an interactive glossary for reasoning
A Posteriori and A Priori
‘A posteriori’ and ‘a priori’ are ways of knowing. A posteriori knowledge is based on experience. A priori knowledge does not require sensory experience to be known to be true. It is based on reasoning rather than observation. For example, I look outside to see whether it is raining, but know by definition that rain is water.
Accent
Accent is a fallacy in which changing stress on a key element interprets the rule narrowly and changes the meaning of the rule. This can seem to make a prohibition more permissive: by stressing the factor to be excluded, it implies that all else is admissible.
1 Mother said we shouldn’t throw STONES at the cat. She didn’t say anything about throwing apples.
2 Perhaps people are BORN equal, but that does not mean they are equal as adults and so there no reason for giving them all an equal vote.
Accident
Accident (also known as “Sweeping Generalization”) is a fallacy of treating a general rule as absolute and using an exception to that rule to draw an absurd conclusion or refute the rule. The process is to treat a general rule [“most birds can fly”] as rigidly absolute [“all birds can fly”], offer an exception [“a penguin is a bird”], then conclude that the general rule is incorrect [“birds can’t fly”]. Accident is fallacious because it treats a conditional rule (most, some, usually) as absolute (all, every, only).
1 The sign says ‘no parking’ so the ambulance should not park here. Well, obviously parking is allowed. [Treats ‘no parking’ as allowed for no one, ignores emergency vehicles as a permissible exception, concludes that parking is allowed implying for everyone.]
2 Keri claims that we have a duty to repay what is owed. But suppose a man lends you a rifle for hunting, goes insane, then asks for the rife back? It would be wrong to put a lethal weapon into the hands of a madman, so Keri is wrong and we don’t really have a duty to repay what is owed.
A note should be made of the phrase “an exception that proves the rule” since some people use it incorrectly. To ‘prove’ used to mean to ‘test’ or establish quality, such as in proving the temper of a sword. The phrase “an exception that proves the rule” uses ‘prove’ in this sense. The exception puts the general rule to the test and, if the exception has merit, than it refutes the rule.
Advice
Advice is a type of non-inference that makes a recommendation about a future decision or course of conduct. Advice is guidance or recommendation concerning prudent future action, typically given by someone regarded as knowledgeable or authoritative. Advice to beware amounts to a warning. It serves to put another person on guard about an impending harm.
1 Don’t try to argue with anyone who has taken a course on reasoning. They just know too much about fallacies.
2 Before purchasing a puppy, visit the breeder. Ask to see the mother, and be suspicious if you are not allowed to do so. Of course, never purchase from a chain store; that only encourages puppy mills.
Affirm the Antecedent
Affirm the Antecedent (or modus ponens) is a deductive argument that if a conditional statement is true and the antecedent of that statement is true, then the consequent of that statement is true. This has the form: A ⊃ B, A, ∴ B. That is:
if A is true, then B is true
A is true
therefore, B is true
1 If two objects have mass, then they mutually attract. The Earth and its moon each have mass. Therefore, the Earth and moon attract one another.
2 When the bananas are ripe, I slice one onto my granola. This bunch seems ripe, so I will have fruit on my cereal.
Affirm the Consequent
Affirm the Consequent is a fallacy in which a true consequent [“an egg is broken”] of a conditional statement [“if I drop an egg, it breaks”] is considered reason to believe the antecedent is true [“I dropped the egg”]. This is a formal fallacy: A ⊃ B, B, ∴ A. That is: if A, then B; B is true; therefore, A is true.
Affirming the Consequent is a corruption of the valid deductive form called Affirm the Antecedent. Affirming the Consequent is fallacious because an event can be produced by different causes. For instance, I might not have dropped the egg; there could have been other causes. Perhaps someone else dropped it or the chick popped out.
1 When cats are bitten by rabid raccoons, the cats die. We found a dead cat by the roadside so there must be a rabid raccoon around here.
2 To have a fire, there must be oxygen. There is oxygen here, so there must be a fire.
Ambiguity
Ambiguity refers to a category of fallacies in which a word or phrased is not used clearly or consistently. The fallacy occurs because the premises cannot support the conclusion since they refer to different senses or interpretations. Ambiguity can include: Accent, Amphiboly, Equivocation.
Amphiboly
Amphiboly is a fallacy of ambiguity in which a premise can be interpreted in more than one way due to the punctuation or arrangement of words and the conclusion is based on the more unlikely interpretation. Amphiboly is fallacious because the premises cannot support the conclusion since they refer to different interpretations.
1 The supervisor told my client that he made a mistake. That shows the supervisor has the courage to admit making mistakes. [Does ‘he’ refer to the supervisor or to the employee?]
2 The will states, ”I leave my house and my dog to my niece and nephew.” The niece claims she gets the house exclusively. The nephew claims the property and pet pass to both relatives collectively.
Analogy
Analogy is an inductive argument which asserts: that which is true of one case is true of another case that is similar in relevant ways. The pattern is roughly: A and B are similar in various ways and have no overriding dissimilarity; their similarities are sufficient and relevant to property C; A has property C; therefore, B likely has property C.
To assess the strength of an analogy, consider whether the number of ways in which the two cases are said to be similar is sufficient and whether they are relevant to the property at issue. Also consider whether the cases are not dissimilar in some manner that overrides the ways in which they are alike. And keep in mind that a conclusion by even the strongest analogy is inductive (likely), not deductive (certain).
1 Weak Analogy: Puzzles and chores are both time-consuming and difficult. I like doing crossword puzzles, so I will like doing chores.
2 Weak Analogy: Just as it was wrong to deny women the vote, it is also wrong to deny the vote to children.
3 Strong Analogy: A pipe resists the flow of water. A hydraulic circuit and electrical circuit both have a source (pump/battery), connections (pipe/wire), and controls (valve/switch). Source, connection, and controls are relevant to the flow of the contained medium. No overriding dissimilarity: both vessels are tubular containers that carry a flow. Therefore, the wire likely resists the flow of electricity.
4 Strong Analogy: The dogs at your kennel ate the kibble. You have yellow labs and I have golden retrievers. The dogs are similar in size, age, health, and breed, which are factors that can affect what they will eat. There is no major difference in the dogs nor in their kennels. We live in the same climate and the dogs get about the same exercise. So, when I give some of that kibble to my retrievers, I suspect they will eat it.
Appeal To Antiquity
Appeal To Antiquity ( argumentum ad antiquitatem ) is a fallacy that draws a conclusion from the mere fact that it does as has always been done. To do as we’ve always done is reason to continue to do so. This fallacy supposes that something is good or right simply because it is old. If something fits with tradition, then it is true or appropriate.
Appeal to Antiquity is fallacious since the age of a belief or practice may make it more familiar, but familiarity is not relevant to whether or not it is correct. A long-held false belief (e.g., the world is flat) or practice (e.g., blood-letting reduces fever) can still be false.
1 The mayor should be a man because we have always had male mayors.
2 We don’t need email. Telegrams were good enough when my father ran the company.
Appeal to Antiquity is similar to Appeal to Novelty (argumentum ad novitatem), which draws a conclusion from the mere fact that it is new and newer is good or true. Being new or newer is reason to believe that it is true or good.
