Syllogisms

Syllogisms are deductive and contain one or more premise statements plus one, and only one, conclusion statement.

Here is the most important fact about syllogisms and all thinking. Most people don’t base their beliefs on rational thought. Sales people know that people don’t really tend to make decisions of whether or not to buy something based on rational thought. They make those decisions based on emotion and then they rationalize those decisions. That’s why fallacies work so well for deceiving people.

One common fallacy is the deception that valid form of a syllogism makes something true. Some students come out of logic classes thinking that if they can reduce a statement to a syllogism that has valid form, that proves the conclusion to be true. It does not, as this page will explain.

Syllogisms are one tool that can be used to expose formal fallacies, that is flaws in the argument’s form. This type of analysis often requires a reconstruction of the argument into a syllogistic form. However, the person trying to do this often must guess the missing information to complete the syllogism, since people tend to leave out parts of their argument. Guessing isn’t an accurate way of knowing.

Many of the books on logic focus on the form of the argument, making it seem as if an argument that has valid form is a sound argument. They obviously don’t state it that way. In fact, they do define the difference between a sound argument and a valid form. Then, they have a very low threshold for truth. They explain a certain number of fallacies as tests for truth, but they don’t go to the root, the very basis for the claim. In the same way that an emotional sales pitch can sway judgment, focusing on the form can give the illusion of sanity when what is being stated is not rational. There are classes and text books that make logic seem as if it were a mathematical problem. Even mathematical logic can be used to lie, as in the many statistical fallacies that are not fallacies of form (informal fallacies). They leave out the most important component: how do you know?

A syllogism only tests the form of the argument to see if it is a valid form. However, an argument can have perfectly valid form and still be unsound. The conclusion can be false. When a syllogism has valid form and both the major and minor premises are true, then the conclusion must also be true and the argument sound.

Premises always make a truth claim. They claim that something is true. When you hear or read something, it is most often claiming that something is true. This is called a truth claim. Here are three questions you can ask to know whether or not the thing is true. Think of A.S.K. to remember the three questions. These stand for Authority, Starting point, and Know.

  • What is the Authority of the claim?
  • What is the Starting point for the claim?
  • How do they Know?

When you ask these three questions and really give it some serious thought, you begin to realize the weakness of almost all reasoning that is used all around you. If someone is going to make a truth claim, every premise statement (the ones that supposedly prove the conclusion to be true) must, of necessity, be absolute. If any of those statements are not absolute, then each non-absolute statement must be proven to be true by another set of absolute premises and a conclusion. If anyone going to prove that anything is true, meaning absolutely true, then they must eventually come to absolute premises. You need to ask, “Are all the premises stated clearly and are they absolute? Rarely is this the case.

Types of Syllogisms

    1. Categorical syllogisms: a conclusion follows from a general statement (the major premise) and a specific statement (the minor premise).
    2. Hypothetical syllogisms:
      1. Conditional Syllogism: If P, then Q. Q. Therefore, P.
      2. Disjunctive Syllogism: either… or. Either P or Q. Not Q. Therefore, P. (Note that this is only valid if both P and Q is not an option. If both P and Q are possible, then the Fallacy of affirming one alternate is committed, though the form may seem valid.)
      3. Conjuctive Syllogism: both…and. Both P and Q are true. Therefore, P is true. Therefore, Q is true.
    3. Dilemma: two hypothetical syllogisms plus a disjunction. (If P, then Q) and (If R, then S). P or R. Therefore Q or S.
    4. Sorites: a series of premises (incomplete syllogisms) where the predicate of each incomplete syllogism’s premise is the subject of the next. The conclusion consists of the subject of the first premise being joined with the predicate of the last premise. All A is B; All B is C; All C is D; All D is E; Therefore, all A is E.
    5. Perfect Syllogisms: syllogisms that are self-evidently or obviously valid. Another way to say this is that perfect syllogisms are syllogisms where it is obvious that the conclusion follows from the premises if the premises are true. This doesn’t mean that the premises are true, that the conclusion is true, or that the argument is sound.
    6. Epichereme: a syllogism in which a proof is joined to one or both of the premises. The proof is often expressed by a casual clause. The premise to which a proof is annexed is an enthymeme.
    7. Enthymeme: an abbreviated categorical syllogism. One of its premises and/or its conclusion is not expressed.

