Stoic logic included rhetoric, dialectic, grammar, epistemology and a philosophy of language. They developed theories of concepts, propositions, perception, and thought. Their logic was propositional, rather than the Aristotelian logic of syllogisms and predicates. They defined five fundamental logical tools:

• if p then q; p; therefore q (modus ponens);
• if p then q; not q; therefore not-p (modus tollens);
• either p or q; p; therefore not-q;
• either p or q; not p; therefore q;
• not both p and q; p; therefore not-q;

They had a strict interpretation of the principle of bivalence (Aristotle's non-contradiction) and the law of the excluded middle. Every statement is either true or false, even statements about the future, as Diodorus Cronus maintained. Aristotlehad denied the present truth or falsity of future statements with his analysis of future contingency (e.g., the Sea Battle).

The Stoic philosophy of language had a theory of signs long before Charles Sanders Peirce's semiotics or Ferdinand de Sausurre's semiology. A signifier is an utterance of a name, a proper noun (onoma). The name-bearer is the object or concept that gets signified. The signification consists of the immaterial qualities that they called lekta, or ‘sayables,’ predicates that are true or false of the signified. The sayables are that which subsists (grows and decays) in an individual.

Stoic physics included a wide range of topics including ontology, cosmology, theology, psychology, and metaphysics. The basic principles of the universe (Aristotle's archai) are two - matter and pneuma - an immaterial breath or psyche. Pneuma combined two of the four fundamental elements, fire and air, representing hot and cold, as the active principle. A passive principle combined earth and water as the basis for material objects. The Stoics regarded matter as "unqualified" and inert. Changes in the material in an object they described as generation and destruction (following Aristotle).

Pneuma is the cause (aition) of change in the peculiar qualities of an individual that constitute growth and decay, corresponding to the Platonic and Aristotelian forms and ideas that shape a material object. Pneuma endows the bodies with different qualities as a result. The pneuma of inanimate object is called a ‘tenor’ (hexis, "having"). What it "has" are qualities. Pneuma in plants has a (phusis, ‘nature’). Pneuma in animals the Stoics called soul (psychê) and in rational animals pneuma includes the commanding faculty (hêgemonikon)

The Stoics saw the identity of an individual as its immaterial bundle of properties or qualities that they called the "peculiarly qualified individual" or ἰδίος ποιὸν.

Zeno of Cytium had formulated a psychological theory of how we acquire beliefs that are justified empirically and not by reasoning. To form a belief is to give one's assent to an "impression" (a phenomenal appearance: phantasia) about the material substrate of an object. Some perceptions are ‘cognitive’ or self-warranting. Assenting to them is a cognition or grasp (katalêpsis) of their objects. Assent should be restricted to these cognitive or kataleptic impressions. Cognitive impressions give us infallible knowledge or wisdom. Our beliefs will then be constituted entirely by self-warranting perceptual cognitions. Zeno argued that a cognitive impression "stamps" the form of the object (its peculiar qualities) on our mind or soul (pneuma).

Following Aristotle, the Stoics called the material substance or substrate ὑποκείμενον (or "the underlying"). This material substrate is transformed when matter is lost or gained, but they said it is wrong to call such material changes "growth (αὐξήσεις) and decay (φθίσεις)." The Stoics suggested they should be called "generation (γενέσεις) and destruction (φθορὰς)." These terms were already present in Aristotle, who said that the form, the essence, is not generated. He said that generation and destruction are material changes that do not persist (as does the Stoic peculiarly qualified individual).

It is therefore obvious that the form (or whatever we should call the shape in the sensible thing) is not generated—generation does not apply to it—nor is the essence generated; for this is that which is induced in something else either by art or by nature or by potency. But we do cause a bronze sphere to be, for we produce it from bronze and a sphere; we induce the form into this particular matter, and the result is a bronze sphere... For if we consider the matter carefully, we should not even say without qualification that a statue is generated from wood, or a house from bricks; because that from which a thing is generated should not persist, but be changed. This, then, is why we speak in this way.

(Aristotle, Metaphysics, Book VII, § vii & viii)

It is important to see that the Aristotelian view is very similar to the Stoic - that individuals are combinations of matter and form. At times Aristotle made the matter the principle of individuation, at other times he stressed the immaterial qualities or "affections," as did the Stoics, with their peculiarly qualified individual (ἰδίος ποιὸν).

Is Aristotle here the source of the four Stoic genera or categories?

The term “substance” (οὐσία) is used, if not in more, at least in four principal cases; for both the essence and the universal and the genus are held to be the substance of the particular (ἑκάστου), and fourthly the substrate (ὑποκείμενον). The substrate is that of which the rest are predicated, while it is not itself predicated of anything else. Hence we must first determine its nature, for the primary substrate (ὑποκείμενον) is considered to be in the truest sense substance.

Aristotle clearly sees a statue as an integral combination of its form/shape and its matter/clay, not two distinct things, as Skeptics would claim.

Now in one sense we call the matter (ὕλη ) the substrate; in another, the shape (μορφή); and in a third, the combination. Both matter and form and their combination are said to be substrate. of the two. By matter I mean, for instance, bronze; by shape, the arrangement of the form (τὸ σχῆμα τῆς ἰδέας); and by the combination of the two, the concrete thing: the statue (ἀνδριάς). Thus if the form is prior to the matter and more truly existent, by the same argument it will also be prior to the combination.

(Aristotle, Metaphysics, Book VII, § iii, 1-2)

. . . since the duality which they say belongs to each body is differentiated in a way unrecognizable by sense-perception. For if a peculiarly qualified thing like Plato is a body, and Plato's substance is a body, and there is no apparent difference between these in shape, colour, size and appearance, but both have equal weight and the same outline, by what definition and mark shall we distinguish them and say that now we are apprehending Plato himself, now the substance of Plato? For if there is some difference, let it be stated and demonstrated

(Anonymous Academic treatise, Oxyrhynchus Papyrus 3008 in Stoic Ontology, The Hellenistic Philosophers, A. Long and D. Sedley, p.167)

Many of the classic metaphysical puzzles are arguments over this dual nature of something as matter and form, especially Dion and Theon, Tibbles, the Cat, The Growing Argument, The Ship of Theseus, and The Statue and the Clay.

Modern metaphysicians mistakenly think that matter alone constitutes an entity.