Contemporary Syllogisms

Author: Timothy Eshing
Category: Logic and Reasoning, Philosophy of Mind and Language
Word Count: 999

Consider this argument:

All humans are mortal.
Socrates is human.
Therefore, Socrates is mortal.

It’s an example of a contemporary syllogism. Contemporary syllogisms are part of modern day quantificational logic, which is widely regarded as an improvement upon the approach originally described by Aristotle about 2,400 years ago.^[1]

This essay discusses the contemporary approach to syllogisms. A companion essay, Classical Syllogisms, discusses the older, Aristotelian approach.

The modern "square of opposition." — The modern “square of opposition.”^[2]

1. Terms

Whereas the classical approach admits only one kind of “term”, the general, the contemporary approach also admits a second kind of term: the singular.^[3] Singular terms are also called particulars; they pick out specific individuals. General terms are also called universals; they describe or put things in a category. Conventionally, we symbolize particulars with lowercase letters and universals with uppercase letters.^[4] Some examples:

Singular Terms	General Terms
s = Socrates	P = philosophers
t = this table	F = furniture
u = the world’s ugliest painting	A = artwork

2. Sentences and Distribution

As with classical syllogisms, only sentences that affirm or deny a predicate of a subject are allowed in contemporary syllogisms.^[5] But now singular terms can be subjects and predicates too, so we can use any of the following twelve kinds of sentences:

All S is P.	No S is P.
Some S is P.	Some S is not P.
s is P.	s is not P.
s is p.	s is not p.
All S is p.	No S is p.
Some S is p.	Some S is not p.^[6]

For the purposes of evaluating syllogisms in the next section, we’ll also need to know when terms are “distributed” or not. Distributed terms refer to every member of a category. Here’s the rule to determine distribution:

Any term immediately after “All” or “not” and every term after “No” is distributed.^[7]

The distributed terms are bold-underlined in the table above.

3. Contemporary Syllogistic Validity

A contemporary syllogism comprises three such sentences, that is, two premises and a conclusion, where every singular or general term is used exactly twice, just like a classical syllogism.

As with classical syllogisms, a contemporary syllogism is said to be “valid” if and only if, were its premises all true, then its conclusion would have to be true, too.^[8]

Gensler’s test is a convenient way to check whether a syllogism is valid:

Mark (i) all the distributed terms in the premises and (ii) all the undistributed terms in the conclusion.
If every general term is marked …
[on the classical approach] … at least once
[on the contemporary approach] … once and only once
and there’s exactly one right-hand mark (mark at the end of a sentence), then the syllogism is valid.^[9]

Notice how step two may be performed two different ways, depending on whether one adopts the classical or contemporary approach.^[10]

Here’s an example of an invalid syllogism:

All doctors are intelligent.

Some fishmongers are intelligent.

Therefore, some doctors are fishmongers.

All D* is I.

Some F is I.

∴ Some D* is F*.

Per step one of Gensler’s test, the relevant terms above are marked with an asterisk (*). But one general term, “I”, is never marked! So, per step two, this syllogism is invalid on both the classical and contemporary approaches.

Here’s an example of a valid syllogism:

All doctors are intelligent.

Some fishmongers are doctors.

Therefore, some fishmongers are intelligent.

All D* is I.

Some F is D.

∴ Some F* is I*.

Every general term is marked at least once and only once, and there’s only one right-hand mark, so this syllogism is valid on both the classical and contemporary approaches.

Let’s revisit the syllogism that opened this essay:

All humans are mortal.

Socrates is human.

Therefore, Socrates is mortal.

All H* is M.

s is H.

∴ s* is M*.

