Warning: I wrote to following paper to maintain a high level of precision when considered formally under the disciplines of mathematical logic and computer science. This allows for the proper consideration of semantics in future discussions. This writing may lead you to believe that you are taken along a platitude of truism devoid of axiomatic purpose that amounts to a march of banality unrivaled in the history of human thought. Let us begin.
Definition : A truth table lists all possible unique inputs to a logic function accompanied by each output value.
A truth table uses the veracity of propositions. Veracity refers to truthfulness, whether the statement is true or false. A proposition is a statement.
e.g. let P be the proposition ‘it is raining’.
We can assign 2 values to P. That is,
|P = true||meaning that ‘the statement P is true’|
|P = false||meaning ‘the statement P is false.|
NOTE: We can use the following symbols for truth values:
|True||True, T, 1|
|False||False, F, 0|
In the above example, the proposition for P can be written as the following table:
Note, the proposition has no meaning when taken by itself. Here we only see the possible circumstances described by the statement as it can be used as an input to a logical scenario or taken s the output of a scenario.
I now introduce…
Truth tables in computer science
In this context, Truth tables can be used to represent the voltages that are applied as input to a circuit and the resulting output.
Though it may seem trivial, we consider a simple circuit with one input or switch , a battery, and a one output or lightbulb:
Approaching this physical construct of discrete possible events using mathematical logic we say:
Let the input be called I , and we use the convention that a value of 1 means that a high voltage is applied to the bulb and a 0 being the falsehood of “a high voltage is applied to the bulb ” or rather we can say a high voltage is not applied to the bulb or simply no voltage is applied to the bulb.
Another school as thought would describe I having a value 1 as the switch is on and I having a value of 0 as the switch is off.
Let the output of the circuit be O using the convention that a value of 1 means that the bulb is lit and a value of 0 means the falsehood of “the bulb is lit” or rather we can say the bulb is not lit or simply the bulb is unlit.
Yet another school of thought would describe O having a value of 1 as a high voltage was observed at the output and O having a value of 0 as low or no voltage observed at the output
Now we are ready to consider these discrete events devoid of timing to form an enumeration or superposition of sorts that is a truth table.
We arrive at the following truth table which list all possible inputs to the “circuit” and the observed outputs:
This table tells us,
For an input of 1:
- If a high voltage is applied to the bulb , the bulb is lit
- If a high voltage is applied to the bulb then the statement “the bulb is lit” is True
- If a high voltage is applied to the bulb , a high voltage was observed at the output.
- If the switch is on, the bulb is lit
For an input of 0
- If a high voltage is not applied to the bulb, the bulb is unlit
- If no voltage is applied to the bulb, then the statement “the bulb is lit” is False
- If no voltage is applied to the bulb , low or voltage was observed at the output.
- If the switch is off, the bulb is unlit.
Our truth table allows for us to perform an abstraction on our circuit so that we no longer consider the events that physically occur in the circuit, rather the ideas on how the circuit’s output behaves given discrete inputs.
We no longer care about how the circuit is physically implemented; rather we only consider the logical implications of using the particular type of circuit as a building block in other circuits.
Thank you for reaching the end of this paper.
Attributions to media used in this post:
يحيى بن علي, CC BY-SA 4.0, via Wikimedia Commons
© 2021 Vedesh Kungebeharry. All rights reserved.