Appeal to Emotion
Appeal to Emotion is a category of fallacies in which the desire to have or avoid a certain feeling is reason for a belief or action. In an effective argument, the premises support the conclusion. In fallacies of emotional influence, however, the premises are not relevant to the conclusion. Instead the arguer uses some sort of emotional appeal to make the premises appear relevant.
Appeal to Emotion regards esteem or sorrow as sufficient reason to accept a certain belief or course of action. Esteem may be desire for flattery, popularity, pride, sense of inclusion, or other personal affirmation. Sorrow can be the wish to avoid confusion, threat, disgust, or other feelings of discomfort. Whether the appeal is to esteem or sorrow, Appeal to Emotion is fallacious because the emotional factor is not relevant to whether the proposition is true or false.
Examples: Appeal to Fear, Appeal to Pity.
Appeal to Fear
Appeal to Fear is a type of Appeal to Emotion fallacy in which the conclusion is based on the desire to avoid likely harm or discomfort. This amounts to coercion; using threat to cause a reluctant person to do something. The coerced party acts out of fear of harm rather than out of reasoned persuasion.
1 In this neighbourhood we give expensive treats for Halloween. It is still cheaper than removing rotten eggs or spray paint.
2 The lawyers handling the estate of my late aunt sent papers for the beneficiaries to sign. The documents are loaded with legal jargon, but my cousin just signed them. The legal language was confusing and he didn’t want to feel awkward asking what this word or that word means.
3 Soda pop sales declined as people switched to milk, So, let’s do an ad against milk repeatedly using the word “chemical” — you know, milk is full of chemicals. Calcium and lactose are still chemicals.
Appeal to Ignorance
Appeal to Ignorance is a fallacy that draws a conclusion from the mere absence of proof to the contrary. A claim needs some support; absence of evidence to the contrary is, by itself, insufficient. Appeal to Ignorance is sometimes used as a tactic to shift the burden of proof.
1 Life must exist somewhere in the universe since, after all, we have not visited and ruled out every possible planet [qualified investigators fail to find it, so this may imply there is nothing to be found].
2 There is no conclusive proof that nuclear power plants pose a danger to people living in their vicinity. Therefore, it is perfectly safe to continue to build nuclear power plants near urban centres.
3 If you can’t prove that you weren’t driving recklessly, then you must be guilty.
4 Whoa! Feel that? There’s a cold spot in this room and no apparent source. It must be the presence of a ghost.
Appeal to Inclusion
Appeal to Inclusion is a type of Appeal to Emotion fallacy in which the desire to not be left out of a group is reason to act or believe as others do. In this fallacy, wanting to belong is reason to act or believe as others do.
1 But mom, everybody is going to be wearing these when school starts. If you don’t buy me a pair, I won’t fit in. I just want to be like the other kids.
2 Country western has got to be the purest form of musical ballad; I mean just listen to the applause from the studio audience.
Appeal to Legitimate Authority
Appeal to Legitimate Authority is a means of inductive inference that amounts to citing an expert in support of a claim within the experts area of competence.
1 The vet says Rocky has hemangiosarcoma and, unless the growth is removed from his gums, it could spread. Rocky could die.
2 Wilson’s pet pig has been sniffing out truffles for years. It rooted out this nugget, so I’m going to eat the delicacy in confidence.
Appeal to Pity
Appeal to Pity is a type of Appeal to Emotion fallacy in which a conclusion is based on feeling sorry for the suffering of others.
1 In asking yourself if this man is to be convicted, ask yourself what it means for him to be locked up in prison, deprived of his liberty, and turned into an outcast from humanity.
2 Listen, I didn’t mean to make you cry, but you have lots of toys at home. No, not tears again. Okay, just the teddy bear, okay?
Appeal to Praise
Appeal to Praise is a type of Appeal to Emotion fallacy in which a conclusion is based on desire for the approval of others or acknowledgement of self-regard (vanity).
1 You’ve worked hard for your money. Anything less than a Brand X sportcar will not do for someone of your class and distinction.
2 What a wonderful wit you have – a good sense of timing and great punch-lines! No doubt you can see the humour in the mayor’s plan to tax employees for parking spaces.
Appeal to Spite
Appeal to Spite is a fallacy in which a conclusion is based on revenge. The fallacy attempts to leverage a grudge, to convince a second person by exploiting the other person’s existing feelings of bitterness or spite toward a third party.
1 The French refused to join us in the military invasion, so don’t buy French wines.
2 Your mom grounded you for staying up too late? You should jump around in that mud puddle before you go home and make sure to put shoe prints all over the carpet. It’ll drive her crazy.
Argument
An argument is a group of statements, some of which (the premises) give reason to believe the one of the other statements (the conclusion). The premises support the conclusion. The statements have an inferential relationship. That is, they reach a conclusion based on the evidence (premises, reasons) and reasoning (rules).
In a deductive argument, true premises guarantee a true conclusion. In an inductive argument, true premises make the conclusion likely, but not certain.
If the statements have no inferential relationship, the non-inference passage wouldn’t be an argument. Even so, it might be informative or persuasive, such as an explanation or command.
Art and Science
Art/artistic: an artifact (object or performance) that has aesthetic merit. For example, Fallingwater house as an example of organic architecture.
Science/scientific: an assertion that stands up to a method of measurement and verification that others can reliably reproduce. For example, belief that water expands as it freezes.
Association
Association is a rule of equivalence that switching the grouping of statements joined by “and” or by “or” does not change whether the overall expression is true or false.
1 Putting the egg and vanilla in the bowl then adding sugar is the same as adding egg to the bowl that already has vanilla and sugar. (A & B) & C ≡ A & (B & C)
2 “Do you want pepperoni or salami — or would you rather have just cheese” is the same as “Do you want pepperoni — or would you rather have salami or just cheese”. (A | B) | C ≡ A | (B | C)
Belief and Justification
Belief is an attitude of accepting something as true, whether or not one consciously considers the assertion. For example, I believe there are seven stars in the Big Dipper.
Justification is an experience that reliably represents an assertion as true. I am justified in believing that I just won the lottery because the numbers on my ticket match those announced. If the announcer misread a 9 for a 6, I am justified in my belief until the error is discovered.
Black Box
A black box refers to a system that has input and output, but whose workings inside the box are not known. “I turn the thermostat and there is cool air, but I have no idea how AC works; it’s a black box to me.” A system where the internal operation is known or knowable is referred to as a clear box.
Begging the Question
Begging the Question is a fallacy that presents a claim as a conclusion when the claim is actually just a restatement of one of the premises. A conclusion is the proposition to be proven. In this fallacy, the conclusion is presumed in the premises, usually in synonyms or paraphrase. Begging the Question is fallacious because the reasons or evidence are not independent of the claim; the conclusion just restates a premise which is presumed.
1 The reason there is such a big demand for the latest technology toys is because everybody wants them.
2 Telepathy cannot exist since direct transfer of thought between individuals is impossible.
Changing the Subject
Changing the Subject (also known as Red Herring) is a fallacy with the pretense of refuting a claim or argument by diverting attention from it. This fallacy responds to the claim of another party by diverting attention from that claim or its evidence and in doing so claims to have defeated the claim. One version of Changing the Subject introduces an anecdote or other bit of humour to divert attention.
