different colours the answer is a red brick. The truth table looks like this:
We can use this truth table to calculate whether two lego truth table columns are equal or not just by looking up the combinations in this EQUALS truth table. Continuing our example we can carrying building our truth table about NOT(NOT P)). To make things clearer first add a column corresponding to P again. That means we will be applying the EQUALS operator to the last two columns. As before, for each row, look up the corresponding pattern for those last two columns in the EQUALS truth table to get the answer for that row. In the first row we have two blue bricks so that becomes a blue brick according tot he EQUALS truth table. In the next row we have two red bricks. That also becomes a blue brick. This gives:
The thing to notice here is that all the entries in the final answer column are blue lego pieces. Switching back from the lego world to the logic world, what does this mean? Blue is true so all rows in the answer are true. That means whatever value of the proposition P the answer to NOT (NOT P) EQUALS P is true. We have proved a theorem that this is always true. We have shown by building with lego that a double negation cancels itself out.
Logical expressions like this that are always true (whatever the values of the variables) are called tautologies. We can tell something is a tautology, so we have proved a theorem, just by the simple manual check that its truth table values are true (or in lego all blue).
The important thing to realise about this is all the reasoning can be done without knowing what the symbols mean, and certainly not worrying about English words, once you have the truth tables. You do it mechanically. You do not need to think about what, for example, red and blue mean until the end. At that point you return to the logical world to see what you have found out. All blue means it is always true! You can also at that point substitute back in actual words of interest into the statements proved. P means “I am happy”. We started by asking what “I am not not happy” means. We converted this to “NOT (NOT (I am happy))”. By swapping in “I am happy” for P in our theorem gives us that NOT (NOT “I am happy”) EQUALS “I am happy”, or that “I am not not happy.” just means the same as “I am happy”
We have been reasoning about English statements, but this kind of reasoning is the basis of all logical reasoning and essentially the basis of formal verification where the meaning of programs and hardware is checked to see if it meets a specification. It tells you what a test in a program like “if (! temperature != 0) …) means so does for example, or what a circuit with two NOT gates does.
And lego logic has even given us a way to prove things just by building with lego.
Lego Computer Science
Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binary and computation.
Lego Computer Science
Part 1: Lego Computer Science: pixel picture
Part 2: Lego Computer Science: compression algorithms
Part 3: Lego Computer Science: representing numbers
Part 4: Lego Computer Science: representing numbers using position
Part 5: Lego Computer Science: Gray code
Part 6: Lego Computer Science: Binary
Part 7: Lego Computer Science: What is computation (simple cellular automata)?
Part 8: Lego Computer Science: Truth tables
Part 9: Lego Computer Science: Logic with truth tables
More on …
EPSRC supports this blog through research grant EP/W033615/1, The Lego Computer Science post was originally funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.