Propositional Logic

This is Unit Four of Logic: Propositional Logic. It runs long, so it comes in two parts, and this is the first. Unit Three took statements apart by their classes. This unit ignores classes entirely and looks at how whole statements get joined together — and that turns out to be enough to build a system that decides validity by calculation rather than by judgment. Five operators do all the joining. Every connective English has ever invented, however elaborate, comes down to those five, and Part One is largely about getting real sentences into them accurately — which is harder than it sounds, because three ordinary English phrasings place the pieces in the opposite order from the one people expect. Then the tables. A table is not a description of what a connective does. It is what the connective is. Once all five are on record, the rest of the unit is computation. By the end you will be able to do five things. One: symbolize a compound statement, choosing the connective the English is actually asserting rather than the one it sounds like. Two: translate the three phrasings that reverse the arrow, without reversing it. Three: tell a necessary condition from a sufficient one, and put each in its proper slot. Four: find the main connective, which is the first move in every technique here. Five: build a full truth table at the right size and in standard order.

Two definitions to start. A statement is simple when nothing inside it is itself a statement, and compound when something is. The important word in that is "inside" — compounds are assembled out of whole statements, which stay whole after assembly.

The distinction.

Logic 116 Simple vs. Compound Statements

Rain on its own is simple. Rain together with a cancelled game is compound, and the cancellation is a complete statement sitting inside another one — built up rather than mixed together.

The whole toolkit.

Logic 117 The Five Connectives

One for denying, one for joining, one for offering alternatives, one for making something conditional, and one for locking two statements together. Five is sufficient — every English connective reduces to them.

Three of the five behave about how you would guess. The one worth slowing on is the alternative-offering operator, because English uses one word for two different meanings and the system commits to only one of them by default.

The first operator.

Logic 118 Negation

True becomes false and false becomes true. The care needed is about reach: denying a whole compound and denying something inside it are different claims, and English blurs them.

The second.

Logic 119 Conjunction

True only when each half holds. And every contrastive word English offers — however, although, yet — becomes this same operator. They differ in feeling; the system tracks only the joint claim.

The third, by default.

Logic 120 Disjunction

An alarm answering to smoke or heat certainly answers to both together. False only when neither holds. That inclusive reading is the default here, and the next entry handles the other one.

And the other reading.

Logic 121 Exclusive "Or"

Soup or salad with the meal, and ordering both tests the exclusion. This meaning is not a sixth operator — it is assembled from the ones you have. Use it only where the context plainly demands it.

Now the operator that does the most work in the rest of the course. It has exactly one false case, and it asserts neither of the statements it joins — only the link between them. That last point is where most confusion about it starts.

The operator.

Logic 122 The Conditional

A cleared cheque and a shipped order: false in exactly the case where the first happens and the second does not. And the sentence does not claim either one occurred.

The two slots.

Logic 123 Antecedent and Consequent

One statement supplies the condition and the other the result. English reorders these on the page constantly — the word "if" marks the condition wherever in the sentence it happens to sit.

And the operator that locks both ways.

Logic 124 The Biconditional

True when the two sides agree. Note what it costs: this makes two separate promises running in opposite directions, and most English uses of "if" make only one of them.

Here is where translation actually goes wrong. Three ordinary phrasings put the pieces in an order that feels backwards, and one of them is a straight echo of the trap from Unit Three — the same English word doing the same reversal in a different system.

The first phrasing.

Logic 125 Translating "Unless"

Walking unless it rains says one of the two happens, and it can be rewritten as a conditional running from the absence of rain. Resist adding more: it does not promise what happens when the rain arrives.

The second.

Logic 126 Translating "Only If"

Boarding permitted only with a ticket means everyone boarding holds one — not that holding one gets you aboard. Same restricting word as Unit Three, same reversal, new notation.

And the pair, side by side.