Every general term is marked once and only once, so this syllogism is valid on the contemporary approach. But “s” is a singular term! So it’s invalid on the classical approach, which doesn’t permit the use of singular terms.^[11]

4. Existential Import

Aristotle required the subjects of sentences to be non-empty terms. Aristotelians — that is, so-called “classical” logicians — wouldn’t use a categorical sentence like “All unicorns are herbivores” in a syllogism, because unicorns don’t exist. The unicorn category is empty.^[12]

This requirement produces a feature called existential import: sentences starting “All S …” or “No S …” imply there exists at least one thing that is S. Basically, by assuming existential import we can infer a particular positive sentence like “Some humans are mortal” from a universal positive sentence like “All humans are mortal”. The former is tantamount to asserting that (mortal) humans exist.^[13]

By contrast, moderns — that is, contemporary logicians — typically don’t assume existential import, which means they’ll let the subjects of universal sentences be empty.^[14] So they’ll use syllogisms to reason about categories of things that may not exist, like unicorns.^[15]

The two different ways to perform step two of Gensler’s test reflect whether or not one assumes existential import.^[16]

This syllogism is valid for Aristotelians, but invalid for moderns:

All fish are aquatic creatures.

All aquatic creatures are swimmers.

Therefore, some fish are swimmers.

All F* is A.

All A* is S.

∴ Some F* is S*.

Notice “F” is marked twice, which violates the contemporary rule. It just so happens that both the premises and the conclusion are true. However, one must assume existential import to infer the conclusion. Since moderns don’t, they’ll deny this syllogism’s validity.

By contrast, this syllogism is invalid for Aristotelians, but valid for moderns:

All unicorns are horned creatures.

All horned creatures are herbivores.

Therefore, all unicorns are herbivores.

All U* is H.

All H* is E.

∴ All U is E*.

Moderns think this syllogism is valid. But Aristotelians will deny it, because “U” is an empty term. There are no unicorns.

5. Beyond Syllogisms

Moderns make other changes too, like letting syllogisms have more or fewer sentences. Gensler’s test will also work for all of these!^[17] And there are many more kinds of syllogisms to explore.^[18] Ultimately, modern logic has moved beyond syllogisms, admitting valid arguments unlike those we’ve discussed herein.^[19]

Notes

^[1] For more information on quantificational logic, see my Formal Logic: Symbolizing Arguments in Quantificational or Predicate Logic. For introductory treatments of classical and contemporary syllogistic, see respectively Rescher 1966 part II (chs. 7–11) and Gensler 2017 ch. 2. For a brief history of logic, including discussions of classical and contemporary syllogistic, see Gensler 2017 ch. 16. For more information and resources, see the notes and references of the companion essay Classical Syllogisms.

^[2] The modern square of opposition depicts how contemporary logicians think differently about the relationship between the four categorical sentences discussed in §2 of the companion essay Classical Syllogisms; see note 3 of that essay for a description of the traditional square of opposition. Moderns do not assume existential import (discussed in §4 of this essay), so the relations depicted in gray (contraries, subcontraries, and subalternates) no longer hold. On the modern approach, only the relations between contradictories remain, depicted in black.

^[3] Furthermore, unlike classical syllogisms, contemporary syllogisms admit empty terms — meaning they can refer to things that don’t exist, like unicorns, talking donkeys, and the present king of France. Discussion of this issue is postponed until §4 of this essay (cf. note 14 below). There is one other kind of term, the indefinite, which can be ambiguous as to whether they are singular or general terms. Aristotle’s original example of an indefinite term was “pleasure”. Neither the classical nor contemporary approaches to syllogisms permit the use of indefinite terms. For more information on terms, including empty terms, see §1 of the companion essay Classical Syllogisms.

^[4] This is just like what we do in quantificational logic. For more information on modern logical symbolism, §1 of my essay Formal Logic: Symbolizing Arguments in Quantificational or Predicate Logic.

^[5] Cf. §2 of the companion essay Classical Syllogisms.

^[6] Here is how we usually translate these twelve sentences into quantificational logic:

“All S is P”: ∀x(Sx→Px)
“No S is P”: ¬∃x(Sx∧Px) or, equivalently, ∀x(Sx→¬Px)
“Some S is P”: ∃x(Sx∧Px)
“Some S is not P”: ¬∀x(Sx→Px) or, equivalently, ∃x(Sx∧¬P(x))
“s is P”: P(s)
“s is not P”: ¬P(s)
“s is p”: s=p
“s is not p”: s≠p
“All S is p”: ∀x(Sx→x=p)
“No S is p”: ¬∃x(Sx∧x=p) or, equivalently, ∀x(Sx→x≠p)
“Some S is p”: ∃x(Sx∧x=p)
“Some S is not p”: ¬∀x(Sx→x=p) or, equivalently, ∃x(Sx∧x≠p)

For more information on the language of quantificational logic, see §2 of my essay Formal Logic: Symbolizing Arguments in Quantificational or Predicate Logic.

