Last Sync: 2022-06-05 10:30:04

Hardware/Logic_Gates/Nand_gate.md

@@ -57,6 +57,7 @@ f    f     f
 We can see that it inverts the value of AND.
 
 ## Significance of the NAND gate: functional completeness
+
 > **Equipped with just a NAND we can represent every other possible logical condition within a circuit.**
 
 In practice, it is more efficient to use specific dedicated gates (i.e OR, AND, NOT etc) for the other Boolean connectives but in principle the same output can be achieved through NANDs alone.  

Hardware/Logic_Gates/Not_gate.md

@@ -7,28 +7,44 @@ tags:
 ---
 # NOT gate
 
-This gate corresponds to the `NOT` Boolean or negation logical connective. It is really simple and derived from the trivial logical fact that `true` is `true` and `false` is `false` also known as **logical identity**.
+This gate corresponds to the negation logical connective.
 
-### Natural language
- > 
- > The negation operator (`¬` or `~` ) switches the value of a proposition from `true` to `false`. When we put `~` before `true` it becomes `false` and when we put `~` before `false` it becomes `true` .
+Negation switches the value of a proposition from `true` to `false`. When we put `~` before `true` it becomes `false` and when we put `~` before `false` it becomes `true` .
 
-### Truth table
+The truth-table for negation:
+````
+P			~P
+T			F	
+F			T
+````
+In terms of gates this logic obviously corresponds to a simple on/off switch. 
 
-![1-w2ILS6M9pgmLcK6V1PEs3Q.png](../../img/1-w2ILS6M9pgmLcK6V1PEs3Q.png)
+In terms of pure logic, negation is an unary connective in contrast to binary connectives like conjuction and disjunction. It doesn't operate on two parts it simply switches the truth value of whichever proposition it is applied to. 
 
-This corresponds to a simple on-off switch.
+How then, can it be expressed in terms of a gate that takes two inputs and returns an output? Well, we can recreate the logic by manipulating a NAND gate. (Demonstrating the point that many logical conditions are expressible purely via NAND gates.)
 
-In terms of logic gates we would create this by using a single NAND gate. Although it can take a total of two inputs, it would be controlled by a single switch, so both inputs would be set to `1 1` or `0 0` when the switch is activated and deactivated. This would remove the `AND` aspect of `NAND` and reduce it to `NOT` .
+Although a NAND can take a total of two inputs, it would be controlled by a single switch, such that both inputs would be set to `1 1` or `0 0` when the switch is activated and deactivated. This would remove the `AND` aspect of `NAND` and reduce it to `NOT` .
 
 A NAND gate simulating NOT logic
 
 ![Screenshot_2020-08-25_at_15.09.01.png](../../img/Screenshot_2020-08-25_at_15.09.01.png)
 
-TO DO: Come back to this with new book and explain better how a NAND can be reduced to NOT. 
+Going back to the truth-conditions for NAND, NAND-as-NOT just means that the conditons on lines 1 and 4 obtain:
+
+```
+A    B     C
+_    _     _
+
+0    0     1      1    
+1    0     1      2 
+0    1     1      3 
+1    1     0      4
+```
 
 ### Symbol for `NOT` gate
 
 NOT has its own electrical symbol to distinguish it from a NAND:
 
 ![Screenshot_2020-08-25_at_15.18.34.png](../../img/Screenshot_2020-08-25_at_15.18.34.png)
+
+Note that as this is an abstraction of the process, there is no need to show two inputs in the representation of the gate. We have a single input and single output reflecting the unary nature of the negation operator. 

Logic/Truth-functional_connectives.md

@@ -5,8 +5,6 @@ tags:
   - truth-tables
 ---
 
-## Truth values: simple and compound sentences, symbolic representation of each
-
 ## Truth-functional connectives
 
 Sentences generated from other (simple) sentences by means of sentential connectives are [compound sentences](Atomic%20and%20molecular%20sentences.md).
@@ -267,7 +265,6 @@ The example above is a key equivalence that you will encounter a lot especially
 
 ## Enforcing binary connectives through bracketing
 
----
 
 If we had a sentence of the form