This article looks at Propositional Logic, also called Statement Calculus, from a combinatorial and algebraic point of view (Sections 3-6), its implementation in software (Section 7), and its application to digital electronics (Section 10). Historical sections cover the shift in viewpoint from classical logic based on Aristotle’s syllogism to modern symbolic logic (Section 2) and the axiomatization of logic (Section 9). (See logic sourcebook for the original source papers (1830-1881) that drove this shift.)
In Section 7, we implement the grammar of the statement calculus in the Symbolic Logic Simulator (SLS), a program written in 28 lines of Forth code, that allows computer-aided verification of any theorem in Propositional Logic (see Appendix 1 for source code). The program makes it straight-forward to explore non-obvious logical identities, and verify any propositional logic theorem or conjecture, in particular see Appendix 2 for key identities in the statement calculus (duality, algebraic, and canonical identities).
The concept of linguistic adequacy is developed in Section 8 and the NAND Adequacy Theorem is proved showing that NAND can generate all logical operations. A corollary is that any digital logic circuit can be built up entirely using NAND gates, illustrated using the free Digital Works software.
Rapid prototyping can be an accelerator for hardware or embedded solutions. But for a successful outcome, follow the recommended checklist / task sequence given below, as all of these steps will likely need to be touched upon at some point in the process. Continue reading this article…
For under £10, you can put together a microcontroller development platform, ready to program directly from your PC over USB using free Arduino software. Once programmed, your microcontroller will run autonomously, untethered from your PC, powered by as small a battery power supply as a single 1.5V AAA or 3V CR2032 coin cell. You can have it interact with its environment using dozens of low-cost sensors and motors. Everything you need to explore the exciting world of embedded systems is available to you, typically for less than a day pass on the London underground.
A homebrew Arduino Nano microcontroller development kit for under £12 (including optional OLED display)
Building a fully analog electronic piano using only resistors, capacitors, and transistors, is an insightful experiment in electronic sound generation from first principles. I designed and built a 13-key analog piano in early 2019 using discrete through-hole components on a breadboard powered off a 9V DC battery. The design creates 13 astable multivibrator oscillator circuits, each able to be tuned to a given note frequency in the C5 to C6 range. The outputs of the oscillators are collected (mixed) to create a polyphonic analog audio signal that is amplified and run through an 8-ohm speaker. The device fits into an 11x25cm footprint. Check out how it sounds! (To hear the explanation of how it works, start at the beginning.)
Electronics, computing, and applied mathematics are gateway subjects to modern technology.
For young learners, we believe that electronics provides an ideal entry point. It is practical, with manipulables. It is easy to see cause and effect. With the right equipment and approach, exploring electronics can begin for children as early as 3 years old.
There are many tangible benefits for young learners getting started in electronics:
fine motor skill development,
an intuition for how technological things work at a component level,
the integration of technology into the palette for imagination and creativity,
improved self-confidence,
strengthening a growth mindset,
building resilience,
raising the threshold of frustration,
better dexterity,
stronger focus.
A three year old wiring his first circuit and the joy at seeing the LED, which he selected, light up!
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