What kind of reasoning compares two cases and argues whats true in one case is true in another case?

The most common forms of deductive argument found in legal reasoning are those known as “syllogisms”. The term comes from Aristotle: “A syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so”. Structurally, a syllogism is any deductive argument that strives to draw a necessary inference on the basis of three propositions or statements: two premises and a conclusion. The four most common deductive syllogisms are categorical syllogisms, hypothetical syllogisms, disjunctive syllogisms and conjunctive syllogisms.

A categorical syllogism is a deductive argument wherein all three of the argument’s statements are “categorical propositions”. Categorical propositions are statements that assert or deny relationships. Every categorical proposition has two “terms” — that is, two things, classes, or categories about which something is said in the proposition. The two terms are called the subject term and the predicate term. The nature of a categorical proposition is to assert or deny that a relationship exists between the subject term of the proposition and the predicate term. For example, the claim, “All kangaroos are marsupials”, is a categorical proposition asserting that each and every member of the category or class represented by the subject term “kangaroos” falls within the predicate class, “marsupials”. The statement, “Aristotle is not a bandicoot”, is also a categorical proposition, though of a different kind. This statement denies that a relationship exists between the particular thing referenced by the subject term (“Aristotle”) and the entire membership of the predicate class (“bandicoots”).

There are four ways that categorical propositions relate classes or categories, one to another. These are often referred to as the four standard forms of categorical propositions. Each standard-form categorical proposition has a name: universal affirmative, universal negative, particular affirmative, and particular negative. By convention, each also goes by a letter, respectively, A, E, I, O. The nature of the claim made by each standard-form categorical proposition also can be represented by a formal logical statement using the letters S and P as place-holders for the proposition’s subject and predicate terms. The claims made by each form of categorical proposition are — universal affirmative: All S is P; universal negative: No S is P; particular affirmative: Some S is P; particular negative: Some S is not P. The four standard forms can be summarised as follows:

[A] Universal affirmative categorical proposition — All S is P: This form of categorical proposition asserts a relationship of containment, such that each and every member of the first-stated class (the subject class) is said to be a member of (included or contained within) the second class (the predicate class).
[E] Universal negative categorical proposition — No S is P: This form of categorical proposition states that no relationship exists between the memberships of the two classes — that is, that no member of the subject class is also a member of the predicate class.
[I] Particular affirmative categorical proposition — Some S is P: This form of categorical proposition expresses a relationship of partial containment, in that some (at least one) but perhaps not every member of the subject class are said also to be members of (included or contained within) the predicate class.
[O] Particular negative categorical proposition — Some S is Not P: This form of categorical proposition asserts a relationship of partial non-containment, such that some (at least one) but perhaps not every member of the subject class are said not to be members of (included or contained within) the predicate class.

Categorical syllogisms: standard-form

The basic requirements for an argument to be a categorical syllogism with the possibility of being valid in form are the following:

both of the argument’s premises and its conclusion must be categorical propositions
collectively, those three categorical propositions must contain exactly three different terms (that is, three different things, classes, or categories about which something is asserted), and
each of the three terms must occur twice in the argument (that is, be present in two different propositions).

Deductive arguments that satisfy these three requirements and are arranged in a certain order are said to be standard-form categorical syllogisms.

Of the three categorical propositions in a standard-form categorical syllogism, one is designated the conclusion, the other two are premises. The identity of each proposition as a premise or conclusion is determined by the terms it contains. The three terms are known as the major term, the minor term, and the middle term. Their identities are determined by their positions within the categorical syllogism, as follows:

Major term: the term that occurs as the predicate of the conclusion.

Minor term: the term that occurs as the subject of the conclusion.

Middle term: the term that does not occur in the conclusion, but appears in both premises.

Every categorical syllogism states in its conclusion that a relationship exists or does not exist between its minor term and its major term. The two premises assert that each of those terms, minor term and major term, stand in a certain relationship to a common third term, the middle term. While it does not figure in the conclusion, the middle term provides the critical link that makes reasoning by categorical syllogism possible.