Logic 128 "If" vs. "Only If"

Passing with a score of sixty makes sixty enough. Passing only with sixty makes it required and nothing more. Those are converse claims, not two ways of saying one thing — drill the pair until it is automatic.

The same distinction in the vocabulary it usually appears in outside a logic class, which is worth naming because mistaking one for the other has a real record of causing harm.

Which slot each takes.

Logic 127 Necessary vs. Sufficient Conditions

Fire requires oxygen, so oxygen belongs in the second slot — and it comes nowhere near guaranteeing fire. Treating something merely required as though it were enough is how this error does damage.

One structural idea before the procedures. Brackets decide what each operator reaches, and whichever operator applies last determines what the whole compound actually is. Every technique in this unit begins with that identification.

The first move, every time.

Logic 129 Scope and the Main Connective

Denying a whole conjunction produces a denial; denying one half produces a conjunction. Different statements, different tables, and the brackets are the only thing telling them apart.

Five sentences taken into notation, and the pair from Topic Four appears here doing its damage — two instructions one word apart that send the arrow in opposite directions.

Watch it run.

Logic 130 Symbolizing Compound Statements

Break out the simple statements, letter each one, pick the operator the sentence actually asserts, and bracket so that one operator ends up governing the whole. Where a restricting word appears, check which slot it is filling before writing the arrow.

Now the definitions. A table here is not a convenient summary of how an operator behaves — the table is the operator, completely. Four of the five will look obvious. One of them looks wrong to nearly everyone, and the entry after it explains why it is not.

The smallest table.

Logic 131 Truth Table: Negation

True flips to false, false flips to true, and that is the entire definition. Two rows, and nothing is being summarised.

Four rows, one of them true.

Logic 132 Truth Table: Conjunction

True on the single row where both halves are, false on the other three. One failing half sinks it, and that lopsidedness does heavy work later.

Its mirror image.

Logic 133 Truth Table: Disjunction

False on the single row where both halves fail, true elsewhere. The top row, where both hold, comes out true — that is the inclusive reading, now official.

And the controversial one.

Logic 134 Truth Table: The Conditional

False on exactly one row: the condition holding while the result fails. True everywhere else, and that includes both rows where the condition never happened at all.

Why those rows are right.

Logic 135 Why the Conditional's Table Looks Wrong

Offer twenty dollars for a mown lawn; the lawn goes unmown. You did not lie. The promise was never triggered, and an untriggered promise has not been broken — which is exactly what those rows record.

The fifth completes the set.

Logic 136 Truth Table: The Biconditional

True on the two rows where both sides agree, false on the two where they differ. All five definitions are now on record, and everything after this is computation.

Two housekeeping facts that stop tables going wrong before any thinking happens: how many rows the table needs, and the order they get filled in — which is not a stylistic convention.

The size, and the fill.

Logic 137 Row Counts: 2ⁿ

The row count doubles with each additional letter, and the standard fill halves the alternation column by column. That pattern is what guarantees every combination appears once and none appears twice.

The procedure, and one warning about reading the result: a completed table has several columns in it and only one of them is the answer. The rest were scaffolding.

Watch a table get built.

Logic 138 Building a Full Truth Table

Count the distinct letters to fix the height, fill the reference columns in the standard pattern, then work outward one column per piece. The column belonging to the operator that governs the whole is the verdict — reading any other column gives you an answer to a different question.

Five things you can now do — and Part Two reads these tables for verdicts. You can symbolize a compound statement, picking the operator the sentence is actually asserting: every contrastive word English has becomes the joining operator, and the alternative operator is inclusive unless the context insists. You can handle the three phrasings that reverse the arrow, and you know the restricting word from Unit Three is doing the same job here in new notation. You can place a necessary condition and a sufficient one in their proper slots, and you know that treating a requirement as a guarantee is where this confusion does real damage. You can find the main connective and say what kind of statement a compound is — the first move in every technique in this unit. And you can build a full table: right height, standard fill, columns from the inside out, and the verdict read off the one column that governs the whole.