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This is a summary from absurdbeing.com:

Unfortunately, as mentioned, Aristotle’s system of logic (based entirely on the deductive syllogism) was incomplete in at least three ways: first, it didn’t account for deductions which relied on propositional logic, where variables stand for whole propositions; second, it couldn’t express relationships between two or more objects; and third, it ignored inductive logic, which sacrifices certainty for the benefit of adding new knowledge.

What are the deductions which rely on propositional logic?

What does he mean 'where variables stand for whole propositions'?

Why can't it express relationships between two more objects.

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    "What are the deductions which rely on propositional?" See Propositional calculus dedicated to the rules involving truth-functional connectives only: negation, conjunction, disjunction. A's Logic used some "propositional rules" but there is no systematic treatment of it. Compare with Stoic Logic. May 18, 2022 at 11:50
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    "Why can't it express relationships between two more objects." Because the analysis of sentence structure provided by A was in terms of subject-predicate: La (animal is Living) and not in terms of binary (or more) relations: aLb (amy Loves bob). May 18, 2022 at 11:55
  • Perhaps you do not recognize that there is a distinct subject called MATHEMATICAL LOGIC. Aristotelian logic had a different purpose. The quote you refer to does not point that out. Math people don't tend to understand what PROPOSITIONS are. The words may be the same but the meaning is different. Philosophers don't need SENTENCES. Aristotelian logic is not about how people do argue. We convert normal language into the syllogism rules which many people are not aware of. Aristotelian logic is not for persuading people. Persuasion is psychological & rhetorical not deductive per se.
    – Logikal
    May 19, 2022 at 23:10
  • Just to be clear: different sentences can represent the same proposition. This is what Psychology, Mathematics & Rhetoric seem to ignore. They are focused on how people actually speak and NOT the idea being expressed. Most cases can be translated into categories but it loses rhetorical impact. We can persuade people to do as we want when we want them to. So rhetorically that is more important in reality so they try to notate how people speak. Mathematical logic has a specific domain that does not always apply to reality as well. So there will always be complaints.
    – Logikal
    May 19, 2022 at 23:21

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Mauro's comments are correct, but just to unpack them a little more...

Propositional logic is the logic of propositions containing connectives such as 'and', 'or', 'not', 'if', 'nor', etc. Aristotle's categorical syllogisms do not cover the logic of these terms, so it is unable to express arguments like, "Mary won the race or Jane won the race; Mary didn't win the race; therefore, Jane won the race". Or, "Russell didn't win both the Nobel prize for literature and the Nobel prize for physics; he did win the Nobel prize for literature; therefore, he didn't win the Nobel prize for physics". Or, "Alice is hungry or Bob is hungry; if Alice is hungry the pie will get eaten; if Bob is hungry the pie will get eaten; therefore, the pie will get eaten".

Propositional logic was invented by the stoic philophers in the third century BCE, particularly by Chrysippus, Diodorus and Philo. The point about variables is that once algebra was invented (the ancient Greeks didn't manage to do this, it was accomplished later by Arabs), it became possible to express propositional arguments using symbols, e.g. "M v J; ¬M; therefore, J" and to treat these symbols as variables. So, in the same way that in algebra we can denote an unknown quantity by x and write down some equations and then solve for the value of x, in propositional logic we can treat symbols such as P, Q, R, as propositional variables, write down some sentences involving them and 'solve' for their truth values.

The point about relationships is that Arisototle's logic is concerned with things and properties of things, but it is unable to express general relations between things. It can express "Some boys like girls", since this can be understood, at a pinch, as some boys have the property of being girl-likers. But as soon as you want to say something a little more complex, such as "Some boys like girls who like them back", it is unable to express this. For this, we require a two-place relation: Likes(x,y) and two quantifiers.

Aristotle's logic can be thought of roughly as a fragment of first order logic with the restriction of having unary predicates only, a single quantifier per sentence, and no propositional connectives.

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  • Is there an implicit logic for negative and disjunctive connectives in Aristotle's laws of noncontradiction and bivalence/excluded middle? I remember Kant complaining about Aristotle cluttering the LNC with stuff like "not at the same time," which makes Aristotle's LNC seem predicate-oriented; but bivalence is surely propositional? May 18, 2022 at 15:19
  • LNC and LEM place some restrictions on propositional logic, but I wouldn't say that two axioms make a logic.
    – Bumble
    May 19, 2022 at 0:59

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