When in standard form, a categorical syllogism presents what is called its major premise first, followed by its so-called minor premise, and then the conclusion. The determining feature that makes one premise the major premise is that it contains the major term (that is, the predicate term of the conclusion). The minor premise is the premise that contains the minor term (the subject term of the conclusion). A standard-form categorical syllogism is thus stated as follows:

Major premise: contains major term and middle term

Minor premise: contains minor term and middle term

Conclusion: contains minor term and major term as, respectively, its subject term and its predicate term.

The classic example of a categorical syllogism is found in the claim, “Since Socrates is human and all humans are mortal, Socrates is mortal”. This argument can be stated as a standard-form categorical syllogism in the following way:

Major premise: All humans are mortal.

Minor premise: Socrates is a human.

Conclusion: Therefore, Socrates is mortal.

This form of categorical syllogism represents a very common form of argument in law and judicial decision-making. Consider the following passage from Marshall CJ in the US Supreme Court’s decision in United States v Bevans:

[T]he jurisdiction of a state is co-extensive with its territory; co-extensive with its legislative power.

The place described [Boston Harbor] is unquestionably within the original territory of Massachusetts. It is then within the jurisdiction of Massachusetts.

This passage precisely mirrors the Socrates argument. Chief Justice Marshall’s reasoning can be stated in the form of a standard-form categorical syllogism:

Major premise: All places within the territory of a State are places within the jurisdiction of the State.

Minor premise: Boston Harbor is a place within the territory of the State of Massachusetts.

Conclusion: Therefore, Boston Harbor is a place within the jurisdiction of the State of Massachusetts.

As an argument identical in form to the Socrates argument, the Bevans argument from Marshall CJ carries the same logical force in terms of validity. This is because the validity of a categorical syllogism is entirely a product of the argument’s form — ie, its formal structure as determined by the types of categorical propositions it contains and the positioning of its terms. A valid categorical syllogism is thus valid solely by virtue of its form. Content, subject matter, and the truth or falsehood of its propositions have no bearing on the argument’s validity. Since validity is entirely a matter of formal structure, categorical syllogisms that take the same form are the same in terms of validity, regardless of content. If we know that a certain form of categorical syllogism is valid (for example, the Socrates argument), then another argument in that same form (Bevans) is valid too. This holds even if one or more of the categorical propositions in the argument are false. Consider this argument:

Major premise: All night parrots are dead.

Minor premise: A certain bird photographed (much alive) and captured on video in early July 2013 in western Queensland was a night parrot.

Conclusion: Therefore, that certain bird photographed (much alive) and captured on video in early July 2013 in western Queensland was dead.

This argument is valid. But it is not sound. The logical form of the argument matches exactly the form of the Socrates and Bevans arguments. All three arguments begin with a universal affirmative categorical proposition followed by two particular affirmative categorical propositions follow (AII sequence of categorical propositions). The terms in the arguments also occupy the same positions. While the three arguments follow the same form (which happens to be valid), the night parrot argument is obviously troublesome. Its trouble lies in falsehood; either its major premise or minor premise is not true. And the conclusion is nonsensical. The lack of truth, however, does not affect the validity of the argument. It is valid, but due to the fact that its major premise is false, the argument is unsound.

Determining the validity of categorical syllogisms

Every categorical syllogism in standard form can be tested for validity by enquiring whether it violates one or more rules which must all be satisfied for a categorical syllogism to be valid. If a form of categorical syllogism fails to satisfy any one (or more) of the rules, that form is invalid; and every argument that takes that form is invalid, regardless of the truth of its content. Each rule has associated with it a particular formal fallacy. The five standard rules of validity, along with their associated fallacies, are set out below:

Rule 1: Three terms rule Every valid categorical syllogism contains precisely three terms, each of which is used in the same sense throughout the argument.

Fallacy: Fallacy of four terms or fallacy of equivocation
Rule 2: Middle term distribution rule In every valid categorical syllogism, the middle term is distributed in at least one premise.