Syllogisms are about discursive reasoning, that is, attempting to think a problem through logically step by step from one premise to another in an attempt to arrive at an acceptable conclusion or explanation, as opposed to intuitive knowledge. One has to ask how this would be possible without revelation.

Categorical Syllogisms:

  • Categorical Syllogisms are deductive and contain two premise statements plus one, and only one, conclusion statement.
  • Each part of the syllogism is a categorical proposition.
  • Each categorical proposition contains two categorical terms.
  • Each of the premises (proof statements), where “S” is the subject term and “P” is the predicate term, is in one of the following forms:
    • “All S are P.” (Universal Affirmatives)
    • “Some S are P” (Particular Affirmatives)
    • “No S are P” (Universal Negatives)
    • “Some S are not P” (Particular Denials)

A proposition is the statement that is part of a syllogism. There are statements that are not part of a syllogism, and those statements are not propositions. A proposition has two terms, one of which is the subject and the other of which is the predicate.

Conversions—Rules for reversal of subject and predicate:

  • Universal Negatives: If it is true that no A is B, then it will also be true that no B is A. If it is true that no frogs are princes, then is will also be true that no princes are frogs.
  • Universal Affirmatives: If all A is B, then some B is A. (If all B is A, then some B is A.) If all frogs are animals, then some animals are frogs.
  • Particular Affirmatives: If some B is A, then some A is B. If some frogs are green, then some green things are frogs.
  • Particular Denials: subject and predicate cannot be reversed.

Each of the premises has one term in common with the conclusion:

  • In a major premise (the general statement), this is the major term; that is, the predicate of the conclusion.
  • in a minor premise (the specific statement), it is the minor term (the subject) of the conclusion

Major premise: All M are P. (P is the major term)

Minor premise: All S are M. (S is the major term)

Conclusion: All S are P.

Notice that “All P are S” or all “All P are M” would be fallacious and invalid form.

Major premise: All M are P.

Minor premise: All S are M.

Conclusion: All S are P.

M = middle, S = subject, P = predicate.

Major premise: All men are mortal.

Minor premise: Socrates is a man.

Conclusion: Therefore, Socrates is mortal.

men/man = middle, Socrates = subject, mortal = predicate.

Mood

Syllogisms are said to have a mood.

The mood of a syllogism is given in three letters: major premise, minor premise, and conclusion sentence types. Each sentence type is represented by a letter: A (all), E (no), I (some), or O (some not).

A = All X are Y

E = No X are Y

I = Some X are Y

O = Some X are not Y

OAO has an O proposition as its major premise, an A proposition as its minor premise, and another O proposition as its conclusion

Major Premise (O): X does not belong to some Y

Minor Premise (A): X belongs to every Y

Conclusion (O): X does not belong to some Y

EIO syllogism has an E major premise, and I minor premise, and an O conclusion.

Major Premise (E): X belongs to no Y

Minor Premise (I): X belongs to some Y

Conclusion (O): X does not belong to some Y

Symbols:

a = belongs to every

e = belongs to no

i = belongs to some

o = does not belong to some

Abbreviations for categorical sentences:

AaB = A belongs to every B (Every B is A)

AeB = A belongs to no B (No B is A)

AiB = A belongs to some B (Some B is A)

AoB = A does not belong to some B (Some B is not A)

Figure

The figure is solely determined by the position in which its middle term appears in the two premises.

The figure can be 1, 2, 3, or 4:

1:  subject in the major premise and predicate in the minor premise. (mp)(sm)(sp)

2:  predicate in the major premise and predicate in the minor premise. (pm)(sm)(sp)

3:  subject in the major premise and subject in the minor premise. (mp)(ms)(sp)

4:  predicate in the major premise and subject in the minor premise. (pm)(ms)(sp).