^[7] To put it a bit more technically: a term is distributed when the context in which it appears includes the whole extension of the term, the whole set of things to which the term applies, and not just some part of it. So, for example, in the universal positive categorical sentence “All rodents are mammals”, “rodents” is distributed, because the sentence says something about every member of the set of rodents; but “mammals” is not distributed, because the sentence does not say something about every member of the set of mammals. For more information on distribution, see Rescher 1964 §7.5.

^[8] In constructing and assessing arguments we want to know whether they’re valid, because valid arguments with true premises must have true conclusions: it’s impossible for the conclusion of a valid argument to be false when all its premises are true. (Also consider the inferences without premises discussed in note 17 below: these are “valid” because it’s impossible for their conclusions, which are theorems, to be false when their premises are true. But there are no premises! Which means it’s impossible for theorems to be false, full stop.) Hence, validity is an incredibly important logical notion. It’s discussed in every introductory logic textbook. For more information on validity, see, e.g., Gensler 2017 §1.2.

^[9] Originally devised by the late Fr. Harry J. Gensler. He called it the “star test”, because (as depicted above) we add an asterisk to mark all the relevant terms. But how we visually keep track of the behavior of our terms doesn’t matter at all, so we’ve opted for a more neutral name here.

For more information on Gensler’s test, including a great many examples and exercises, see his 2017 §2.2. For even more information on Gensler’s test, including a lengthy discussion of antilogisms (which are like syllogisms except that their conclusions are contradictory, so they’re always invalid) on which Gensler’s test is based, as well as a link to the original 1973 article in which he first published his test, see Gensler 2009.

^[10] The importance of this difference is discussed in §4 of this essay.

^[11] Note: what’s here called “classical” is restricted to Aristotle’s assertoric syllogistic as originally set out in his Prior Analytics 1.1–2&4–7 and its codification in the medieval period, all other developments aside. The kind of argument at issue above is sometimes known as a “quasi-syllogism”, in contrast to a proper “categorical” syllogism, because one of its premises contains a singular term and is thus not a proper categorical sentence. For more information on categorical sentences, see §1 of the companion essay Classical Syllogisms, and of course see that essay as a whole for more information on what are here called “classical” syllogisms.

^[12] This is briefly discussed in §1 of the companion essay Classical Syllogisms.

^[13] Likewise, we can infer a particular negative sentence like “Some humans are not amphibians” from the universal negative sentence “No humans are amphibians”. The former is tantamount to saying that (non-amphibious) humans exist. Both of these examples concern subalternation from a universal to a particular categorical sentence. For more information on how assuming existential import affects the relationships that obtain between categorical sentences, see first note 3 of the companion essay Classical Syllogisms on the traditional square of opposition, and then note 2 of this essay on the modern square of opposition.

^[14] As aforementioned in note 3 above.

^[15] In §4 of the companion essay Classical Syllogisms, nine of the twenty-four classically valid syllogistic forms are singled out for special attention. These nine forms are classically valid, but invalid on the contemporary approach, because their validity depends on the assumption of existential import. While these nine classically valid forms are invalid for moderns, nevertheless a large number of forms that would be invalid on the classical approach are valid on the contemporary approach, including forms with empty or singular terms, forms with more or fewer premises, and many others.

^[16] For more information on using Gensler’s “star test” with or without the assumption of existential import, see Gensler 2017 §2.8.

