Predicate Logic

This is Unit Six of Logic: Predicate Logic. The last two units treated every simple statement as an unbreakable unit — a letter, with a truth value and no interior. That is enough for a great deal, and it fails on the single most famous argument in the subject. Everyone mortal, Socrates human, therefore Socrates mortal. In the previous notation those are three unrelated letters, and the argument is invalid. But it is not invalid. What makes it work is happening beneath the level the notation can see. So this unit goes down a level: individuals, properties they have, relations between them, and two quantifiers that say how many. Nothing from Units Four and Five gets discarded — every rule you have keeps working, and four new ones let you get at the material inside. Two of those four are relaxed and two are severely restricted, and nearly every mistake in this unit is a restricted rule used as though it were a relaxed one. By the end you will be able to do five things. One: say why an argument nobody doubts can come out invalid in the old notation. Two: translate a sentence with its quantifier and its connective correctly paired. Three: state the restrictions on the four new rules, and say which two bite. Four: say why stacking two quantifiers in the other order changes the claim. Five: refute a predicate argument by building a small world where it fails.

Begin with the failure that motivates everything else. An argument nobody doubts comes out invalid under the previous notation, and the reason is not a defect in the argument. It is that the machinery cannot see small enough.

The motivating failure.

Logic 256 Why Propositional Logic Isn't Enough

Symbolized the old way, the Socrates argument is three separate letters with no connection, so nothing follows. What drives it sits inside the statements, below the level the previous units could reach.

Two pieces of vocabulary, and the distinction between them is load-bearing — the strictest restriction in the whole unit is enforced entirely on the difference between naming something and merely holding a place for it.

The two kinds of term.

Logic 257 Constants and Variables

One kind picks out a particular individual; the other stands in for individuals without settling which. Blur the two and the restrictions later in the unit will catch you out.

What gets said about them.

Logic 258 Predicates

A property letter attached to a name produces something with a truth value. Attached to a placeholder it does not — that is not yet a statement, and will not be until a quantifier reaches it.

Two quantifiers, and that is the entire inventory — every claim about how many gets built from these. Once they exist, Unit Three's four categorical forms can be rewritten in this finer notation, and the rewriting exposes a pattern the next topic turns into a law.

The first.

Logic 259 The Universal Quantifier

A claim covering everything under discussion. What counts as everything is fixed by the domain you are working in, not by the notation.

The second.

Logic 260 The Existential Quantifier

Asserting that something has the property — at least one, in exactly the sense Unit Three used. A false claim of this shape is still perfectly well formed.

Unit 3, rebuilt.

Logic 261 The Four Forms, Upgraded

The four categorical forms written out with quantifiers. Watch the pattern that appears: the universal ones take the conditional and the existential ones take the joining operator. That pattern is about to become a rule.

Three entries that between them account for most translation errors. What a quantifier reaches, which connective it must take, and then six sentences worked through. The pairing is not a stylistic habit — get it wrong and you produce something grammatical, meaningful, and not remotely what you meant.

What the quantifier reaches.

Logic 262 Free vs. Bound Variables

A placeholder is covered when a quantifier's reach extends over it, and loose otherwise. A statement has none loose. Brackets set that reach exactly as they set an operator's, with more at stake.

The law.

Logic 263 The Pairing Rule

Pair a universal with the joining operator and you have claimed everything in the world has both properties. Pair an existential with the arrow and it comes out true the moment anything fails the first property. Both are nonsense you can read aloud.

Six sentences, worked.

Logic 264 Translation Practice

Pick the quantifier the sentence needs, apply the connective that goes with it, then read the symbols back into English as a check. Reading back is the step that catches a mispairing, because the mistake is inaudible in the symbols and obvious in the sentence.

Denying a quantified statement, and the same flip you met in Unit Five happens here one level down: as the denial crosses a quantifier, the quantifier changes into the other one. Forgetting the change is the same error in a new place.

The rule.

Logic 265 Quantifier Negation

Denying that everything has a property says something lacks it. Denying that anything has it says everything lacks it. The denial moves inward and the quantifier switches as it passes.

Watch it applied.

Logic 266 Applying Quantifier Negation

Drive the denial inward past the quantifier, flipping it, then let the propositional rules finish the work inside the brackets. The flip happens only on an actual crossing — a denial already sitting inside the brackets never crossed anything.

Four new rules for getting quantifiers off and putting them back on. Two of them will let you do almost anything. Two are hedged about with conditions, and the asymmetry is not arbitrary — each restriction blocks a specific way of proving something false.

The first, and it is generous.

Logic 267 Universal Instantiation

From a claim about everything, write it about anything you like — a name already in play, or a placeholder. What holds universally holds of each particular case.

The second is not.