The 4 figures of syllogistic arguments for each mood:
1-(mp)(sm)(sp). 2-(pm)(sm)(sp). 3-(mp)(ms)(sp). 4-(pm)(ms)(sp).

Valid Forms

There are 512 logically distinct types of syllogisms, or 256. depending on how you count them. Only 24 types are valid. Of the 24 valid forms, 15 are unconditionally valid, and 9 are conditionally valid. Here are all the valid forms.

Fifteen Unconditionally Valid Forms:

Barbara / AAA-1(mp)(sm)(sp)

All M are P.

All S are M.

Therefore, All S are P.

This is the only valid form of syllogism where conclusions are universal affirmative propositions.

Baroco / AOO-2(pm)(sm)(sp)

All P are M.

Some S are not M.

Therefore, Some S are not P.

Bocardo / OAO-3(mp)(ms)(sp)

Some M are not P.

All M are S.

Therefore, Some S are not P.

Camenes / AEE-4(pm)(ms)(sp)

All P are M.

No M are S.

Therefore, No S are P.

Camestres / AEE-2(pm)(sm)(sp)

All P are M.

No S are M.

Therefore, No S are P.

Celarent / EAE-1(mp)(sm)(sp)

No M are P.

All S are M.

Therefore, No S are P.

Cesare / EAE-2(pm)(sm)(sp)

No P are M.

All S are M.

Therefore, No S are P.

Darii  / AII-1(mp)(sm)(sp)

All M are P.

Some S are M.

Therefore, Some S are P.

Datisi  / AII-3(mp)(ms)(sp)

All M are P.

Some M are S.

Therefore, Some S are P.

Disamis / IAI-3(mp)(ms)(sp)

Some M are P.

All M are S.

Therefore, Some S are P.

Dimaris / IAI-4(pm)(ms)(sp)

Some P are M.

All M are S.

Therefore, Some S are P.

Ferio  / EIO-1(mp)(sm)(sp)

No M are P.

Some S are M.

Therefore, Some S are not P.

Festino / EIO-2(pm)(sm)(sp)

No P are M.

Some S are M.

Therefore, Some S are not P.

Fresison / EIO-4(pm)(ms)(sp)

No P are M.

Some M are S.

Therefore, Some S are not P.

Ferison / EIO-3(mp)(ms)(sp)

No M are P.

Some M are S.

Therefore, Some S are not P.

Valid only if S exists:

Barbari / AAI-1(mp)(sm)(sp)

All M are P.

All S are M.

Therefore, some S are P.

Celaront / EAO-1(mp)(sm)(sp)

No M are P.

All S are M

Therefore, some S are not P.

Camestros / AEO-2(pm)(sm)(sp)

All P are M.

No S are M.

Therefore, some S are not P.

Cesaro / EAO-2(pm)(sm)(sp)

No M are P

All S are M

Therefore, some S are not P

Calemos / AEO-4(pm)(ms)(sp)

All P are M

No M are S

Therefore, some S are not P

Valid only if M exists:

Darapti / AAI-3(mp)(ms)(sp)

All M are P

All M are S

Therefore, some S are P.

Felapton  / EAO-3(mp)(ms)(sp)

No M are P.

All M are S.

Therefore, some S are not P.

Fesapo  / EAO-4(pm)(ms)(sp)

No P are M.

All M are S.

Therefore, some S are not P.

Valid only if P exists:

Bamalip  / AAI-4(pm)(ms)(sp)

All P are M.

All M are S.

Therefore, some S are P.

Hypothetical Syllogisms

Kinds of Hypothetical Syllogism:

  • Conditional Syllogism (“If…, then…”)
  • Disjunctive Syllogism (“Either…, or…”)
  • Conjunctive Syllogism (“Not both…, and…”)

II.  Mixed Hypothetical Syllogisms:

One premise is conditional.

One premise affirms or denies either the antecedent or consequent of that conditional statement.

Four Forms of Mixed Hypothetical Syllogisms:

VALID

(AA) Affirming the Antecedent or Modus Ponens

If p, then q.

p.

q

(DC)  Denying the Consequent or Modus Tollens

If p, then q.

Not q.

Not p.

