Boolean Algebra

[steve_bank]

Philosophical arguments can reduced to Boolean Algebra for which a truth table can be constructed.

When working I would see truth tables instead of words, it became subconscious. No verbal or translation to words. Beyond simple logic words become impossible.

For the AND for x to be true it is necessary but not sufficient that6 A be true

In technology international standards for logic symbols.

Boolean algebra - Wikipedia

en.wikipedia.org

 

[Swammerdami]

Steve's post looks a bit lonely, so I'll offer some hand-drawn examples. He shows the primitives of COMBINATORIAL logic. I will show the simplest and most common element of SEQUENTIAL logic -- a one-bit memory, aka a "flip-flop." Three circuits are shown. Each is a flip-flop. (Or rather a "latch", the term preferred back in the day.)



In the first example two NOR gates feed into each other. When both +Set and +Reset are negative (false) the flip-flop remembers its last setting. Activate +Set to force +Q to be true (and -Q negative or false). Activate +Reset for the vice versa.

As shown in the second example you can do the same thing with NAND gates. Or with one AND gate and one OR gate as in the third example.

When I was a circuit designer I worked primarily with TTL and ECL devices, both built from bipolar transistors. The highest speed computers used ECL (NPN). Two anecdotes about those devices may be of interest.

(1) In an ECL OR-gate (or NOR-gate), the inputs connect to the bases of transisters wired in parallel, but in an AND-gate (or NAND-gate) transistors are in series. Thus for the AND-gate a 5.2 volt differential was needed, while about a volt less might suffice if only OR-gates are used. (I write "might suffice" since AFAIK nobody relied on 4.2 volts unless the chip supplier's specs specifically allowed this.) IBM's chips needed only 4.25 volts so the "add-on" people had to route an extra voltage wire to their circuitry -- unless they were using only OR-gates, and Fairchild chips.

Another consequence of the AND-gate internals is that one of the two inputs had about 250 picoseconds of extra margin! I doubt if many designers were perfectionistic enough to take advantage of this.

(2) In the third flip-flop depicted above, the OR-gate is shown as a "Wire-Or." With ECL(NPN) devices, connecting two outputs together provides an OR; no transistors are needed. This can reduce the chip count for a given circuit, and can be convenient for other reasons, but notably The wire-Or reduces delay. In the example circuit, the input data and the output data are the very same wire -- this is a "zero delay latch"!

 

[Jarhyn]

the OR-gate is shown as a "Wire-Or."

Interestingly, this means that all of logic can be accomplished with a hypothetical NOT gate and mere wires, which is how original Minecraft redstone circuitry worked (as powered redstone torches disabled, creating a NOT).

 

[steve_bank]

NAND and NOR gates are considered universal gates, meaning that any Boolean logic function or digital circuit can be implemented using only one type of gate (either all NANDs or all NORs

In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and negation (not) denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations in the same way that elementary algebra describes numerical operations.

Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847),[1] and set forth more fully in his An Investigation of the Laws of Thought (1854).[2] According to Huntington, the term Boolean algebra was first suggested by Henry M. Sheffer in 1913,[3] although Charles Sanders Peirce gave the title "A Boolian [sic] Algebra with One Constant" to the first chapter of his "The Simplest Mathematics" in 1880.[4] Boolean algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming languages. It is also used in set theory and statistics.[5]

Boolean algebra is a branch of abstract algebra. It is the study of a specific type of algebraic structure that captures the essential properties of logical operations and set theory operations.

Boolean algebra is a form of formal logic, specifically the mathematical foundation and an algebraic model for classical propositional logic. It provides a formal system for representing and manipulating logical statements using algebraic rules.

https://www.analog.com/en/resources/design-tools-and-calculators/ltspice-simulator.html

LTspice is a free electrical simulator that will draw and simulate logic. Plenty of online tutorials on logic.

Boole was a mathematician. One of the first applied uses was before digital logic. Relays were used. Boolean algebra was used to design relay logic. Like elevator controls.

There are methods and algorithms to go from a truth table to logic expressions,.

So if you can reduce a verbal argument to a binary true-false truth table you can synthesize a logical expression for the argument. And then test its validity.

Some specialize in logic synthesis.

 

[steve_bank]

De Morgan is an important part of Boolean logic and apparently modal logic as well..

De Morgan's laws - Wikipedia

en.wikipedia.org