A categorical syllogism is a logical argument consisting of three categorical propositions: two premises and one conclusion. Each proposition relates two of three terms — the minor term (S), major term (P), and middle term (M). First systematized by Aristotle in his work Prior Analytics (c. 350 BCE), the syllogism remains a cornerstone of deductive reasoning in philosophy, mathematics, and computer science.

The figure of a syllogism is determined by the position of the middle term (M) in the two premises:

Figure 1: M-P, S-M → S-P (M is subject in major, predicate in minor)
Figure 2: P-M, S-M → S-P (M is predicate in both premises)
Figure 3: M-P, M-S → S-P (M is subject in both premises)
Figure 4: P-M, M-S → S-P (M is predicate in major, subject in minor)

Each figure constrains how terms relate, affecting which moods are valid.

Figure 1: AAA (Barbara), EAE (Celarent), AII (Darii), EIO (Ferio)
Figure 2: EAE (Cesare), AEE (Camestres), EIO (Festino), AOO (Baroco)
Figure 3: IAI (Disamis), AII (Datisi), OAO (Bocardo), EIO (Ferison)
Figure 4: AEE (Camenes), IAI (Dimaris), EIO (Fresison)

Weakened moods (deriving particular conclusions from universal premises, traditionally accepted): Barbari (AAI-1), Celaront (EAO-1), Cesaro (EAO-2), Camestros (AEO-2), Darapti (AAI-3), Felapton (EAO-3), Bramantip (AAI-4), Fesapo (EAO-4).

Rule 1: The middle term must be distributed in at least one premise.
Rule 2: If a term is distributed in the conclusion, it must be distributed in its premise.
Rule 3: At least one premise must be affirmative.
Rule 4: A negative conclusion requires a negative premise (and vice versa).
Rule 5: If both premises are universal, the conclusion cannot be particular (traditional rule; modern logic treats this as a "weakened mood").
Rule 6: No valid syllogism has two particular premises.

Distribution: In A (All X are Y), X is distributed. In E (No X are Y), both X and Y are distributed. In I (Some X are Y), neither is distributed. In O (Some X are not Y), Y is distributed.

Validity concerns the logical form: if the premises are true, the conclusion must necessarily follow. A syllogism can be valid even with false premises. Soundness requires both validity and true premises. For example:

All birds are mammals (false), All penguins are birds (true), ∴ All penguins are mammals (false) — This AAA-1 syllogism is valid but unsound because the major premise is false.

This tool checks validity, not soundness — it evaluates logical form, not factual truth.

The middle term (M) is the bridge connecting the premises. For a syllogism to be valid, M must be distributed (refer to all members of its class) in at least one premise. If M is undistributed in both premises — a fallacy called the "fallacy of the undistributed middle" — the premises fail to establish a necessary link between S and P. For instance, in AAA-2 (All P are M, All S are M, ∴ All S are P), M is the predicate in both A-propositions and thus undistributed — making this form invalid despite its intuitive appeal.

Medieval logicians assigned mnemonic names to the 24 valid moods. The vowels indicate the mood (A/E/I/O), and the consonants encode reduction rules to Figure 1:

Figure 1: BarBaRa (AAA), CeLaReNT (EAE), DaRii (AII), FeRio (EIO)
Figure 2: CeSaRe (EAE), CaMeSTReS (AEE), FeSTiNo (EIO), BaRoCo (AOO)
Figure 3: DiSaMiS (IAI), DaTiSi (AII), BoCaRDo (OAO), FeRiSoN (EIO)
Figure 4: CaMeNeS (AEE), DiMaRiS (IAI), FReSiSoN (EIO)

Yes. Traditional Aristotelian logic assumes that all categories have at least one member (existential import). Modern predicate logic does not make this assumption. As a result, weakened moods — such as Barbari (AAI-1) and Darapti (AAI-3) — which derive a particular conclusion ("Some S are P") from universal premises ("All M are P, All S are M"), are considered valid in traditional logic but invalid in modern logic when the subject class may be empty. This tool identifies weakened moods and flags them separately so you can interpret results in either framework.