Fallacy: Fallacy of the undistributed middle term
Rule 3: Conclusion distribution rule In every valid categorical syllogism, any term distributed in the conclusion is also distributed in the premise where it appears.

Fallacy: Fallacy of the illicit process of the major term or minor term
Rule 4: Negative premise rule No valid categorical syllogism has two negative premises.

Fallacy: Fallacy of exclusive premises
Rule 5: Negative conclusion rule Every valid categorical syllogism with one negative premise has a negative conclusion.

Fallacy: Fallacy of an affirmative conclusion from a negative premise

With these rules in hand it is possible to confirm the validity or prove the invalidity of any categorical syllogism. The form represented by the Socrates/Bevans/night parrot arguments is valid for it satisfies all the rules. It is, as a matter of fact, the most common form of categorical syllogism used in law and ordinary life. Yet just one slight modification from that argument form yields an invalid argument. In the familiar valid form, the middle term (the term that appears in each premise) is positioned as the subject term in the major premise and the predicate term in the minor premise. If instead of that placement, the middle term serves as the predicate term in each premise, the argument is invalid. Consider:

(1) All Hulme Supercars are orange.
(2) Betty’s car is orange.
(3) Therefore, Betty’s car is a Hulme Supercar.

The fallacy this form of categorical syllogism commits is that of the undistributed middle term. This is the most vexing fallacy in reasoning by way of categorical syllogisms. It vexes because it can be difficult to detect and is committed in arguments that are extremely close in formal structure to the most common form of categorical syllogism.

Another invalid categorical syllogism that is close in form to the Socrates/Bevans/night parrot form results from beginning an argument with a particular affirmative major premise instead of the universal affirmative categorical proposition found in the major premise of that most familiar form. This would be to argue like:

(1) Some controversial historical figures from the 19th century were scientists.
(2) Ned Kelly is a controversial historical figure from the 19th century.
(3) Therefore, Ned Kelly was a scientist.

Again, the error of reasoning here is the fallacy of the undistributed middle term. For a categorical syllogism to be valid, the middle term must be distributed in at least one of the premises. Each term in a categorical proposition is either distributed or not. For a term to be distributed means that what is said about it in a categorical proposition involves a claim of knowing something to be true about each and every individual in the entire class or category to which the term refers. Thus, the universal affirmative statement, “All kangaroos are marsupials”, asserts a claim of knowing that each and every member of the category represented by the subject term “kangaroos” falls within the predicate class “marsupials”. The subject term “kangaroos” is thus distributed. The statement does not, however, imply knowing anything to be true about every member of the class “marsupials”; hence, that predicate term is not distributed. A universal affirmative categorical proposition accordingly distributes its subject term but not its predicate term. By contrast, in the particular affirmative categorical proposition form (for example, “Ned Kelly was a scientist”), neither subject nor predicate term is distributed.

Failure to distribute the middle term renders a categorical syllogism invalid regardless of the argument’s content. Were Charles Darwin substituted for Ned Kelly in the argument above, it would read:

(1) Some controversial historical figures from the 19th century were scientists.
(2) Charles Darwin is a controversial historical figure from the 19th century.
(3) Therefore, Charles Darwin was a scientist.

All three propositions in this syllogism are true. Still the argument remains invalid. The fallacy remains the undistributed middle, as it will for every argument that takes this form, no matter the content. For the middle term (here “controversial historical figures from the 19th century”) must be distributed for a categorical syllogism to be valid. This is because the middle term is the argument’s yeoman. While the purpose in arguing by way of a categorical syllogism is to draw a necessary inference about how the minor term relates to the major term, it is the middle term that does the inferential work. The two premises assert relationships between the major and minor terms and the middle term. That middle term is the class or category that they share in common. If the middle term is not distributed in at least one premise, it cannot do its work of bringing the minor and major terms together by necessary inference.