Invalid forms or pretenders:

(AC)  Affirming the Consequent (AC)

If p, then q.

q.

p.

(DA)  Denying the Antecedent (DA)

If p, then q.

Not p.

Not q.

You can perhaps see why these forms are valid or invalid by considering a very simple example. Think of the following four syllogisms:

1.  Affirming the Antecedent (AA)

If Fido is a dog, then Fido barks.

Fido is a dog.

Fido barks

2.  Denying the Antecedent (DA)

If Fido is a dog, then Fido barks.

Fido is not a dog.

Fido doesn’t bark.

3.  Affirming the Consequent (AC)

If Fido is a dog, then Fido barks.

Fido barks.

Fido is a dog.  

4.  Denying the Consequent (DC)

If Fido is a dog, then Fido barks.

Fido doesn’t bark.

Fido is not a dog.

The following structure is used:

If A, then B.

If B, then C.

Therefore, If A, then C.

Modus ponens

If A, then B

A

Therefore, B

Modus tollens

If A, then necessarily B (this demands that B is necessarily true if A is true.)

Not B

Therefore, not A

Hypothetical syllogism

If A, then B

If B, then C

Therefore, if A, then C

Disjunctive syllogism

A or B (A or B must be the only choices.)

Not A

Therefore, B

Constructive dilemma

A or B

If A then C

If B then D

Therefore C or D

The Conditional Syllogism:

Conditional syllogisms have a conditional statement as the major premise. There are two types of conditional syllogisms:

  1. Mixed Conditional (the minor premise is a categorical proposition)
  2. Purely Conditional (both of whose premises are conditional propositions)

Conditional syllogisms have a major premise, a minor premise and a conclusion. However, conditional syllogisms are often not completed with all three sentences. Sometimes only the major and minor premises are given. Sometimes only the major premise is given. In these cases, the conclusion of the conditional syllogism is by innuendo and left for the audience to infer for themselves.

An antecedent is the statement that leads to the consequent, which follows from the antecedent. The “if clause” is the antecedent. The “then clause” is the consequent.

Mixed Conditional:

 

Affirm antecedent; then affirm consequent (valid)

If A, then B
And A; therefore B

Deny consequent; then deny antecedent (valid)

If A, then B
But not B; therefore not A

Purely Conditional:

If A is a B, then C is a D;

but if X is a Y, then A is a B;

therefore, if X is a Y, then C is a D.

The invalid forms and their respective fallacies are:

Fallacy of Affirming the Consequent (not valid: ACq)

If A, then B
And B; therefore A

Fallacy of Denying the Antecedent (not valid: DA)

If A, then B
Not A; therefore not B

The Disjunctive Syllogism

This syllogism presents two alternatives in an “either . . . or” form; one of the alternatives is for formal reasons assumed to be necessarily true, so that to deny one leaves the other as the only possibility. A disjunct is something disjoined and distinct from some other thing. This relationship is an either this or that but not both relationship. The two possibilities, called disjuncts, are stated in the major premise; one is and must be denied in the minor premise; and the other is affirmed in the conclusion:

Denying the first disjunct and affirming the second  (valid)

Either A or B
Not A; therefore B

Denying second disjunct and affirming the first (valid)

Either A or B
Not B; therefore A

Anyone using this kind of logic must prove that A and B are mutually exclusive, that they are the only two choices, and that one of the absolutely must be true. Many times, the form is right but the logic falls down in a false premise or a premise that is based on nothing of substance.

Fallacy of Affirming a Disjunct (AD):

A possible fallacy is to first affirm one disjunct and then to deny the other. This is known as affirming the disjunct:

Affirming the Disjunct (not valid: AD)

Either A or B
And A; therefore not B

Either A or B
And B; therefore not A

The Conjunctive Syllogism

In the major premise of this syllogism two propositions, called conjuncts, are presented, both of which cannot be true simultaneously. The minor premise proceeds to affirm the true conjunct and the conclusion then denies the remaining one, which must be false by definition. The valid form is:

A cannot be both B and C
A is B; therefore A is not C

(Affirm the first conjunct; deny the second)

A cannot be both B and C
A is C; therefore A is not B

(Affirm the second conjunct; deny the first)

Bill cannot be both a follower of Christ and a non-follower of Christ.