1 We’ve all heard the argument that too much television is the reason our children can’t read and write. Yet, many of today’s TV shows are excellent. Reality shows require ordinary people to use their wits, sit-coms provide comedy relief from everyday stress, and drama programs add a sense of excitement and adventure. Today’s TV is just great!
2 Is nursing a worthwhile occupation? Believe me, we not only work as hard as anyone else, but harder. The hours are long, the demands are many, and you have to keep current with medical changes. [Which changes the subject: it may be hard work, but is it worthwhile?]
Commutation
Commutation is a rule of equivalence that switching the order of statements joined by “and” or by “or” does not change whether the overall expression is true or false.
1 A dollar and a dime has the same value as a dime and a dollar. A & B ≡ B & A
2 Whether you put on the left glove first or right glove first, the result will be the same. A | B ≡ B | A
Complex Question
Complex Question (or loaded question or plurium interrogationum) is a fallacy in which the conclusion is presumed within a question so that answering the question affirms the conclusion. The question is used to imply more than to inquire. This fallacy is a variation of Begging the Question because a complex question contains an assumption that the concealed question has already been answered affirmatively. It is this unjustified presumption which constitutes the fallacy.
1 Where did you hide the jewels you stole? [Even answering “nowhere” presumes that you stole the jewels, but just did not hide them.]
2 Timmy, do you want to put your toys away in the box or on the shelf? Would you prefer to take your nap now or after your cocoa? [Timmy is being told to ‘put away your toys’ and ‘take a nap’ even though he appears to have a choice in the matters.]
Respond to a complex question by dividing it into its component questions and answer each separately.
Composition
Composition is a fallacy that concludes that something has a property because its parts have that property, even though the attribute cannot be transferred from part to whole. The error lies in transferring attributes that cannot be transferred.
1 Every part of the model can fit in its package, but it does not follow that the assembled airplane can fit in its package.
2 You like cola. You like corn flakes. So you will like cola on corn flakes.
3 Every atom of the apple is invisible, but it does not follow that the apple is invisible.
4 Oxygen and hydrogen are gases at room temperature, water consists of oxygen and hydrogen, so water is a gas at room temperature.
Expansive Property is an exception to the fallacy of Composition.
Comprehend
The ability to infer from supporting evidence, transfer content to another context, and predict what comes next.
1 I know that water forms a sphere when floating free of gravity.
2 A sphere has minimum surface area per volume.
3 A flame lit in zero gravity will likely form a sphere.
Conditional Proof
Conditional Proof is a valid argument that if A is assumed to be true and B results, then the conditional statement “if A, then B” is true. This method consists of assuming the antecedent of the required conditional statement and deriving the consequent.
1 Whenever it is snowing, it is cold outside and there are clouds. Whether it is cold or warm, you should dress appropriately. Therefore, if it is snowing, you should dress appropriately.
2 If I had wealth, then I’d take care of my health. If I had both wealth and health, then I would be happy. Therefore, if I had wealth, I would have happiness.
3 If the mouse is still in the house, then it has to be hiding in the attic or basement because they’re the only places still not checked. No wait, it can’t be the attic. We sealed the attic last summer. The basement then. If the mouse is still in the house, then it has to be in the basement. Which means either it has left the house or it’s hiding in the basement.
Let M = mouse in house, A = in attic, B = in basement:
1. suppose M
2. if M, then A | B
3. ~A
4. so, B
5. therefore: if M, then B, which is to say ~M | B
Conditional Statement
Conditional Statement is a type of non-inference that asserts an antecedent and its consequent, but doesn’t assert that they are true. A conditional statement expresses belief in a connection, but does not assert that either part (the antecedent or consequent) is true.
A conditional has an if-then structure: if A, then B. For example: “if you are a bachelor, then you are not married”. The antecedent in a conditional is asserted in the if part (or a synonym, such as “when” or “provided”). The consequent is asserted in the then part (or a synonym, such as “so” or “subsequently”). A conditional statement is not an argument, but could be added to other premises to become part of an argument.
Protases (pronounced: PROT uh sis) is the clause expressing the condition in a conditional sentence, in English usually beginning with “if.” Apodoses (pronounced: uh POD uh sis) is the clause expressing the consequence in a conditional sentence, often beginning with “then.”
1 If art is an artifact that evokes emotion, then an arsonist is an artist.
2 Suppose the air is removed from a solid closed container; the container will weigh less than it did.
Conjunctive Addition
Conjunctive Addition is a valid argument that if two or more statements are true on their own, then they are true together. Any two true statements can be joined to form a true conjunction. A is true; B is true; so, A is true and B is true. A, B, ∴ A & B. This is true and that is true, then “this and that” is true. The order of conjuncts does not matter, so B & A is also true.
1 The sky is blue; the grass is green; so it is true that the sky is blue and the grass is green.
2 The rain stopped; the sun is shining; so it is true that the rain stopped and the sun is shining.
Constructive Dilemma
Constructive Dilemma is a valid argument that, given two or more true conditionals, if one of the antecedents is true, then one of the consequents must also be true. If A is true, then B is true and if X is true, then Y is true; A or X is true; so, B or Y is true. (A ⊃ B) & (X ⊃ Y), A | X, ∴ B | Y. If at least one antecedent is true, then at least one consequent is true.
1 If you play the ace, then you win the hand; but if you play the deuce, your partner will win. You must play either the ace or the deuce. Therefore either you win or your partner wins.
2 If there is a red sky at night, then the weather will be clear. However, if there is a red sky in the morning, then the weather will be stormy. There will be a red sky either tonight or in the morning. Therefore the weather will be either clear or stormy.
Contextualism and Textualism
Contextualism and Textualism are types of interpretations.
Contextualism takes into account the intention of the author or the social context of when and where a composition was written.
Textualism is based on the ordinary meaning of the text without taking into account external sources, such as what the author intended.
For example, according to Harry Styles, his song “Sign of the Times” is about a mother dying after childbirth. By the lyrics, the song could be about a criminal’s regret to reform.
Corresponding Cause
Corresponding Cause (or Mill’s Methods of Induction) is an inductive argument that one event is the cause of the other based on their correspondence. A causal relationship exists when one event (the cause) brings about or influences the other event (the effect). Corresponding Cause tests whether events match in presence (when Bart’s here, cookies disappear), absence (when he’s away, the cookies stay), and variation (the longer he’s here, the more that disappear).
Deduction Guarantee
The form of the conclusion is in the premises. The rules of deduction transform or extract the conclusion from the premises. For example:
The light is on or off;
it is not on; so, it is off.
A or B;
if you cover up A, what’s left is B.
Deductive Argument
In a deductive argument the premises have a formal connection to the conclusion: if the premises are true, the conclusion must be true. Said another way, it isn’t possible for the premises to be true and yet the conclusion to be false.
In a valid deductive argument, true premises guarantee that the conclusion is true. That is, the structure of the argument fits one of the recognized forms or patterns of inference, such as Affirm the Antecedent.