^[17] Thus, on the contemporary approach, we can construct syllogisms with zero, one, three, or any greater number of premises, so long as we otherwise follow all the rules laid out above. Example of a valid syllogism with no premises (i.e., a theorem): “∴ All A is A*.” Example of a valid syllogism with only singular terms and only one premise: “a is b. ∴ b* is a*.” Example of a valid syllogism with three premises: “All A* is B. All B* is C. All C* is D. ∴ All A is D*.” Notice how, in each example, every term occurs exactly twice, every uppercase letter is marked exactly once, and there is exactly one right-hand mark. This method generalizes to contemporary syllogisms with any number of premises and any number of singular or general terms.

Readers may be surprised to learn that, on the contemporary approach, we can have syllogisms that simply consist of a concluding sentence and no premise sentences whatsoever. Yet we can! We normally think of an “argument” as consisting of at least one premise plus a conclusion. But modern logicians render our pre-theoretical notion of an “argument” more precisely with their technical notion of a logical consequence relation, sometimes symbolized as a “sequent”. A sequent looks like this: P ⊢ C, where “P” is a set of zero or more premises, “⊢” expresses from left to right the relation of logical consequence (similar to how “∴” expresses “therefore”), and “C” is the conclusion, which is a formula that is entailed by and thus a logical consequence of the premises. Now there exists a special class of formulas, called “theorems”, that are entailed by the null set of premises — that is, they are logical consequences of any premise set whatsoever, even the empty set, the set of zero premises. This means one can prove theorems without assuming any premises at all. An example acknowledged by Aristotle is the law of non-contradiction: ⊢ ~(P&~P). Notice there’s nothing to the left of the “⊢”, signifying that this formula is a theorem. It says that, for any sentence P, it’s not the case that both P and not-P are true. It’s a theorem that all contradictions are false. We may regard this sequent as a kind of argument with zero premises. Granted, some people prefer not to use the word “argument” in this way; but whether or not we call such a sequent as this an “argument” is simply a matter of convention. It’s still correct to say that the law of non-contradiction, just like any other theorem, is a logical consequence of the null set of premises. Accordingly, in the proof theory for “propositional” or sentential logic, the law of non-contradiction, just like any other sentential theorem or “tautology”, can be derived without making assumptions about the truth of any premises at all. (For more information on sentential logic, see Thomas Metcalf’s essay Formal Logic: Symbolizing Arguments in Sentential Logic.)

Now the contemporary approach to syllogisms actually comprises but a fragment of a more robust theory, called “quantificational” or predicate logic. (For more information on that theory, see my essay Formal Logic: Symbolizing Arguments in Quantificational or Predicate Logic.) Contemporary syllogisms are really a kind of shorthand for certain special inferences in quantificational logic, which (just like sentential logic, modal logics, and other modern logical systems) includes theorems that may be proved without assuming any premises. Regardless whether or not one elects to call contemporary syllogisms with zero premises “arguments”, there are exactly two such syllogisms that pass Gensler’s test (discussed in the main text above), which means that they correspond to valid inferences from the null premise set in quantificational logic. I gave the first of these as an example in the first paragraph of this note: “Therefore, all A is A*” [⊢ ∀x(Ax→Ax)]. The other is: “Therefore, a* is a*” [⊢ a=a]. Also, if we were to use the classical rule in step two of Gensler’s test, thereby permitting existential import (discussed in §4 of this essay), then a third syllogism would be validated: “Therefore, some A* is A*” [⊢ ∃x(Ax)]. But modern logicians don’t usually do that, so that syllogism is still invalid on the contemporary approach. (For more information on this topic, see notes 6, 8, 9, and 16 above.)

^[18] Especially those containing conditional or disjunctive sentences, those featuring statistical claims, and those involving modal terms (e.g., necessity and possibility, obligation and permission, belief, knowledge, etc.). For an introduction to modality, see Possibility and Necessity: An Introduction to Modality by Andre Leo Rusavuk.

^[19] Following the work of Gottlob Frege and others in the late nineteenth and early twentieth centuries, syllogistic logic was superseded by quantificational (also known as “predicate” or “first-order”) logic, which is much more powerful and flexible. And many more interesting logical systems followed, all of which deserve their own entries. As aforementioned in note 1 above, for more information on quantificational logic, see my essay Formal Logic: Symbolizing Arguments in Quantificational or Predicate Logic.