Bill is following Christ; Therefore, Bill is not a non-follower of Christ.

 

Note that we have avoided the term, Christian, because it has so many meanings.

This form requires the following condition: both conjuncts cannot be true.

Fallacy of Denying a Conjunct (not valid: DCj)

A cannot be both B and C
A is not B; therefore A is C

A cannot be both B and C
A is not C; therefore A is B

 

Bill cannot be both an Atheist and a follower of Christ.

Bill is not and Atheist; therefore, Bill is a follower of Christ.

Bill could be a Buddhist, a Muslim, or a religious person who is not following Christ.

Dilemma

A dilemma presents a disjunction. However, the form is different from the hypothetical in that a dilemma is a syllogism that is disjunctive and also conditional. The major premise is conditional, while the minor premise is disjunctive. The major premise will have two or more conditional propositions. The minor premise either affirms the antecedents or denies the consequents. If it affirms the antecedents, then it is called a constructive dilemma. If it denies the consequents, then it is called a destructive dilemma.

A dilemma is sometimes called a syllogismus cornutus, which means a horned argument. The alternatives that are given in the dilemma are called the horns of the dilemma. If you can show that there is another alternative that was not mentioned, it is then said that you have escaped the horns of the dilemma.

Constructive Dilemmas:

Simple Constructive Dilemma:

Either A or B

But, if A, then Z; if B, then Z.

Therefore Z.

Either I must be perfect to get to Heaven or I must be very religious.

But if I try to be perfect, I don’t seem to be good enough so I fail; if I am very religious, I don’t know when I have been religious enough so I fail.

Therefore, I fail in my attempt to go to Heaven.

This person needs to hear the Gospel. It is truly good news.

Complex Constructive Dilemma:

Either A or B.

But, if A, then X; if B, then Y.

Therefore either X or Y.

Destructive Dilemmas

Simple Destructive Dilemma:

If A, then X and Y

But, either not X or not Y.

Therefore not A.

Complex Destructive Dilemma:

If A then X; and if B, then Y.

But, either not X or not Y.

Therefore either not A or not B.

 

If a man is wise, he would not speak irreverently of holy things in jest;

and if he is good, he would not do so in earnest.

But this man does it either in jest or in earnest.

Therefore this man is either not wise or not good.

Rules for Dilemmas

  • No viable alternatives can be left out of the either-or. Beware of false dichotomy.
  • Beware of non sequitur. In the minor premise, the consequent must follow from the antecedents.
  • The dilemma must be logically sound in every way.

 

 Sorites

A sorites is a polysyllogism, that is many syllogisms. It is also called a multi-premise syllogism, a climax, or a gradatio. The syllogisms are all simple. All the conclusions of those syllogisms are sometimes unstated. Each syllogism (or partial syllogism) in the heap is a prosyllogism except for the last one, since the last syllogism is not a premise of another syllogism. To be sound, the sorites must go all the way back to something that is known to be true. The questions that need to be asked about every prosyllogism is, “How do you know?” and “How can I test this so I can know it?” A thought chain is as strong as its weakest link. If one of these prosyllogism cannot be absolutely known, then the entire argument is unsound.

A is B

B is C

C is D

D is E

Therefore, A is E

 

If A, then B.

If B, then C.

If C, then D.

If D, then E.

Therefore, if A, then E.

If the conclusions are stated, it would be:

A is F.

A is B; therefore, B is F

B is C; therefore, C is F

C is D; therefore, D is F

D is E; therefore, E is F

Rule 1: All but the last premise must be affirmative. If a premise is negative, the conclusion must be negative.

A is B

B is C

C is D

D is not E

Therefore, A is not E

Rule 2: All but the first premise must be universal. If the first premise is particular, the conclusion must be particular.