1 Squares are rectangles; rectangles are quadrilaterals; so squares are quadrilaterals.
2 All humans have genes; I am human; therefore, I have genes.
In an invalid deductive argument, the conclusion is claimed to — but does not actually — follow from the premises. It is defective.
If an argument has one of the following deductive forms, then that argument is valid. These forms guarantee that if the premises are true, the conclusion must be true.
1. Affirm the Antecedent (A ⊃ B), A, ∴ B
2. Conditional Proof presume A, B follows, ∴ A ⊃ B
3. Conjunctive Addition A, B, ∴ A & B
4. Constructive Dilemma (A ⊃ B) & (X ⊃ Y), A | X, ∴ B | Y
5. Deny the Consequent A ⊃ B, ~B, ∴ ~A
6. Destructive Dilemma (A ⊃ B) & (X ⊃ Y), ~B | ~Y, ∴ ~A | ~X
7. Disjunctive Addition A, ∴ A | B
8. Disjunctive Syllogism A | B, ~A, ∴ B
9. Hypothetical Syllogism A ⊃ B, B ⊃ C, ∴ A ⊃ C
10. Simplification A & B, ∴ A
The following argument is valid: (1) if every animal can fly, then pigs can fly; (2) a pig is an animal; (3) therefore, a pig can fly. It is valid because it can be expressed in one of the deductive forms, in this case Affirm the Antecedent. Even so, the argument is not sound.
An argument is said to be sound only if it is valid and the premises are true. The following is sound: (1) if every animal has DNA, then a pig has DNA; (2) a pig is an animal; (3) therefore, a pig has DNA.
Definition
Definition by genus and difference is a type of non-inference that describes the category to which something belongs and its distinctive features.
1 A triangle is a type of figure [its closest category] that is two-dimensional and has three straight sides [these features distinguish it from other figures].
2 Legally blind: a measure of vision less than 20/200 or less than 20 degrees diameter (10 degrees radius) and that cannot be improved with corrective lenses.
A definition is a description that identifies the closest category to which something belongs and the features that distinguish it from other members of that category. The list of features is sufficient if it identifies all and only the members to be included.
3 “Engine: a type of machine that converts gasoline into motion” is too exclusive. A truck might run on diesel fuel.
4 “Bachelor: a male who is unmarried” is too inclusive. A baby boy is not really a bachelor.
A definition should not be circular (Hill: land lower than a mountain. Mountain: land higher than a hill). State the definition in positive terms if possible, although sometimes negative terms are unavoidable, such as in defining “bald” or “darkness” or “silence”.
DeMorgan’s Rule
DeMorgan’s Rule is a rule of equivalence that a negative can be distributed to a conjunction or disjunction. That is, ~(A & B) ≡ ~A | ~B. Likewise ~(A | B) ≡ ~A & ~B.
1 “He isn’t tall, tanned, and handsome” means he is not tall or not tanned or not handsome.
2 “The subway does not run north or south” means the subway does not run north and it doesn’t run south.
Deny the Antecedent
Deny the Antecedent is fallacy that a false antecedent is reason to believe the consequent is false. This is a formal fallacy: A ⊃ B, ~A, ∴ ~B. That is: if A is true, then B is true; A is not true; therefore, B is not true.
1 If I am in the ocean, then I am in water; I am not in the ocean, therefore I am not in water. [I could be in a river, pool, or bathtub.]
2 If capital punishment deterred murder, then it would be justified. It does not deter murder and so capital punishment is not justified.
Deny the Antecedent is a corruption of the valid deductive form called Deny the Consequent.
Deny the Consequent
Deny the Consequent (or modus tollens) is a valid argument that if the consequent in a conditional statement is false, then the antecedent in that statement is false. If A is true, then B is true; but B is not true; so, A is not true. A ⊃ B, ~B, ∴ ~A.
1 If there is fire, then there must be oxygen; the sun has no oxygen; therefore the sun is not on fire.
2 If you love me, you would not leave me; you did leave me; therefore, you do not love me.
Destructive Dilemma
Destructive Dilemma is a valid argument that, given two or more true conditionals, if one of the consequents is false, then one of the antecedents must also be false.
Start with two conditionals: (if A is true, then B is true) and (if X is true, then Y is true). In symbols, this is: (A ⊃ B) & (X ⊃ Y). One consequent is false: B is false or Y is false. ~B | ~Y. Therefore, one of the antecedents must be false: A is false or X is false. ∴ ~A | ~X. If at least one consequent is false, then at least one antecedent is false.
1 If we are going to paint the deck, then we need to buy brushes; but if we are to stay within budget, then we must borrow brushes from a neighbour. We will either not purchase brushes or we will not borrow them. Therefore we will either not paint the deck or we will not stay within budget.
2 If the model car is under-oiled, it will squeak; but if it is over-oiled, it will start to smoke. Either the model car did not squeak or it did not smoke; so it wasn’t under-oiled or it wasn’t over-oiled.
Disjunctive Addition
Disjunctive Addition is a valid argument that disjunction is true as long as at least one of its simple statements is true. Starting with a true statement, any other statement can be connected by “or” and the resulting disjunction will also be true. In other words, if a statement is true, then any statement in which it is a disjunct is also true. A is true; therefore, A is true or B is true even if B is false or unrelated to A. A, ∴ A | B. In other words, “this or that” is true as long as one of them is true.
1 Any mass has inertia, so it is true that “any mass has inertia or the Earth is flat”
2 Since 2+2=4, it is true that “2+2=4 or I am 6 cm tall”
Disjunctive Syllogism
A disjunction is an exclusive statement of alternatives: this or that, not both, not something else. Disjunctive Syllogism (also know as excluded middle) is a valid argument that if one of the disjuncts is false, then the other has to be true. A is true or B is true; but A is not true; so, B is true. A | B, ~A, ∴ B. That one disjunct is false is reason to believe its alternative is true.
1 We know that she paid either Pete or Paul. Our sources show that she didn’t pay Pete, so she must have paid Paul.
2 The king held out a basket with two notes. If the knight picks the one marked Yes, he may marry the princess. The knight, suspecting both notes are marked No, grabbed a note and swallowed it, proclaiming that he picks the note remaining in the basket.
Distribution
Distribution is a rule of equivalence that disjunction is distributive over conjunction, and conjunction is distributive over disjunction.
1 Ice cream on cake or pie is the same as ice cream on cake or ice cream on pie. [A & (B | C)] ≡ (A & B) | (A & C)
2 “Jogging or sitting and reading” is the same as “jogging or sitting and jogging or reading.” [A | (B & C)] ≡ (A | B) & (A | C)
Division
Division is a fallacy that concludes that the parts have a property because the whole has that property. Division assumes that whatever is true of a whole must be true of each of its parts. As with the fallacy of Composition, the error lies transferring properties that cannot be properly transferred.
1 Smith, you claim to be bankrupt, but you work for a wealthy company, so you must be wealthy.
2 Human beings are made of cells. Human beings are conscious, so cells must be conscious.
Double Negation
Double Negation is a rule of equivalence that a double negative is equivalent to a positive. That is, ~~A ≡ A.