Unfortunately, categorical syllogisms that commit the fallacy of the undistributed middle term are all too common in legal reasoning. Many cases discuss the fallacy. One is the US Court of Customs and Patent Appeals’ judgment in Atlantic Aluminum & Metal Distributors, Inc v United States. This case concerned a dispute over the appropriate classification of an importer’s merchandise for the purpose of assessing import duties under the federal Tariff Act of 1930. The merchandise consisted of aluminum tubes. The importer sought to have the tubes classified with bars and rods, which would significantly lower the import duty from the assessment resulting from the classification used by the government. The principal evidence submitted by the importer were definitions from various dictionaries meant to establish the common meaning of the terms “bars” and “rods”. Finding the definitions inconclusive, the court reasoned:

These definitions establish that a bar or a rod is “long in proportion to its breadth and thickness”. The evidence establishes that the imported tubes are also “long in proportion to [their] breadth[s] and thickness[es]”. From these premises the importer asks us to find that tubes are bars and rods. This constitutes an invalid syllogism. The undistributed middle term prevents reliance upon the premises to support the importer’s conclusion. If we were to agree with this argument, we would then be required logically to hold that every item having length would be a rod or bar, since every item having length is, by definition, long in proportion to its breadth and thickness.

Essentially, the argument the court here rejects is the following:

(1) Some items long in proportion to breadth and thickness are bars/rods.
(2) The importer’s aluminum tubes are long in proportion to breadth and thickness.
(3) Therefore, the importer’s aluminum tubes are bars/rods.

Just as the court rules, this argument commits the fallacy of the undistributed middle term. The explanation Smith J gives for rejecting the argument nicely illustrates how the fallacy is here implicated.

Another noteworthy judicial discussion of the fallacy of the undistributed middle is found in the judgment of the NSW Court of Appeal in Bishop v Electricity Commission of NSW. The plaintiff in Bishop suffered from substantial tinnitus allegedly caused by two decades’ exposure to industrial noise while employed by the defendant. The evidence showed that in addition to his time in the defendant’s employ, the plaintiff had been exposed to industrial noise in previous workplaces. Further, the plaintiff also had a history of exposure to noise through his hobbies of rifle shooting and pistol shooting. The trial judge found the evidence insufficient to establish that the plaintiff’s tinnitus could be causally attributed to his employment with the defendant. In dismissing the appeal, Handley JA reasoned:

In my judgment [plaintiff’s] submission should not be accepted. The evidence earlier referred to indicates that tinnitus has other causes, apart from exposure to industrial noise. One common cause is the exposure to noise made by firearms, but some persons suffer from tinnitus without having any history of exposure to industrial noise, or the noise made by firearms. In these circumstances, the fact that a significant proportion of persons suffering from industrial deafness also suffer from tinnitus does not permit the Court to draw the conclusion that in this, or any case, the inference is available that proved tinnitus must have been, more probably than not, caused by exposure to industrial noise.

There is, of course, a well-established logical fallacy in that proposition known as the undistributed middle.

Handley JA’s argument can be formally stated as:

(1) Some tinnitus is caused by exposure to industrial noise.
(2) Bishop’s injury is tinnitus.
(3) Therefore, Bishop’s injury is caused by exposure to industrial noise.

This argument precisely matches the form of the Ned Kelly argument. All three propositions (major premise, minor premise, conclusion) in both arguments are particular affirmative categorical propositions. That type of categorical proposition does not distribute either its subject or predicate term. The middle term accordingly is left undistributed. The excerpt here from the judgment of Handley JA admirably describes the logical problem that attends trying to draw an inference without distributing the middle term.

There are three valid types of hypothetical syllogisms: the pure hypothetical syllogism and two forms of “mixed” hypothetical syllogisms: modus ponens and modus tollens. The distinguishing feature of hypothetical syllogisms is that they rely on “hypothetical propositions” — that is, conditional or “if/then” statements.

Using the letters A and C as placeholders for any two simple propositions, it is possible to represent the basic form of a hypothetical proposition as:

If A is true, then C is true.

In the hypothetical proposition, the assertion that follows “if” is called the antecedent. The assertion that follows “then” is the consequent. So every hypothetical proposition states that:

If antecedent A is true, then consequent C is true.