Some A is B

B is C

C is D

D is not E

Therefore some A is not E

C. S. Lewis: A man who was merely a man and said the sort of things Jesus said would not be a great moral teacher. He would either be a lunatic–on the level with a man who says he is a poached egg–or he would be the devil of hell. You must take your choice. Either this was, and is, the Son of God, or else a madman or something worse. You can shut him up for a fool or you can fall at his feet and call him Lord and God. But let us not come with any patronizing nonsense about his being a great human teacher. He has not left that open to us.

 

Perfect Syllogism

Perfect syllogisms are syllogisms that are self-evidently or obviously valid. Another way to say this is that perfect syllogisms are syllogisms where it is obvious that the conclusion follows from the premises if the premises are true. This doesn’t mean that the premises are true, that the conclusion is true, or that the argument is sound.

The perfect syllogisms are not provable and are taken as axioms in Aristotelian logic. Aristotle uses these axioms to prove the imperfect syllogisms with the use of conversion. The imperfect syllogisms are demonstrated to be valid by some means, usually reduction to one of the perfect syllogisms.

Aristotle said that Barbara, Celarent, Darii, and Ferio are the perfect syllogisms.

Barbara / AAA-1(mp)(sm)(sp)

All M are P.

All S are M.

Therefore, All S are P.

Celarent / EAE-1(mp)(sm)(sp)

No M are P.

All S are M.

Therefore, No S are P.

Darii  / AII-1(mp)(sm)(sp)

All M are P.

Some S are M.

Therefore, Some S are P.

Ferio  / EIO-1(mp)(sm)(sp)

No M are P.

Some S are M.

Therefore, Some S are not P.

 

Epichereme

An epichereme is a syllogism in which a proof is joined to one or both of the premises. The proof is joined to the premise by a connective such as “for,” because,” “since,”  or “due to.”

Proof: A reason that the premise is believed to be true

(Check proofs to see if they really prove what they say that they are proving. Make sure that you don’t need to prove the proof before going forward. This is almost always a problem.)

Casual Clause: If there is a causal relationship between two things, one thing is responsible for causing the other thing

The cause will be connected to the premise by a connective that shows cause, like “because.” Always check that when cause is shown, the cause fully proves the claim that is being made. You will be surprised to find out that it usually does not prove the claim.

Enthymeme: The premise to which a proof is attached

A is B because X

C is A because Y

Therefore, C is B

 

Every person who follows Christ is led by Christ and comes to know Him personally, and we know this because God reveals it to us through Scripture and by the testimony of the Holy Spirit through all of those who experience this, and I also personally know this by the revelation of Christ to me as He leads me moment-by-moment.

You can follow Christ if you are willing, since Christ died and rose again to pardon you and set you free from your own sins just ash he has done for everyone else who follows Him.

Therefore, you can also prove Christ to yourself and come to know Him personally simply by seeking Him in sincerity, humility, and with a will to do His will.

Enthymeme

An enthymeme is a syllogism in which one of the premises or the conclusion is omitted. One of the problems of the enthymeme is that the audience must often guess the missing information. Guessing is not the best way to know. There are three orders of enthymeme:

First Order: major premise is omitted

You can follow Christ if you are willing, since Christ died and rose again to pardon you and set you free from your own sins just ash he has done for everyone else who follows Him.

Therefore, you can also prove Christ to yourself and come to know Him personally simply by seeking Him in sincerity, humility, and with a will to do His will.

Second Order: minor premise is omitted

Every person who follows Christ is led by Christ and comes to know Him personally, and we know this because God reveals it to us through Scripture and by the testimony of the Holy Spirit through all of those who experience this, and I also personally know this by the revelation of Christ to me as He leads me moment-by-moment.

Therefore, you can also prove Christ to yourself and come to know Him personally simply by seeking Him in sincerity, humility, and with a will to do His will.

Third Order: conclusion is omitted

Every person who follows Christ is led by Christ and comes to know Him personally, and we know this because God reveals it to us through Scripture and by the testimony of the Holy Spirit through all of those who experience this, and I also personally know this by the revelation of Christ to me as He leads me moment-by-moment.

You can follow Christ if you are willing, since Christ died and rose again to pardon you and set you free from your own sins just ash he has done for everyone else who follows Him.

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