1 I wouldn’t ask if this were not so important ≡ I am asking since this is important.
2 Never have the bees been without a queen ≡ The bees have always had a queen.
3 That white hat of yours is hardly inconspicuous ≡ Your hat is conspicuous.
4 There isn’t a day when I don’t think about it ≡ I think about it every day.
Double Negation can be used with an inherently negative term.
5 It is not true that the performance was boring (not interesting) ≡ The performance was interesting.
6 The building is not in total darkness (lacking light) ≡ Some light is showing in the building.
Equivalence
Equivalence is a relationship between two statements having the same truth value. One statement is another form of the other statement. This transformation does not make a new argument. It just puts components of a statement in another form. In the following deductive transformation rules, one expression can be replaced with the other.
Equivocation
Equivocation is a fallacy of Ambiguity in which a word or phrase is used in one sense in the premises and in a different sense in the conclusion. The premises cannot support the conclusion since they refer to different senses.
1 An athlete is a human being, so a good athlete is a good human being [equivocation on “good” as “successful” or “virtuous”].
2 Emeralds are seldom found in this country, so you should be careful not to misplace your emerald ring or it is not likely to be found here [equivocation on “found” as “indigenous to” or “located”].
Essay examples
The following essay examples are organized around the deductive argument form Affirm the Consequent in order to be more persuasive.
1 Hero or Zero: a matter of fairy tale perspective
For any conflict, the victor writes the history and the vanquished becomes the bad guy. That’s true for the little old lady that Hansel and Gretel called a witch. It is true for the lonely wolf bullied by the three pigs-and trust me they weren’t little. In other words, if you examine fairy tales from the victim’s viewpoint, then you’ll see that the so-called hero is really the villain.
Consider the case of Jack and the Bean Stalk. My name is Jolly Green and this is how Jack vanquished my brother, André XXXXL. Jack’s mother, a shrewd lady, decided to pawn off their aging cow on some fool before it started pushing up daisies. A traveling salesman suckered Jack into trading the cow straight up for a pocket full of kidney beans. His mother, upon seeing his take, was furious. She tossed the beans out the window and gave Jack a tongue-lashing. The next morning, to everyone’s surprise, the beans have turned into some sort of mutated freak-stalk. Jack decided to climb this floral staircase to see what might be at the top. Upon spying my brother’s abode, the boy sneaks into the kitchen. Here the crimes begin: (1) trespassing, followed by (2) breaking-and-entering. From the point of view of the fairy tale giant, he is the real victim.
As an amateur alchemist, André developed a process to turn ordinary animal feed into gold using only the reproductive tract of a goose. Jack stealthily crept up and pilfered the goose. Crime (3) grand theft goose. Springing from his chair André chased the evil little snot onto the clouds and towards the beanstalk. My brother, as a result of his dense bone structure and massive torso, clambered much more carefully down the vines. At the bottom, Jack stashed the goose and started up his chainsaw. With a loud crack the stalk separated from its base, sending it and its rider crashing. Crimes (4) and (5): willful destruction of property and homicide (or giant-icide). No, the giant (or wolf or troll or witch) does not lives happily ever after. Moral of the story: the fairy tale ‘hero’ is really the villain.
Affirm the Consequent is the basis of the essay. The conditional is the last sentence of ¶1. The antecedent is the last sentence of ¶2. The conclusion is the last sentence of ¶3. In each paragraph, sentences prior to the last serve to establish the credibility of that last sentence. They give evidence for believing it.
2 Cats Make Healthy House Pets
People need pets. The common saying that “dog is man’s best friend” reflects the fact that people have a basic need for companionship. It is well known, for instance, that the elderly feel better and live longer when caring for and interacting with a pet. Dogs are not the only animals that give companionship, however. Some people keep goldfish, parrots, hamsters, turtles, lizards, Guinea Pigs, and even ferrets. As long a pet has many advantages and few disadvantages, then it makes a good pet.
Despite what some dog lovers believe, cats make excellent house pets. They are affectionate, playful, and quiet. They will snuggle up and purr to be petted or scratched under the chin. Who can resist a kitten chasing a balls of yarn. Bathing is rarely needed since cats take care of their own grooming. As a bonus, they do not have to be walked, getting plenty of exercise playing in the house. Cats don’t bark; most don’t even meow very often. They generally lead a quiet existence. Cats do need a litter box, but the mother trains her kittens and from then on most will use the litter box without fail. Cats also need a scratching post, but will use it safely and leave the furniture alone. Clearly, the benefits of cats as pets are many, with no significant concerns.
Conclusion: cats make excellent house pets. Cats are low maintenance, civilized companions. People who have small living quarters or less time for pet care should value these characteristics of cats. However, many people who have plenty of space and time still opt to have a cat because they love the cat personality. In a variety of ways, a cat is an excellent, healthy choice for a house pet.
This essay is also structured as an Affirm the Consequent argument. If a pet has many advantages and few disadvantages, then it makes a good pet. The benefits of cats as pets are many, with no significant concerns. Therefore, a cat is an excellent, healthy choice for a house pet.
Expansive Property
Expansive Property is an exception to the fallacy of Composition. If every part of a whole has an expansive property, then the whole will, too. Plastic and yellow are expansive properties. If all the parts of a model airplane are plastic and yellow, then the model will also be plastic and yellow. Light and cheap are not expansive properties. Individual parts might all be light and cheap, but the product as a whole might be heavy and expensive.
Explanation
Explanation is a non-inference account intended to clarify why something happened or is a certain way, to make sense of an event or phenomenon. The event or phenomenon is known or accepted as a matter of fact and the explanation provides understanding as to why it happened or why is as it is. By comparison: an argument tries to prove that something occurred or that it is a certain way; an explanation tries to clarify why something occurred or why it is a certain way. Some explanations are based on conjecture or hypothesis rather than on direct observation.
1 The sky appears blue from the earth’s surface because light rays from the sun are scattered by particles in the atmosphere.
2 Without regular maintenance, a crack in the car’s brake line went undetected. The fluid leaked out and this awful auto accident ensued.
Exportation
Exportation is a rule of equivalence that a series of antecedents is equivalent to their conjunction. A ⊃ (B ⊃ C) ≡ (A & B) ⊃ C.
1 If the firecracker explodes, if it makes a loud noise, it will wake the baby ≡ If the firecracker explodes and makes a loud noise, that will wake the baby.
2 If you put water in the tray and put the tray in the freezer, then there will be ice cubes ≡ if you put water in the tray, then put the tray in the freezer, then there will be ice cubes.
Exposition
Exposition is a non-inference that explains by expanding or elaborating upon a main point. An expository passage develops a topic sentence. Other sentences in the passage expand or elaborate upon the topic sentence. If the other sentences attempt to prove the topic sentence, however, then the passage is argumentative and not classified as expository.
1 The three familiar states of matter are solid, liquid, and gas. Solid objects keep their shape and volume. A liquid has a definite volume, but takes the shape of its container. Gas doesn’t hold a specific shape or volume. It expands to fill its container.
2 The pace of reading depends on the reader. One may stop and reread or seek clarification before continuing. The reader can accelerate the pace when the material is easy or uninteresting, and can slow down when it is difficult or enthralling. If the content is moving, one can put down the book for a moment to reflect without fear of losing anything.
