The Modern Classification: What It Is For
Chapter Thirty-Three
Syllabus topic 2.6, "Modern classification of Propositions - Aim of modern classification, Kinds of simple and compound propositions and basic truth tables for compound propositions."
Pages 159 to 162 of 334
In one line
The modern classification divides propositions by their structure rather than by their subject and predicate, so that the parts which decide an inference are the parts the classification names.
In the wording a student can write in an examination: the aim of the modern classification is to divide propositions in a way that reflects the features on which their logical behaviour depends. It therefore asks first whether a proposition contains another proposition as a part, dividing propositions into simple and compound, instead of asking about quantity and quality.
The aim, stated properly
MU asks for the aim and not for the kinds, so the aim has to be answerable on its own. It has four parts.
To classify by what matters to inference. Quantity and quality are real features, but they are not the only ones that decide what follows from what. Whether a proposition is compound decides it too, and the traditional scheme never asked.
To reach the propositions the older scheme could not. Relational propositions, compounds, and propositions with more than one quantifier: the seven failures of the last chapter are the specification the modern classification was built against.
To make the test mechanical. Once propositions are classified by structure and the connectives are defined truth-functionally, whether a compound is true can be computed rather than argued about. That is what a truth table is, and it is possible only because of the way the classification is drawn.
To separate form from content completely. The traditional scheme keeps the subject and predicate in view, which keeps the subject matter in view. The modern scheme replaces a whole simple proposition with a single letter, so nothing of the content survives into the analysis at all.
The first cut: simple and compound
The modern classification's first question is the one prepared at sequence 260.
A simple proposition contains no other proposition as a component. It has constituents only. "The notice was served."
A compound proposition contains at least one component, that is, at least one part which is itself a proposition. "The notice was served and the rent was unpaid."
This single cut is what makes the rest possible. A simple proposition is written as a single letter, p, q, r, because there is nothing inside it that a truth table needs to see. A compound proposition is written as its components joined by connectives, and the truth table then works on the components.
Notice the contrast with the traditional first cut. Traditional logic asks whether the proposition asserts absolutely or conditionally, which puts hypotheticals in one box and everything else in another. Modern logic asks whether the proposition has propositional parts, which puts hypotheticals, disjunctions and conjunctions together in one box, since they behave alike, and leaves the categoricals in the other.
The Modern Classification: What It Is For
Why the modern cut is the better one
Because the members of each modern class behave alike and the members of each traditional class do not.
All compounds are governed by their connectives and can be tested by truth tables. All simples must be analysed internally, into subject and predicate or into a relation and its terms, and quantified where necessary.
The traditional class of "conditional" propositions contained hypotheticals and disjunctions, which is right, and excluded conjunctions, which is arbitrary: "p and q" is exactly as compound as "if p then q". The traditional class of "categorical" propositions contained singular, universal and relational propositions together, though they behave quite differently.
A classification is good when knowing which class a thing is in tells you how it behaves. That is a criterion from Module IV, at sequence 630, applied here, and it is the honest answer to why the modern classification replaced the older one.
What the modern scheme does with quantity
It does not throw quantity away; it relocates it.
In the modern treatment, quantity is not a feature of the proposition as a whole. It is a quantifier attached to a variable inside a simple proposition, and it can occur more than once, with a scope. That is what answers failure four of the last chapter.
"All contracts are agreements" becomes, in the modern reading, a general proposition: for anything whatever, if it is a contract then it is an agreement. Written that way, it says nothing about whether any contracts exist, which answers failure two, and it makes the universal into a conditional, which is where existential import goes.
This is MU's topic 2.7, at sequence 360, and it is the deepest single difference between the two schemes.
A worked example
Take three propositions and classify each in both schemes.
"No agreement made by a minor is enforceable."
Traditional: categorical, universal negative, E.
Modern: simple, and generally quantified. For anything whatever, if it is an agreement made by a minor then it is not enforceable.
"If the notice was served, the tenancy ended."
Traditional: conditional, hypothetical.
Modern: compound, an implication with two components.
"The notice was served and the rent was unpaid."
Traditional: no class. The scheme has nowhere to put it and treats it as two propositions.
Modern: compound, a conjunction with two components.
What the comparison shows. The second and third are treated as quite different things by the traditional scheme and as the same kind of thing by the modern one, and the modern grouping is the useful one, because both can be tested by the same method. Meanwhile the first is treated as a single indivisible form by the traditional scheme and is opened up by the modern one into a quantifier, a variable and a conditional, which is what allows the modern scheme to say what the older one could not about existence.
