Propositional Logic

This is Unit Four of Logic, Part Two: reading the tables, and the argument forms. Part One built the five tables and put sentences into notation. Everything here is what the tables tell you once they exist. The first thing they tell you is about single statements, and about pairs of them. Then comes the payoff: a test for validity that does something no method so far in this course has managed. The counterexample hunt in Unit One could prove an argument invalid and never prove one valid. This test does both, because working every row leaves nothing unexamined. Then the named forms — and they arrive in pairs. Each valid one has a partner that is invalid always, and the partner sounds right, because it reuses the same sentences. Confidence is no guide here at all. By the end you will be able to do five things. One: classify a statement as true no matter what, false no matter what, or dependent on how things happen to be. Two: test two statements for equivalence, and a set of them for consistency. Three: certify validity by full table, and reach the same verdict the short way. Four: name the four conditional forms, and say which two of them are valid. Five: spot the disjunction forms, including the one that quietly changes the operator it was handed.

Start with what a single statement's own column can be. There are three possibilities and no others: it holds on every row, it fails on every row, or it does some of each. The first two are the interesting ones, and each buys its certainty at a price.

The first.

Logic 139 Tautologies

The store is open or it is not — true whatever happens, and for exactly that reason it tells you nothing about the store. Certainty here is bought with emptiness.

The second.

Logic 140 Self-Contradictions

Open and not open, one store, one moment. This is not merely false as things stand — it fails under every possible arrangement, and that totality is what powers indirect proof later.

And the third.

Logic 141 Contingent Statements

True on some rows and false on others. Nearly everything worth saying falls here — the table sorts the claim, and the world settles it.

Now stop reading one column and start comparing two. Two relations matter, and they demand very different amounts: one wants agreement everywhere, the other wants agreement in a single place.

The demanding one.

Logic 142 Logical Equivalence

A conditional and a particular disjunction produce identical columns, which makes them interchangeable anywhere. They can read as completely different sentences — the table has no opinion about that.

The undemanding one.

Logic 143 Consistency

A set qualifies if there is even one row making every member true together. That is a low bar on purpose — it asks whether the statements can coexist, not whether any of them is so.

Both verdicts read straight off the rows, and the point to watch is how differently strict they are — one falls to a single disagreeing row, the other survives on a single agreeing one.

Watch both run.

Logic 144 Testing Equivalence and Consistency

Table the two statements side by side and check row against row. Agreement has to hold everywhere — one differing row ends it, however close the rest came. For the other relation, one row where everything holds together is the whole requirement.

Here is the payoff. Unit One's counterexample method could show an argument invalid and could never show one valid — failing to find a counterexample only meant you had failed to find one. This method examines every possibility, so exhausting them settles it.

The test.

Logic 145 The Truth-Table Test for Validity

Scan for a row where every premise holds and the conclusion fails. Find one and the argument is finished. Get through all of them without finding one and validity is not guessed at — it is certified.

Watch the full test run.

Logic 146 Full Validity Test Walkthrough

Table both premises and the conclusion together, then scan for the one fatal pattern. Every row gets checked, including the ones that look beside the point at a glance — a single row left unexamined can hide the counterexample.

The full table gets expensive fast — each extra letter doubles it. The short method replaces the whole thing with one row of reasoning, and it costs something in exchange: it needs real care at the moments where the values are not forced.

The idea.

Logic 147 The Short Truth-Table Method

Suppose the argument invalid — premises holding, conclusion failing — and push the forced values through. Run into a contradiction and no such row exists. Complete the assignment instead and you are holding the counterexample.

Watch it run, and watch the care.

Logic 148 Short Method Walkthrough

Set the conclusion false and the premises true, then follow what that forces. The contradiction has to be forced rather than wandered into — where the premises leave a value open and you pick one freely, every option has to fail before validity is earned.

Four forms, arranged as two pairs. In each pair one is valid every time and one is invalid every time, and they are built from the very same sentences — which is why the invalid member sounds so convincing. This is the heart of the unit.

The first valid form.

Logic 149 Modus Ponens

Cheque clears, order ships; the cheque cleared; so it shipped. The conclusion cannot fail while the premises hold, and the table certifies it.

Its twin.

Logic 150 Affirming the Consequent

Rain wets the street; the street is wet; therefore it rained — or a hydrant burst. This is where a great many real diagnostic mistakes live, wearing a lab coat.

The second valid form.

Logic 151 Modus Tollens

Blown fuse means darkness; the lights are on; so the fuse held. This runs the conditional backwards, and it is the only backwards move the conditional permits.

And its twin.

Logic 152 Denying the Antecedent

No rain, therefore a dry street — but nothing ever said rain was the only way a street gets wet. The conditional never claimed exclusivity, and this form quietly assumes it did.

Three more forms. Two work, one does not, and the failing one is worth watching closely because of how it fails: it takes the inclusive operator it was given and treats it as the exclusive one, without ever paying for the upgrade.

Conditionals end to end.

Logic 153 Hypothetical Syllogism

Rates up, borrowing down; borrowing down, growth down; so rates up, growth down. Note what the conclusion is: still a conditional. Nothing here says the rates ever moved.

Narrowing a disjunction.

Logic 154 Disjunctive Syllogism

The outage is hardware or software; not hardware; so software. It works by ruling one side out — and only by ruling out, as the next entry shows.

The third twin.

Logic 155 Affirming a Disjunct

A veteran or a teacher; he is a veteran; therefore not a teacher. But the operator was the inclusive one, so he may be both — the argument silently swapped in the exclusive reading.

One more valid form, and its interest is in what the conclusion does not tell you. Two conditionals, a guarantee that one of the two conditions occurs, and a result that names no particular outcome.

The form.

Logic 156 Constructive Dilemma

Litigation costs counsel, settlement costs damages, and one of the two is happening — so something gets paid. What gets paid is left open, which is the form's whole character.

The unit closes on a drill rather than a single problem: short arguments in plain English, at speed, with the valid forms and their twins deliberately mixed together — because in conversation the members of a pair are indistinguishable.

The capstone. Symbolize, name, and rule.

Logic 157 Capstone: Form-Spotting Drill

Take each argument into notation, name the form, and give the verdict. Every invalid one here sounds right because it is built from the same sentences as its valid partner — so how confident you feel tracks nothing at all.

Five things you can now do, and Unit Four is complete. You can classify a statement from its own column — always true, never true, or left to the world — and you know the first two buy certainty by saying nothing. You can test two statements for equivalence, which needs every row to agree, and a set for consistency, which needs only one. You can certify validity by full table — the first method in this course that proves validity rather than only refuting it — and reach the same verdict the short way, provided the contradiction was forced rather than chosen. You can name the four conditional forms: the two that work run from the condition forward and from the failed result backward, and their twins run the other way and fail every time. And you can spot the disjunction pair, including the one that treats an inclusive operator as exclusive without paying for it.