Join a noun and a verb and you get a sentence (λόγος). But not every sentence carries a truth-value. In chapter 4 Aristotle does the sorting.
What a sentence (logos) is
First, the definition. Unlike a name, a sentence has parts that mean something on their own.
Λόγος δέ ἐστι φωνὴ σημαντικὴ κατὰ συνθήκην, ἧς τῶν μερῶν τι σημαντικόν ἐστι κεχωρισμένον, ὡς φάσις, ἀλλ᾿ οὐχ ὡς κατάφασις ἢ ἀπόφασις.
A sentence is a spoken sound significant by convention, some part of which is significant when separated — as an expression, though not as an affirmation or a negation. (ch. 4, 16b; Greek quotations throughout, Japanese and English are L/LAB working translations.)
And here Aristotle enters a small rebuke to Plato. A sentence signifies by convention, “not as an instrument (οὐχ ὡς ὄργανον).” The Cratylus treated a name as “a tool for dividing things” (article 038). Aristotle rejects that tool-model. Language is not an implement for carving up objects but a sign that carries meaning by convention.
Where the sentence branches
As an example of a sentence with no truth-value, Aristotle gives the prayer. “A prayer is a sentence, but neither true nor false.” So too a command, a question. They mean something and act on a hearer, but they are not the kind of thing that can be “on target” or “off” against the world.
Οἱ μὲν οὖν ἄλλοι ἀφείσθωσαν· ῥητορικῆς γὰρ ἢ ποιητικῆς οἰκειοτέρα ἡ σκέψις.
Let the others be set aside — their study belongs rather to rhetoric or poetics. (ch. 4, 17a)
The simplest assertion says “something of something” (τὶ κατά τινος) — or separates something from something. “A human is white,” “a human is not white.” Only here is the sentence held up against the world and made true or false.
One assertion says one thing
An assertion needs a verb. Pile up nouns as you like — “animal, two-footed, walks” — and it is not one assertion.
Ἀνάγκη δὲ πάντα λόγον ἀποφαντικὸν ἐκ ῥήματος εἶναι ἢ πτώσεως ῥήματος.
Every declarative sentence must be made of a verb or an inflection of a verb. (ch. 5, 17a)
And Aristotle asks what “one assertion (εἷς λόγος)” is: it states one thing, or is bound into one by a connective. Stringing loose subjects and predicates together makes several sentences, not one. This question of “the unity of the proposition” recurs in twentieth-century analytic philosophy (Frege, Russell). Truth lives in this “one assertion” — and its exact shape is what Aristotle is carving here. To be true is for the combination or separation to match a combination or separation on the side of things (chapter 1; more sharply formulated in Metaphysics Book IV).
The invention of “the proposition” — and the reaction
The bearer of truth is neither the word nor the mental image but a sentence of a particular form — an idea carried, by way of Frege’s “thought (Gedanke)” and “assertion sign (⊢),” into Wittgenstein’s Tractatus and “the proposition is a picture of reality”. That modern logic begins from “the proposition” is a descendant of Aristotle’s sorting.
Meanwhile, what Aristotle set aside “to rhetoric” was theorised in its own right. The Hellenistic Stoics classified the “complete lekton (what is meant)” into the assertible (ἀξίωμα, which has a truth-value), the question, the inquiry, the command, the oath, the prayer and more — they had already sorted the non-declarative sentences Aristotle lumped together. In the twentieth century, J. L. Austin took utterances like “I promise” and “I order” as speech that does not measure as true or false but performs something (performatives), and Searle organised them into “assertives, directives, commissives, expressives, declarations.” Prayers and commands turned out not to be defective sentences but sentences that do a different job — confirmed on the far side of the line Aristotle drew.