Fair Sample
Fair Sample is an inductive argument which claims that which is true of a representative sample is true of the general population. A generalization about a population is valid if based on a random (or at least not biased) sample whose composition is similar to that of the population.
1 If one strand of spaghetti is cooked al dente, then the pot of pasta (all the spaghetti strands) are equally cooked and firm when bitten.
2 According to a recent poll, the conservative party is favoured in the next election. Pollsters contacted 1000 homeowners as well as 1000 who rent since homeowners are, on average, wealthier than non-homeowners, and the more wealthy tend to be more conservative than the less wealthy.
Fallacy
A fallacy is a type of argument that is defective, meaning it has an error in reasoning. The following is a list of fallacy patterns that appear in these exercises.
False Dichotomy
False Dichotomy is a fallacy that presents two options as if they were the only ones available, asserts that one option is undesirable, and concludes the best choice is the alternative preferred by the arguer. False Dichotomy has the form of a disjunctive syllogism (A or B, not A, therefore B), but is fallacious because these choices might not be the actual options or they might not be the only options. In other words, the fallacy is intended to limit choice; it leaves out relevant alternatives.
1 Use Brand X deodorant or risk perspiration odour. Nobody wants to stink, so use brand X.
2 There are two types of people in this world: the rich and the suckers. Do you want to get rich, or are you happy to remain a sucker?
Formal fallacy
A formal fallacy can be recognized by a defect in its logical form. That is, the structure of the argument does not fit one of the deductive patterns. By comparison, an informal fallacy has no recognizable deductive form and can only be detected by examining the content of the argument, for instance to find that the premises are not relevant to the conclusion.
Gambler’s Fallacy
Gambler’s Fallacy is a fallacy in which a streak of events is considered sufficient reason to believe a contrary result is due to happen. The belief is that a streak of random events affects the likelihood of future independent events. The longer the run of a random event, the stronger the belief that the opposite outcome is due to occur. Gambler’s Fallacy is fallacious in so far as the results of previous events have no statistical bearing on the outcome of the next event.
1 The value of company stock has gone up for several days. Sell now, because what goes up and up must come down.
2 I know your last three blind-dates turned out to be miserable, but that is all the more reason to suppose you’ve use up your share of bad luck so the one will be wonderful.
Hasty Generalization
Hasty Generalization (or converse accident) is a fallacy that concludes a generalization from a sample that is too small or not typical of the group. A sample needs to be large enough to represent the population’s diversity. Strands of spaghetti boiling in a pot are fairly homogenous, so testing whether one strand is done is sufficient. People, however, are more diverse.
1 Age 70 is certainly too old to drive. My mother became reckless on the road in her mid-60s.
2 An election poll wrongly predicted a victory for one political party because it surveyed by telephone, not realizing that fewer members of the opposing political party owned telephones.
Hypothetical Syllogism
Hypothetical Syllogism (or conditional syllogism) is a valid argument that if the first antecedent in a series of overlapping conditionals is true, then the final consequent is true. There could be two, three, or more conditionals as premises. If A is true, then B is true; if B is true, then C is true; so, if A is true, then C is true. A ⊃ B, B ⊃ C, ∴ A ⊃ C. A true first-antecedent is reason to believe the last-consequent.
1 If you sleep in, you’ll miss the bus, then you’ll have to walk. If you sleep in, then you’ll have to walk.
2 If the valve is closed, water won’t flow, so the tub can’t fill. Without a tub of water, I cannot wash the dishes and no dishes mean no plates mean no dinner. Therefore, if the valve is closed, no dinner.
3 Squares are rectangles and rectangles are quadrilaterals, so squares are quadrilaterals.
Illustration
Illustration is a non-inference example serving to clarify (but not prove) a point. An illustration states a point and includes an example as a case in point. This can be a general rule along with an instance or single occurrence of that general rule. In any event, the use of an instance is not to prove the point, but to make the point easier to understand. If examples in the passage can be interpreted as providing evidence to infer or support a conclusion of the point, then the passage is classified as argumentative and not merely illustrative.
1 Chemical elements and compounds can be represented by molecular formulas. For instance, water is H2O, and sodium chloride is NaCl.
2 Whenever a force is exerted on an object, the shape of the object can change. For example, when you squeeze a rubber ball or drop your sleepy head on a feather pillow, the ball and pillow are deformed to some extent.
Inductive
An inductive argument is a method for inferring a generalization from particular instances or premises, provided there is no decisive, overriding reason to the contrary.
1 Many people saw the movie, so it must be good [but did most like it?]
2 The flashlight won’t shine, so the batteries must be dead [but is it switched on?]
An induction is strong if the premises are relevant and sufficient to believe the claim.
3 Some of the egg salad is mouldy, so the rest of the salad is probably not good to eat.
4 Around here, people tend to stay indoors on rainy days and many watch movies, so rainy days promote local movie rentals.
In a strong inductive argument the premises support, but don’t guarantee, that the conclusion is true. The strength of an inductive argument depends not on its form or structure, but on the relevance of the premises to the conclusion.
5 It rained yesterday and it rained today, so it will likely rain tomorrow.
6 You were seen fleeing the scene of the crime and the stolen jewels were found in your apartment, so you are likely the thief.
A cogent argument is a strong induction with true premises.
The following are some of the inductive methods:
Insignificant Cause
Insignificant Cause is a fallacy that focuses on a genuine but minor cause instead of more important factors. In this fallacy, a cause is identified that is insignificant in comparison to other more dominant or decisive causes. The identified cause is not likely to produce or prevent the result.
1 Painting the restaurant’s dining room may attract more customers. [Yes, but advertising and lower prices are more effective.]
2 You shouldn’t leave on your porch lights at night. It contributes to global warming. [True, but the effects of automobile and industrial emissions are more significant.]
Instruction
Instruction is a non-inference expression that directs behaviour based on authority (command) or respect (request).
1 I left explicit instruction that I was not to be disturbed before 7 in the morning.
2 My instructions are for you to take two pain relievers and call me in the morning.
Intuition and Wish
Intuition (hunch) is belief based on a feeling of confidence that something is true. One might have a feeling of confidence without evidence for knowledge. By comparison, a wish (hope) is desire for something to be true without the feeling of confidence that it is true.
1 I wish I could kiss it and make it all better.
2 I am confident that you will get the job.
Joint Effect
Joint Effect is a type of Post Hoc fallacy that concludes A is the cause of B, but in fact both are effects of an underlying cause. It concludes a false causal connection from an observed correlation. Joint Effect is fallacious since the two events are not related as cause-and-effect, even though they correlate as both “symptoms” of an underlying cause.
1 You had a fever, then broke out in spots, so the fever caused the spots. [No, both fever and spots are symptoms of the measles virus.]
2 When I eat chocolate, I get a headache, so chocolate causes a headache. [No, there could be an underlying cause of both. For instance, I only eat chocolate when I feel stress and stress causes muscle tension, which results in a headache.]