Pure hypothetical syllogism: A pure hypothetical syllogism is a deductive argument containing hypothetical propositions exclusively. Every valid pure hypothetical syllogism takes the following form:

(1) If A is true, then C is true.
(2) If C is true, then E is true.
(3) Therefore, if A is true, then E is true.
Modus ponens: The deductive argument form known as modus ponens is one of two valid forms of “mixed” hypothetical syllogism. A mixed hypothetical syllogism is a deductive argument wherein one premise (the “conditional premise”) takes the form of a hypothetical proposition, while the other premise and the conclusion are simple or categorical propositions. Arguments that take the form of modus ponens claim:

(1) If A is true, then C is true.
(2) A is true.
(3) Therefore, C is true.
Modus tollens: The other valid form of mixed hypothetical syllogism is known as modus tollens. As a mixed hypothetical syllogism, modus tollens, like modus ponens, is an argument that contains one conditional premise (that is, one hypothetical proposition), while the other premise and the conclusion take the form of simple or categorical propositions. Arguments in the form of modus tollens assert:

(1) If A is true, then C is true.
(2) C is not true.
(3) Therefore, A is not true.

Fallacies in reasoning by hypothetical syllogism

There are three fallacies that are committed all too frequently when reasoning by way of hypothetical syllogisms:

Fallacy of the illicit conditional: The fallacy of the illicit conditional is associated with the pure hypothetical syllogism form. This fallacy occurs whenever an argument contains three hypothetical propositions but takes a form different from the valid form set out above. As noted, a valid pure hypothetical syllogism contains three propositions, A, C, and E, each of which appears twice in the argument. Those propositions must be positioned in the argument precisely as stated. If they are positioned differently the argument is invalid. This can be by adding a fourth proposition, by positioning the three propositions differently, or by negating one or more of the propositions the second time it occurs in the argument. If any of these errors occur, we have the fallacy of the illicit conditional.
Fallacy of affirming the consequent: The fallacy of affirming the consequent is a mixed hypothetical syllogism fallacy. It begins with a standard-form conditional proposition — If antecedent A is true, then consequent C is true. The second premise then affirms the truth of the consequent, leading to an attempt to establish the truth of the antecedent. That is, an argument that commits the fallacy of affirming the consequent reads:

(1) If A is true, then C is true.
(2) C is true.
(3) Therefore, A is true.

Note that this fallacious form of mixed hypothetical syllogism contains only affirmative propositions. In that respect, the fallacy of affirming the consequent resembles modus ponens. Essentially it modifies modus ponens by inverting the propositions asserted in the minor premise and the conclusion. While modus ponens affirms the truth of the antecedent in the second premise so as to necessarily prove the truth of the consequent in the conclusion, the fallacy of affirming the consequent inverts the ordering of those propositions.
Fallacy of denying the antecedent: The final hypothetical syllogism fallacy, that of denying the antecedent, is another mixed hypothetical syllogism fallacy. Like the fallacy of affirming the consequent, the fallacy of denying the antecedent begins with a conditional proposition — If antecedent A is true, then consequent C is true. Then in this fallacious form, the argument, in its second premise, denies the truth of the antecedent in an effort to prove, in the conclusion, that the consequent cannot be true. Accordingly:

(1) If A is true, then C is true.
(2) A is not true.
(3) Therefore, C is not true.

Given that the fallacy of denying the antecedent asserts negative propositions in its second premise and conclusion, this fallacious form of mixed hypothetical syllogism quite closely resembles modus tollens. It essentially modifies modus tollens by inverting the propositions asserted in the minor premise and the conclusion. While modus tollens denies the truth of the consequent in the second premise in order to necessarily deny the truth of the antecedent in the conclusion, the fallacy of denying the antecedent reorders those propositions, producing an invalid syllogistic form.

