Fundamentals of Logic

The Binary Concept — Yes or No, Nothing in Between

Let me start with something you already know. When you flip a light switch, the light is either on or off. There is no “kind of on”. When you press a doorbell, it is either pressed or not pressed. When a door is closed, it is either closed or open.

This is the binary concept: the idea that a lot of things in the world have only two states, and we can call those states 1 and 0. The 1 might mean ON, the 0 might mean OFF — but the same two digits can represent any pair of opposites:

• 1 = ON, 0 = OFF
• 1 = OPEN, 0 = CLOSED (or the other way round — depends on the device)
• 1 = TRUE, 0 = FALSE
• 1 = HIGH voltage (like +24 V), 0 = LOW voltage (0 V)
• 1 = “I see something”, 0 = “I see nothing” (a sensor)

Why is this useful? Because there is no middle ground. A computer or PLC never has to ask “is this signal 47 % on?” — the answer is always a clean yes or no. That is the secret behind how digital systems can be so fast and so reliable.

A logic gate is a tiny electronic circuit that takes one or more binary inputs and produces a single binary output, based on a fixed rule. The rule decides what combinations of inputs will turn the output ON. The rest of this chapter is about understanding those rules — because they are the same rules a PLC uses to decide whether a motor should run, a valve should open, or an alarm should sound.

The Four Basic Gates — AND, OR, NOT, and XOR

Every digital system in the world — every PLC, every smartphone, every laptop — is built from just a handful of basic logic gates. Let’s meet the four most important ones. Each gate has:

• A name describing what it does (“AND”, “OR” and so on).
• A standard symbol used in circuit diagrams.
• A truth table — a small table that shows the output for every possible combination of inputs.
• A Boolean expression — a tiny mathematical formula that describes the gate.

The AND Gate — “Both Conditions Must Be True”

An AND gate looks at all its inputs and outputs a 1 only when every single input is a 1. If even one input is 0, the output is 0. The Boolean shorthand is Y = A · B (the dot is “AND”) or simply Y = AB.

Where you’d use it: a machine should only start when the safety guard is closed AND the start button is pressed. A mixer should only run when the temperature is high enough AND the lid is closed. The word “and” in plain English almost always becomes an AND gate in the PLC program.

The OR Gate — “Any Condition Will Do”

An OR gate outputs a 1 if at least one of its inputs is a 1. The output is 0 only when every input is 0. Boolean shorthand: Y = A + B (the plus sign is “OR” in Boolean — don’t confuse it with addition).

Where you’d use it: an alarm should sound when the door OR the window is open. A pump should run when tank A is full OR tank B is full. Whenever you hear “either … or …”, you’re hearing an OR gate.

The NOT Gate — “Flip the Answer”

The NOT gate is the simplest of all — it has only one input. If the input is 1, the output is 0. If the input is 0, the output is 1. It just flips whatever you give it. Boolean shorthand: Y = A̅ (a bar over the letter means “NOT this”).

Where you’d use it: a “DOOR OPEN” warning light should turn on when the door is NOT closed. The NOT gate is also called an inverter because it inverts (flips) the signal.

The Exclusive-OR (XOR) — “One but Not Both”

The XOR gate is the picky cousin of the OR gate. Its output is 1 only when its inputs are different from each other (one is 1 and the other is 0). If both inputs are the same — both 0 or both 1 — the output is 0. Boolean: Y = A ⊕ B.

Where you’d use it: the classic example is a stair light with a switch at the top and bottom. The light turns on when one switch is up and the other is down — and turns off when both are up or both are down. XOR is also the heart of binary addition (since 1 + 1 = 10 — the result bit is 0, not 2).

NAND, NOR, and XNOR — The “Inverted” Versions

Three more gates worth knowing — each one is just an “inverted” version of one of the four above:

• NAND = “AND followed by NOT” — gives the opposite of AND. Drawn as the AND symbol with a small bubble at the output.
• NOR = “OR followed by NOT” — opposite of OR. Drawn as the OR symbol with a bubble.
• XNOR = “XOR followed by NOT” — output is 1 when inputs are the same (the opposite of XOR).

NAND and NOR are special: any logic circuit can be built using only NAND gates, or only NOR gates. They are called universal gates for that reason.