The Modern Classification: What It Is For
Distinctions that carry marks
| Traditional classification | Modern classification | |
|---|---|---|
| First question asked | Does it assert absolutely or on a condition? | Does it contain another proposition as a part? |
| First cut | Categorical and conditional | Simple and compound |
| Basis | Manner of assertion | Structure |
| Unit of analysis | The term | The whole proposition |
| Where conjunctions go | Nowhere | With the other compounds |
| Where quantity lives | In the proposition, once | In a quantifier, with a scope, possibly more than once |
| Test available | Rules stated in words | Truth tables, mechanical |
| Simple proposition | Compound proposition | |
|---|---|---|
| Components | None | At least one |
| Symbolised as | A single letter | Components joined by connectives |
| Analysed by | Quantifiers, relations, predicates | Truth tables |
| Examples | The notice was served. All contracts are agreements. | If p then q. p and q. Either p or q. |
What this does not mean
The modern scheme is not a rival account of the same material. It is a different division of a wider field. Every A, E, I and O proposition is a simple proposition in the modern scheme, so the two classifications intersect rather than compete.
"Simple" does not mean short or easy. "Every creditor of an insolvent company incorporated before 1956 has a remedy against the directors personally" is a simple proposition: long, difficult, and containing no component.
Compound does not mean complicated. "p and q" is compound and trivial.
Quick revision
The aim, four parts: classify by what actually decides inference; reach the propositions the older scheme could not; make the test mechanical; separate form from content completely.
First cut: simple, containing no component, against compound, containing at least one.
A simple proposition becomes one letter. A compound becomes components joined by connectives.
Quantity is relocated, from a feature of the proposition to a quantifier with a scope inside it.
"All S is P" becomes a conditional: for anything whatever, if it is S then it is P. This is where existential import goes.
Why the modern cut is better: the members of each class behave alike, which is the test of a good division.
Test yourself
1. State the aim of the modern classification.
To divide propositions according to the features on which their logical behaviour actually depends, rather than according to quantity and quality alone. That means classifying by structure, so that propositions which behave alike fall together; reaching the propositions the traditional scheme could not handle, such as relational and compound propositions; making the testing of inferences mechanical, which truth tables allow; and separating form from content completely, so that a whole simple proposition can be replaced by a single letter.
The Modern Classification: What It Is For
2. What is the first cut the modern classification makes, and how does it differ from the traditional first cut?
The modern scheme first asks whether a proposition contains another proposition as a component, dividing propositions into simple and compound. The traditional scheme first asks whether the assertion is absolute or conditional, dividing them into categorical and conditional. The difference matters because the traditional cut separates hypotheticals from conjunctions, which behave alike, and groups singular, universal and relational propositions together, which do not.
3. Why is the modern division said to be a better one?
Because knowing which class a proposition belongs to tells you how it behaves, which is the test of a good division. Every compound is governed by its connectives and can be tested by a truth table; every simple proposition has to be analysed internally into predicates, relations and quantifiers. The traditional classes do not have this property, since conjunctions have no class at all and the categorical class contains propositions of quite different logical character.
4. What happens to quantity in the modern scheme?
It is relocated rather than discarded. Instead of being a feature of the proposition as a whole, occurring once, it becomes a quantifier attached to a variable inside the proposition, and it may occur more than once with a scope. This is what allows the modern scheme to represent propositions such as "every creditor has some remedy" and to distinguish the two readings of it, which the traditional form cannot express.
5. How does the modern scheme read "All contracts are agreements", and what follows about existence?
As a general proposition: for anything whatever, if it is a contract then it is an agreement. The universal has become a conditional, and a conditional asserts neither of its parts. It follows that the proposition asserts nothing about whether any contracts exist, which is precisely the denial of existential import and the answer to the difficulty raised by "all trespassers will be prosecuted".
6. Is "Every creditor of an insolvent company incorporated before 1956 has a remedy against the directors personally" simple or compound?
Simple. Length and difficulty are irrelevant: what matters is whether any part of it is itself a proposition capable of being true or false on its own, and none is. "Of an insolvent company incorporated before 1956" qualifies the subject term and asserts nothing separately. The proposition is a single quantified statement, however complicated its terms may be.
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself, or the past papers, for the same subject.