The judgment of the US Court of Appeals for the Second Circuit in Crouse-Hinds Co v InterNorth Inc, provides an excellent study of hypothetical syllogisms in the law. InterNorth, desiring to complete a hostile takeover of Crouse-Hinds, filed suit to block a proposed merger between Crouse-Hinds and another company. At trial, InterNorth claimed that the Crouse-Hinds directors, who would remain in control after the merger, were pursuing it out of self-interest and bad faith. InterNorth succeeded in temporarily enjoining the merger by fashioning a hypothetical syllogism on the basis of an earlier Second Circuit judgment. In Treadway Companies v Care Corp, the Second Circuit had declined a request to block a merger on the basis of self-interest or bad faith on the part of the directors. Under the facts in Treadway, the directors were not going to remain in control after the merger. The reasoning of the Second Circuit in Treadway thus took the form of modus ponens:

(1) If the directors of a company are not going to remain in control after a merger, then perpetuating their control cannot be their motive in pursuing the merger.
(2) The Treadway directors were not to remain in control after the merger.
(3) Therefore, perpetuating their control could not be the Treadway directors’ motive in pursuing the merger.

Before the trial court in Crouse-Hinds, InterNorth successfully argued, following Treadway:

(1) If the directors of a company are not going to remain in control after a merger, then perpetuating their control cannot be their motive in pursuing the merger.
(2) The Crouse-Hinds directors were to remain in control after the merger.
(3) Therefore, perpetuating their control must be the Crouse-Hinds directors’ motive in pursuing the merger.

Though the starting-point of this argument is the same as Treadway’s, the Second Circuit correctly observed that the syllogism was very different. Instead of modus ponens, the form of the InterNorth argument was that of the fallacy of denying the antecedent. Reversing the trial court that had accepted the fallacious inference crafted by InterNorth, the Second Circuit in Crouse-Hinds astutely ruled that, “This inference has no basis in either law or logic”.

Disjunctive syllogisms are deductive arguments wherein one premise takes the form of a disjunctive proposition, while the other premise and the conclusion are simple propositions that either deny or affirm part of the disjunctive proposition. Disjunctive propositions are “or” or “either/or” statements. They are compound propositions in that every disjunctive proposition, or disjunction, contains two component propositions called “disjuncts”. One disjunct comes before the “or”; the other appears after it.

Two disjunctive syllogism “moods”

It is often said there are two forms or “moods” that disjunctive syllogisms can take. The two moods differ from one another in two important respects:

Exclusivity of the disjuncts: The first mood does not assume that the disjuncts are mutually exclusive; the second mood requires that the disjuncts be fully exclusive of one another — that is, that it not be possible for both disjuncts to be true.
Conclusions they draw: The first mood denies the truth of one disjunct in its second premise and then affirms, in its conclusion, the truth of the other; the second mood affirms the truth of one disjunct in the second premise and then concludes that the other disjunct must be false. Only the first mood is, in a pure logical sense, a disjunctive syllogism. The second mood presumes, in its disjunctive premise, a proposition that is more complex than a simple disjunctive proposition. Nevertheless, arguments in the second mood do occur in certain situations in legal reasoning. When they do, they are typically presented as simple disjunctive syllogisms.

The two moods can be stated symbolically as follows:

Mood which by denying affirms:

(1) A is true or B is true.
(2) A is not true.
(3) Therefore, B is true.

Mood which by affirming denies:

(1) A is true or B is true.
(2) A is true.
(3) Therefore, B is not true.

While the second mood represents a fairly common argument form, it must be used with care. It is a valid argument form only when the assumption holds that the disjuncts, A and B, are mutually exclusive. The California Court of Appeal addressed this point well and expertly compared the logical validity of the two disjunctive syllogism moods in the case Danzig v Superior Court. The court was asked to consider, in a class action suit, whether unnamed class members are, under the applicable State statutory scheme, “parties” to whom interrogatories may be propounded. An earlier California Supreme Court judgment had held that, under the same statutory scheme, unnamed members of a class are “persons for whose immediate benefit an action or proceeding is prosecuted”. The petitioners in Danzig argued that in ruling that unnamed class members are “persons”, the Supreme Court had impliedly held that they were not “parties”. Justice Feinberg for the Court of Appeal in Danzig rebuffed this argument as unsound:

The argument suffers from a logical fallacy. When a proposition is in the form of two alternatives, if one alternative is false, then the other alternative must be true. But, if one of the alternatives is true, nothing can be said about the truth or falsity of the other alternative except in the situation when the two alternatives are mutually exclusive.