Logic 268 Universal Generalization

Going from one case back up to everything requires that the case was genuinely arbitrary — no name, and nothing smuggled in from a live assumption. Generalising from a name is how one proves the universe is Socrates.

The third is the strictest thing here.

Logic 269 Existential Instantiation

Knowing something has a property lets you name it — but the name has to be one the proof has never used. Reuse an existing name and you have quietly claimed two different somethings are the same thing.

And the fourth is generous again.

Logic 270 Existential Generalization

From a claim about a named individual, conclude that something has the property. Note the shape of the asymmetry: the two rules that put quantifiers on differ enormously in strictness, and so do the two that take them off — but not in the same direction.

Two worked proofs. The first shows the basic shape — strip the quantifier, use the rules you already have, put a quantifier back. The second shows the fresh-name requirement forcing the order of the whole proof, which is where people actually lose marks.

The basic shape.

Logic 271 First Predicate Proof

Take the quantifier off to get a claim about a named individual, finish with the propositional rules, and you are done. Stripping a universal is free with names and stripping an existential is not — treating them alike is the error the next entry is built around.

And the order it forces.

Logic 272 Proof with EI Restrictions

Because the name has to be unused, the existential premise must be handled before any universal introduces a name. Reverse the order and the name is already taken. This is not bookkeeping — reusing one asserts an identity nothing gave you.

Now the capability that puts this system permanently beyond categorical logic. Predicates can take two terms and express a relation between them — and once two quantifiers can stack, the order they stack in changes the claim entirely.

Two terms instead of one.

Logic 273 Relational Predicates

A property letter taking two names expresses something holding between them, and reversing the two names states a different thing altogether. Ask anyone.

Stacking two quantifiers.

Logic 274 Overlapping Quantifiers

Everyone loving someone, against someone being loved by everyone. Same letters, different order, wildly different claims — and swapping mixed quantifiers is an invalid move that is very hard to resist.

Translating them.

Logic 275 Translating Relational Sentences

Take the outermost quantifier from the sentence's subject and nest inward from there. The pairing law applies at every level of the nesting, not only at the top one.

One more relation, and it is special: it lets the system say how many. Exclusivity, exceptions, superlatives, and the ordinary word "the" all turn out to need it, and the translations get long.

The relation.

Logic 276 Identity

Asserting that two names pick out the same thing, so anything true under one name is true under the other. The relation holds between individuals, not between names — the names differ and the man does not.

What it unlocks.

Logic 277 "Only," "Except," and Superlatives

Exclusivity becomes an existence claim plus a clause ruling everyone else out. These run long, so build them a clause at a time rather than trying to write the whole thing in one pass.

And the ordinary word "the".

Logic 278 Definite Descriptions

Something fits the description, nothing else does, and it has the property — three assertions packed into one small word. When nothing fits at all, the whole statement comes out false rather than meaningless.

One proof carrying two variables at once, which is where both restrictions become live at the same moment — the order of stripping and the condition on generalising, competing inside a single derivation.

Watch it run.

Logic 279 Relational Proof Walkthrough

Prove that if someone is loved by everyone, then everyone loves someone. Generalising on a name that arrived from an existential turns a claim about one unnamed thing into a claim about all of them — which is exactly the move the restriction exists to stop.

The unit's last idea completes the system. A proof certifies validity and cannot certify invalidity — a proof you failed to find might still exist. So there is a second method that works the other way: build a small world where the premises hold and the conclusion does not.

The other half.

Logic 280 The Finite-Universe Method

Shrink the domain to one or two individuals and the quantified statements become plain compounds — universals into joinings, existentials into alternations. A tiny world is a single row of a truth table for this system.

Watch one get built.

Logic 281 Proving Invalidity

Choose the world, then assign the properties so every premise holds and the conclusion fails. The work is in the premises — a model refutes nothing unless all of them genuinely come out true in it, so checking the conclusion first wastes effort.

The capstone. Staff, keys, and a vault.

Logic 282 Capstone: Full Predicate Analysis

Translate a passage carrying a relation and an exclusivity clause, prove it, then weaken a premise and refute the result with a two-member world. A proof that stalls is evidence and not a verdict — you may be looking at an invalid argument, or at your own missing step.

Five things you can now do. You can say why the Socrates argument fails in the old notation: three unconnected letters, with the whole engine sitting inside the statements where the previous units could not look. You can translate with the quantifier and connective correctly paired — the universal one takes the arrow and the existential takes the dot — and you know to read the symbols back into English, because that is what catches the error. You can state the restrictions: taking a universal off and putting an existential on are both free, generalising to everything needs a genuinely arbitrary case, and naming an existential needs a name nobody has used. You can say why stacking quantifiers the other way round changes the claim, and that swapping mixed ones is invalid however natural it feels. And you can refute an argument by building a two-member world where its premises hold and its conclusion fails — because proofs only certify in one direction.