Knowledge
Knowledge is justified prevailing belief. That is, to know an assertion is to believe it, have reason to believe it (there are factors relevant and sufficient to believe it), and have no over-riding reason to believe otherwise. What we know is subject to revision when a more compelling, comprehensive understanding arises.
Knowledge comes in degrees. I might be convinced, highly confident, fairly sure, or simply suspect that something is the case. That which is known a priori (e.g., by definition) is known for certain.
Logic
Logic is the application of patterns (rules) to assess whether one statement (the conclusion) follows from (is justified permitted by) other statements (premises).
Material Equivalence
Material Equivalence is a rule of equivalence that two items are equivalent when they imply one another. A ≡ B is the same as (A ⊃ B) & (B ⊃ A) and also equivalent to (A & B) | (~A & ~B).
1 A vixen is a female fox. If you saw a vixen, then you saw a female fox and if you saw a female fox then you saw a vixen.
2 Hesperus (the evening star) is Phosphorus (the morning star), since both are the planet Venus.
Material Implication
Material Implication is a rule of equivalence that a conditional is the same as saying the “antecedent is false or the consequent is true”. A ⊃ B ≡ ~A | B ≡ ~(A & ~B).
1 If you hit the bulls-eye, you win a Kewpie doll ≡ You didn’t hit the bulls-eye or you got a Kewpie doll = It is not the case that you hit the bulls-eye yet didn’t get the doll.
2 If Spain and New Zealand are antipodes, then noon in one is midnight in the other ≡ They are not antipodes or noon in one really is midnight in the other.
Maybe Both
Maybe Both (or Affirming a Disjunct) is the fallacy that one alternative is true is reason to believe the alternative is false when in fact both may be true. In other words, the list of alternatives does not exclude the possibility of both A and B being true.
1 I am at home or in the city. I am at home, so I am not in the city. [Could be both: my home is a city apartment.]
2 My puppy is house-trained or there is an accident to clean up when we get home. My puppy is house-trained, so there will not be pee on the carpet [Could be both: the pup is house broken, but has a bladder problem or drank too much water.]
Maybe Both is a corruption of the valid deductive form known as Disjunctive Syllogism.
Maybe Neither
Maybe Neither (or Denying a Conjunct) is the fallacy that one alternative is false is reason to believe the alternative is true when in fact both might be false. The fallacy reasons as follows: statements A and B cannot both be true; statement A is known to be false; therefore the statement B must be true. It is a fallacy where there is a third possibility: statements A and B are both false.
1 It can’t be both sunny and overcast. It is not sunny; therefore, it is overcast. [Could be neither: clear sky in the middle of the night.]
2 A suspect cannot be guilty and have an alibis. Since you don’t have an alibis, you must be guilty. [Could be neither: innocent, but without evidence that you were elsewhere.]
Disjunctive Syllogism and Maybe Neither are similar in appearance, but it is important to distinguish them since the one is valid and the other invalid.
Missing the Point
Missing the Point (also known as ignoring the issue) is a fallacy that draws a conclusion different from the one logically implied by the premises. The evidence implies one conclusion, but the arguer draws a conclusion not implied by the premises.
1 Members of the jury, clearly the defendant is guilty. He has been in and out of trouble. As a youth he was suspended for truancy, as a teen he was ticketed for speeding, and as an adult he has had his taxes audited. Now he stands here guilty of murder. [The issue is whether the accused is guilty of murder; the argument ignores this and proves that the accused had some concerns growing up.]
2 Nuclear power plants generate electricity, but electricity can be dangerous no matter where it comes from. Every year people are accidentally electrocuted. Most accidents are the result of carelessness and could easily be avoided by using common sense. So no, nuclear power isn’t dangerous. [Makes a connection that being careful can reduce the number of electrical accidents, but misses the issue of whether nuclear power is dangerous.]
Non-inference
Non-inference refers to a passage in which the statements do not and were not intended to have an inferential relationship. An argument or inference purports to prove something; a non-inferential passage does not. The following are examples of non-inferential passages:
Opinion
Opinion is a non-inference expression of what someone happens to think or believe. It express a point of view, personal judgment, or matter of taste. An opinion is of ones own making, but a belief can be accepted or inherited from others. A belief is an assumption made about ourselves, about others, about how we think things really are, or about how we expect things to be. An opinion or belief is held with confidence, but not substantiated with proof.
1 We believe that our company must produce products that fulfill a need for our customers, that our business must be run at an adequate profit, and that the services and products we offer must be better than those offered by competitors.
2 In my judgement, our country must support peoples of other nations seeking to overthrow oppression by offering economic and financial aid, rather than military intervention, so that they can work out their own destinies in their own way.
Oversimplification
Oversimplification is a fallacy that selects one contributing factor and represents it as being the only cause. It looks for a single cause when, in fact, several factors contribute to the result.
1 The sales of video games are down, so software piracy must be the cause [slumping sales could be due to economic recession, no new games, or competing technology].
2 Our airplanes are so well built that the aviation accident must have been the result of pilot error [the accident could have been caused by collision with a bird, bad weather, terrorism].
Particulars and Universals
Particulars refer to individuals considered as a whole, such as an apple or government. A universal is what particulars have in common, at least in name, such as redness or democracy.
Personal Attack
Personal Attack (or ad hominem) is fallacy that rejects a belief merely because it is held by someone disliked. In brief: I don’t like you, so I don’t agree with what you say. Personal Attack is fallacious because the character or conduct of the source is not related to whether the proposition is true or false. Arguing against the person is not arguing against his or her premises. The other’s person’s argument may be valid regardless of his or her faults, agenda, or consistency.
1 Electric automobiles? As every school child knows, gasoline drives the engine of industry. Even a fool can see that electric vehicles will put people out of jobs.
2 My opponent’s speech is like a Texas longhorn: a point here, a point there, but a whole lot of bull in between.
A personal attack is considered Abusive if it claims the other person has faults and thereby so does his or her argument. There is also a variation called “Poisoning the Well” to discredit anyone in advance, such as “everyone but an idiot knows that not enough money is spent on education”.
A personal attack is considered Circumstantial if it claims the other person is motivated by a bias or hidden agenda.
3 Smith argues that the fairest kind of income tax is flat-tax: everybody pays the same amount. But Smith is a software billionaire and stands to save millions of dollars if a flat-tax is enacted. Therefore, we can hardly take Smith’s argument seriously.
4 You can’t trust those so-called consumer protection magazines. They’re set against homeopathy because homeopathic remedies are often home-made and not a product they can buy and test.
Post Hoc
Post Hoc is a fallacy that concludes cause from sequence. Post Hoc is from the Latin phrase post hoc, ergo propter hoc which means “after this, therefore because of this.” Because one event follows another, it is claimed that the second has been caused by the first. Post Hoc is fallacious because sequence does not entail causal connection; the two events are coincidence.
1 Cola cures the cold. Enjoy a bottle of the soft drink; your cold will be gone in only a couple weeks.
2 Star athletes are paid ⬆︎ salaries, so the best way to ensure that the rookie will become a star athlete is to raise her salary to the top.
Post Hoc is the basis of good-luck charms and other superstition, sports rituals, and magical thinking.