In Southern California Edison, the Supreme Court holding that unnamed members of a class … were persons for whose benefit the action was being prosecuted tells us nothing as to whether unnamed members of a class in a class action are “parties” within the meaning of [the applicable] section … unless a “party” and “a person …” are mutually exclusive concepts. Since it appears obvious that the two concepts are not mutually exclusive, we conclude that Southern California Edison is not authority for the resolution of the issue at bar.

Fallacies associated with disjunctive syllogisms

There are three fallacies associated with reasoning by way of a disjunctive syllogism:

Fallacy of non-exclusivity: The fallacy of non-exclusivity applies only to the second disjunctive syllogism mood — the mood which by affirming denies. The logic behind this fallacy is that if the disjuncts in a disjunctive proposition are not contradictory or mutually exclusive, then it is possible for both disjuncts to be true. Hence, to affirm one disjunct in a non-exclusive disjunctive proposition provides no basis for denying the truth of the other. Arguments which employ the second disjunctive syllogism mood when it is possible for both their disjuncts to be true commit this fallacy of non-exclusivity. This is the fallacy committed by the petitioners in Danzig.
Fallacy of missing disjuncts: While the fallacy of non-exclusivity can be committed only when an argument takes the second disjunctive syllogism mood, there are two other fallacies associated with disjunctive syllogisms that can occur in either mood. Both of these fallacies go to the truth of the disjunctive proposition, not to the form or structure of the disjunctive syllogism. The first, the fallacy of missing disjuncts, goes to the incompleteness of a disjunctive proposition. This fallacy is committed whenever a disjunctive proposition asserts the truth of one disjunct taken from a pair or set of disjuncts when in fact other disjuncts not enumerated are possible.
Fallacy of false disjunction: The other disjunctive syllogism fallacy that can arise in either mood is the fallacy of false disjunction. This fallacy is committed whenever an argument rests on a disjunctive proposition that sets in opposition two disjuncts that are not in truth alternatives to one another.

Conjunctive syllogisms are deductive arguments wherein one premise is stated in the form of the negation of a conjunctive proposition. Conjunctive propositions are compound propositions that pull together, usually with the word “and”, two or more component propositions known as “conjuncts”. In a simple two-part conjunctive proposition, one conjunct comes before the “and”, while the other appears after it (for example, “The auto dealership sells BMWs and it sells Holdens”; “The plaintiff filed the complaint and the defendant counterclaimed”).

In a conjunctive syllogism, one premise (the conjunctive premise) negates the truth of a conjunctive proposition — that is, it denies the possibility that both conjuncts could be true. The other premise then affirms the truth of one of the conjunctive premise’s conjuncts, while the conclusion denies the truth of the other conjunct. That is, a valid conjunctive syllogism takes the logical form:

Premise 1: The negation of a conjunctive proposition. (the conjunctive premise)

Premise 2: A simple or categorical proposition affirming the truth of one of the conjunctive premise’s two conjuncts.

Conclusion: A simple or categorical proposition denying that the conjunctive premise’s other conjunct could be true.

Symbolically, the standard claim made by a valid conjunctive syllogism can be expressed as follows:

(1) A and B are not both true.
(2) A is true.
(3) Therefore, B is not true.

Note that the conjunctive syllogism form can be used interchangeably with the second disjunctive syllogism mood. Consider the argument:

(1) Ernest Hemingway could not have been born both in Idaho and in Illinois.
(2) Hemingway was born in Illinois.
(3) Therefore, Hemingway was not born in Idaho.

As stated, this argument is a conjunctive syllogism. The same reasoning can be presented as a disjunctive syllogism in the second mood if the first premise is changed to read: “Ernest Hemingway was born either in Idaho or in Illinois”. The resulting disjunctive syllogism would be valid since the disjuncts are mutually exclusive.