8

In Aristotle's Logic Darapti is a valid figure. I suspect that the issue is with the so-called "existential import" : From a modern standpoint, [we infer] "Some monsters are chimeras" from [...] "All chimeras are monsters"; but the former is often construed as implying in turn "There is something which is a monster and a chimera", and thus that there ...


7

A good introduction is the second volume of Gabbay & Woods (2008), Handbook of the History of Logic, where you'll find the mnemonics explained on pp 331ff. The mnemonics themselves seems to have originated in 13th century textbooks. For the original, see: de Rijk (1967), Logica Modernorum, vol 2, pp 362ff. I would also recommend Kretzmann, Kenny & ...


7

This is one of the classic 24 valid syllogisms, which means: It's a correct logical argument. In first-order logic, the premises can be written as ∀x(P(x)→Q(x)) and ∀x(R(x)→P(x)), and this implies ∀x(R(x)→Q(x)). So, whenever the premises are true, then the conclusion is also true. Except if you cheat. What does "cheat" mean? Well, for instance, words in a ...


6

You are "missing" The Traditional Square of Opposition. As you say : ‘Every S is P’ and ‘Some S is not P’ are contradictories. The "traditional" symbolization is : SaP for "all S are P" SeP for "no S is P" SiP for "some S is P" SoP for "some S is not P". o and i are the negations of a and e respectively. Thus : not SaP will be "not all S are P" i.e. "...


6

Explanation of the Mnemonic Brody, Boruch A. "Logical Terms, Glossary of." Encyclopedia of Philosophy. Ed. Donald M. Borchert. 2nd ed. Vol. 5. Detroit: Macmillan Reference USA, 2006. 533-560. Gale Virtual Reference Library. Web. 19 May 2016.: mnemonic terms The names that the medieval logicians introduced for the valid syllogisms. One such term is "...


6

Comment on similar example. The example is not from Aristotle. Categorical propositions with singular terms are used in Medieval logic; see Peter of Spain's Summulae logicales (XIII century): I,8 Propositio singularis est illa [...], ut "Sortes currit" [Socrates runs]. Into this textbook we can find many examples of them in the discussion of loci and ...


6

Yes there is. The first thing to note is that there are four "figures" of syllogism which are defined by where the major, minor and middle terms fall in the syllogism. Then what you do is memorize a variety of ways in which various valid syllogisms can be "reduced" to one of these four valid figures. These rules are codified in a little Latin song made of ...


6

The syllogism is correct, according to Aristotle's doctrine, exactly because of Existential import: Aristotle's logic system does not cover cases where there are no instances. See The Traditional Square of Opposition: ‘Some S is P’ is a subaltern of ‘Every S is P’. A proposition is a subaltern of another iff it must be true if its superaltern is true,...


5

In origin, sllogism was defined with categorical proposition i.e. for proposition like "All men are mortal" with class terms. The extension to singular terms can be managed considering the "singleton" formed by the class containing the single individual: Socrates. In this way, we can translate "Socrates is a man" as "Every member of the class containing ...


5

See Enthymeme : An enthymeme is a logical fallacy in which a categorical syllogism omits a premise that is necessary for the conclusion to be true or omits the conclusion itself. The missing proposition is considered to be implied. The fallacy is a syllogistic fallacy and a formal fallacy. Formal fallacy because a formal deductive arguments is a ...


5

It is the Fallacy of the undistributed middle: All Apples are Fruit All Oranges are Fruit Therefore, all Apples are Oranges is clearly invalid.


4

The term syllogism is due to Aristotle (originally sullogismos). Aristotle defined it as: an argument (logos) in which, certain things having been laid down, something different from what has been laid down follows of necessity because these things are so. (Source) So by Aristotle's definition, all syllogisms are valid. But consider this. In Aristotle'...


4

It seems that the list of categories from Marcus Friedrich Wendelin's book: Logicae institutiones (1654) [see page 14-on] comes from Petrus Ramus's Dialectique (1955). See also: Walter J. Ong, Ramus, Method, and the Decay of Dialogue: From the Art of Discourse to the Art of Reason (1958), page 183. See also: William & Martha Kneale, The Development of ...


4

Think of some examples. Here's a classic Thomist "analogical" term: "healthy". Properly speaking it is only bodies that are healthy, and for a body to be healthy is for it to be in good working order. For medicine to be healthy isn't for the medicine to be in good working order, it is for the medicine to have the power to put bodies into good working order. ...