Boolean Algebra — A Tiny New Math Just for Logic

You already know regular algebra: 2 + 3 = 5, x · y = y · x, and so on. Boolean algebra is exactly the same idea, but instead of numbers we use the binary values 0 and 1, and instead of plus, minus, and times we use AND, OR, and NOT.

Why bother? Because Boolean algebra lets us write down a complicated logic problem as a short formula, then simplify it into a smaller circuit. A simpler circuit means fewer gates, fewer lines of code, fewer chances for bugs, and a faster scan time.

The Boolean Symbols

Notice that the AND dot is often left out — AB means the same as A · B, just like in regular algebra xy means x · y. The plus sign is OR in Boolean — don’t accidentally read it as ordinary addition.

Three Laws You Need to Know

Most Boolean laws look exactly like the laws of ordinary algebra. There is one important exception — the second part of the distributive law, which only works in Boolean algebra.

De Morgan’s Theorems — Two Rules You Will Use Every Day

One more pair of laws is so important to PLC work that I want to call it out separately. Augustus De Morgan figured out two rules that link AND, OR, and NOT together:

Why does this matter? Because in ladder logic, AND becomes “contacts in series” and OR becomes “contacts in parallel”. De Morgan’s theorems let us convert a NAND into “NC contacts in parallel” and a NOR into “NC contacts in series” — which is exactly how we’ll build them in the worked programs at the end of this chapter.

Building a Logic Circuit from a Boolean Expression

Now let’s put it together. Suppose someone hands you a Boolean expression and asks you to draw the circuit. The trick is to read it from the inside out — handle the innermost operation first, then work outward.

Try a slightly bigger one. The expression Y = A(BC + D) needs three gates:

The pattern: read the Boolean expression from the inside out (deepest parentheses first), pick a gate for each operation, and chain the outputs together. With practice this becomes almost automatic.

Going the Other Way — Reading a Circuit into a Boolean Expression

You also need to do the reverse: someone shows you a logic circuit and asks “what does this do?” To answer, you trace the circuit from the inputs forward to the output, writing down the result of each gate as you go.

The key skill here is patience: take it one gate at a time. Label each intermediate signal with its expression. By the time you reach the output, you’ve assembled the whole formula.

Hardwired Logic vs. Programmed Logic — Why Factories Switched

This is the section that explains why PLCs took over the world.

Before PLCs existed, every logic decision in a factory was made by physical electrical equipment — relays, switches, timers, contactors — all wired together with copper. We call this hardwired logic. The “program” was the wiring itself. To change the logic, an electrician had to physically rewire the panel.

With a PLC, the same logic is stored in software inside the CPU. We call this programmed logic. To change the logic, you just edit the program on a laptop and download it. The wiring to the field devices does not change.

Why Programmed Logic Wins

• Easy to change. A logic change in software takes minutes; the same change in a relay panel takes hours of rewiring.
• Smaller and cheaper. One PLC the size of a textbook can replace a wall full of relays.
• More reliable. No moving contacts to wear out, no coils to burn, no contacts to pit.
• Self-documenting. The ladder logic on a screen shows exactly what the program is doing — much easier to follow than tracing wires through a panel.
• Easier to troubleshoot. The engineer can watch the logic execute live on a laptop, with energised rungs lit up in green.

Hardwired logic isn’t completely dead, though. Some safety circuits — emergency stops, machine guarding interlocks — are still hardwired in parallel with the PLC, because regulators don’t trust software for life-and-limb decisions. But for everyday control, programmed logic has won the argument.

Word-Level Logic Instructions — Operating on 16 Bits at Once

Up to now we’ve talked about logic working on single bits — one input, one output, one yes/no answer at a time. But PLCs can also do logic on entire words (groups of 16 or 32 bits) in a single instruction. These are called word-level logic instructions.

The four basic word-level instructions are:

• AND (mask) — perform a bit-by-bit AND between two 16-bit words. Used to “mask” off bits you don’t care about — set them to 0.
• OR — bit-by-bit OR between two words. Used to set certain bits without disturbing the others.
• NOT — invert every bit in a word.
• XOR — bit-by-bit XOR between two words. Used to detect which bits have changed between two readings.

How does it actually work? The PLC takes the two 16-bit words, lines them up bit by bit, and applies the gate’s rule to every bit position independently. The result is a new 16-bit word.