4

We are talking about expressing statements in two different systems. The first one is the classical syllogistic of Aristotle ("All dogs are mammals"), with categorical syllogisms, whereas the latter ("If it is a dog then it is a mammal") is in the form of a hypothetical syllogism. It is only with the latter that one can speak of something like modus ponens, ...


4

Not exactly. We can consider the propositional valid argument called Hypothetical syllogism as a (derived) rule of inference. We call it "derived", because in standard presentations of propositional logic we can derive it from more basic ones, like Modus Ponens. In modern terms, syllogism is a fragment of first-order logic, the so-called Monadic predicate ...


4

See Syllogism: Aristotle's Theory: terms can be combined in different ways to form three figures (skhemata), which Aristotle presents in the Prior Analytics. When the four categorical sentences are placed into these three figures, Aristotle ends up with the following 14 valid moods [...] A fourth figure was discussed in ancient times as well as ...


4

You are correct that syllogistic, which corresponds to monadic predicate calculus in modern terms, is insufficient for doing mathematics. Modern formalisms use polyadic calculus. However, Euclid does not use syllogistic alone (in fact, he hardly uses it at all). Recent studies of Euclid's method, especially Manders's classic Euclidean Diagram, show that his ...


4

No, it is not. It is a valid argument : The premise : All teenagers are impulsive is equivalent to : for every x, if x is not impulsive, then x is not a teenager. Thus, using the second premise : John is not impulsive we can correctly conclude with : John is not a teenager.


4

Celaront was not in the original Aristotle's list of valid syllogistic figures (or : moods). It was added later (during the Middles Ages ?) as one of the two subalternate moods in the first figure (Barbari and Celaront). If we agree (as Aristotle does) that every term has reference, from Cesare : No reptiles have fur. (MeP) All snakes are reptiles....


3

Because I is particular affirmative and Rule 4 states : A Negative Premise Requires a Negative Conclusion, i.e. if one of the premises is negative, also the conclusion is.


3

Why in modern logic: does “All S is P” contradict “Some S is not P”? Because “All S is P” is ∀x(Sx → Px); negating it, we get: ¬∀x(Sx → Px). Due to equivalence between ¬∀ and ∃¬, this in turn is equivalent to: ∃x¬(Sx → Px). Now, in propositional logic, ¬(R → Q) is equivalent to: (R & ¬Q), and thus we finally get: ∃x(Sx & ¬Px). ...


3

No; reduction to absurdum is not expressible with a syllogistic form of argument. Reductio is a more "basic" argument: a propositional one. We can express it either as a "law" or axiom: ⊢ (¬ϕ → ¬ψ) → ((¬ϕ → ψ) → ϕ) or as a rule: if we have the derivation of a contradiction from ¬ϕ, we may infer ϕ. In more formal way: if Γ, ¬ϕ ⊢ ψ and Γ, ¬ϕ ⊢ ¬ψ,...


3

Why do we use the capital "K" for both "kill papers" and "cars that kill papers"? "[K]ill papers" is actually shorthand for "cars that kill papers". The statement is saying that all members of one set are also members of a second set; so the content of the second set must be defined.


3

The classical syllogisms consist of propositions which use only simple (ie, one "variable") terms in the subject and predicate, so any argument containing propositions which use complex terms isn't a valid (classical) form, although it can still be handled within term logic by extended inference rules. A complex term can be any Boolean expression. Being a ...


3

Irving Copi calls propositions such as "James' son is a man" or "Socrates is mortal" singular propositions. They are nonstandard-form propositions. They need to be translated into standard form categorical propositions which relate classes before being used in categorical syllogisms. He recommends the following: (page 239) To every individual object there ...


2

For "A is B" the explanation is simple. Gensler's language has two types of "basic" formulae: (i) formulae expressing relation between sets ("general categoris"): "All logicians are charming", translated as "all L is C" (ii) formulae expressing the fact that an individual belongs to a set: "Gensler is a logician", translated as "g is L". In this ...


2

Here is an excerpt from Russell's fifth lecture on Logical Atomism: "This notion, of course, of general propositions not involving existence is one which is not in the traditional doctrine of the syllogism. In the traditional doctrine of the syllogism, it was assumed that when you have such a statement as “All Greeks are men,” that implies that there are ...


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