Plain Text
Plain text consists only of characters, such as letters, spaces, line or page breaks. It does not contain information about the text, such as typeface, styles, or text alignment. Examples of plain text editors: BBEdit (Mac OS) or Notepad++ (Windows).
Proof by Example
Proof by Example is a form of the Hasty Generalization fallacy that concludes a generalization from one or few examples. It proceeds from instance to generalization. Stereotypes fit in this category.
1 Rover likes carrots, so dogs like carrots. [It would be valid only of it proceeded from an example to an instance: Rover likes carrots, so some dog or there is a dog who likes carrots.]
2 I saw a couple city construction workers just leaning on their shovels, so city workers must be lazy.
Reasoning
Reasoning is making inferences. An inference connects information to make a decision or to consider whether there are reasons to support a particular belief or claim. For instance, a detective inspects the clues to figure out who is the culprit. A doctor checks symptoms in order to make a diagnosis. A lawyer cites evidence to support a client’s claim of innocence. A meteorologist examines weather current conditions for a forecast.
We communicate our reasoning by means of an argument. An argument expresses an inference. In everyday arguments, parts are often taken for granted and not stated. For instance, it is unlikely that you’ll hear “the mower is running; running requires fuel; therefore, the mower has fuel.” More likely, the argument would be abbreviated to “there must still be fuel in the mower since it started.” An enthymeme is an argument in which some part is understood, but unstated. Enthymemes are useful shorthand when the implied assertion is obvious.
1 All insects have six legs, so all wasps have six legs [unstated premise: all wasps are insects]
2 Your editorial is racist and racism is wrong [unstated conclusion: your editorial is wrong]
The following all have an unstated conditional statement (the if-then part).
1 You’ve got your hands full, so let me hold the door
2 We have to arrive on time, so we can’t stop for lunch
Report
Report is a non-inference group of statements that convey information about some topic or event. A report is an account that describes what one has done, seen, observed, or investigated. Such information could be used in or as the premises of an argument, but because the author makes no claim that they support or imply anything, there is no argument.
1 A powerful bomb blew up outside the regional telephone company headquarters, injuring several people and causing extensive damage to nearby buildings, police said. A police statement said the 50 kilogram bomb was packed into a milk churn hidden in the back of a stolen car.
2 Cancer is not one disease, but many. Some forms are particularly susceptible to radiation therapy. Radiation is carefully aimed at the cancerous tissue, and exposure of normal cells is minimized. If the cancer cells are killed by the destructive effects of the radiation, the malignancy is halted.
Simplification
Simplification is a valid argument that if two statements are true together, then each statement is true on its own. A conjunction is a statement made joining other statements with “and”. The statements joined are called conjuncts. If a conjunction is true, then each conjunct on its own is true. A is true and B is true; so, A is true. A & B, ∴ A
1 If it is true that the union went on strike when negotiations failed, then it is true that negotiations failed and it also true that the union went on strike.
2 If it is true that the movie is short yet funny, then it is true that the movie is short and also that is funny.
Slippery Slope
Slippery Slope is a fallacy that claims an option must be avoided because it inevitably leads to an undesirable result yet the chain of events is unlikely and is presumed, not proven.
1 I am against lowering the drinking age from 21 to 18. Lowering it will only lead further demands to lower it to 16. Then it will be 14. Before we know it our newborns will be suckled on wine rather than mother’s milk.
2 If we ban smoking, people will turn to soft drugs, then move on to hard drugs and the crime rate will go up; so to prevent crime we should allow smoking.
If the chain of events were likely, then the argument would be a valid hypothetical syllogism. For example: slip on a banana peel, succumb to gravity, strike the ground. Slippery Slope is a fallacy because the chain of events is presumed, not proved, and unlikely.
Statement
A statement isn’t the same as a sentence. Some sentences express a command or request, such as ‘leave the cat alone’ or ‘please pass the pepper’. Some sentences ask a question, such as ‘what does it cost?’ or ‘where is the washroom?’. Other sentences declare an idea or opinion, such as ‘I believe in magic’ or ‘Pluto is a planet’. A statement (or its synonym, a proposition) is that which a declarative sentence asserts and it is either true or false.
1 ’All humans have genes’ is a true statement; ‘pigs can fly’ is false statement.
2 ’Je t’aime’, ‘I love you’, and ‘I am in love with you’ express the same statement.
The truth value of a statement depends on the truth value of its simple statements and how they are connected. A simple statement asserts a fact, such as ‘fire requires oxygen’. Statements can be connected into more complex statements with: not, and, or, if-then.
Negation (not): ~A is read ‘not A’ or ‘it is not true that A’ or ‘A is not true’. When A is true ~A is false and when A is false ~A is true.
Conjunction (and): A & B is read ‘A and B’ or ‘it is true that A and B’. A and B are called conjuncts. A & B is true only when A is true and B is true; otherwise A & B is false.
Disjunction (or): A | B is read ‘A or B’ or ‘A is true or B is true’. A and B are called disjuncts. A | B is true as long as one or both are true.
Implication (if-then): A ⊃ B is read ‘A implies B’ or ‘if A is true, then B is true’. A is called the antecedent, B is called the consequent, and the expression is a conditional statement. A ⊃ B is true under all truth-value assignment except when A is true and B is false.
Equivalence (if and only if): A ≡ B is read ‘A is equivalent to B’ or ‘A has the same value as B’. A ≡ B is true when A and B have the same value (that is, both true or both false)
Logical form makes the structure of a statement more clear.
Logical Form
The logical form of a statement shows its structure using connectives (not, and, or, if-then, is equivalent to) with capital letters representing the parts to be connected. Parentheses are used for grouping indicators and ∴ is shorthand for ‘therefore’.
The statement ‘the light is on or the light is off’ has the form: A or B.
The statement ‘the baby is a boy or the baby is a girl’ also has the form: A or B.
Connectives have an order of priority and this order can be used to reduce parentheses. From highest to lowest:
negation (not), ~A
conjunction (and), A & B
disjunction (or), A | B
implication (if-then), A ⊃ B
equivalence (if and only if), A ≡ B.
So A | B & ~C means the same as (A | (B & (~C)))
Statistical Syllogism
Statistical Syllogism is an inductive argument which claims that which is true in general is likely true in a particular instance. Statistical Syllogism applies a statistical generalization about a group to an arbitrary member of that group. A statistical generalization is a statement which is usually true. The closer the generalization is to 100%, the stronger the induction.
1 Bob is a mechanic. Most mechanics have dirty fingernails, so Bob probably does too.
2 The first card dealt from a well-shuffled deck is probably not going to be an Ace.
Statistical Syllogism can be quantified with percentages or relative descriptions, such as: most, usually, commonly, frequently, generally, seldom, rarely, scarcely. At times the quantifier is unstated, but implied.
3 Lions are (usually) faster than zebras.
4 Barley is (commonly) used in making beer.
Straw Man
Straw Man is a fallacy that presumes to refute an opposing belief by refuting a misconstrued and easier-to-attack version of it. The process is to create a distorted version of what your opponent is claiming, refute the distorted version, then concludes that the original position is refuted. Straw Man is a fallacy because rejecting a misrepresented or oversimplified version of a position does not reject the original position. One version, called “Runaway-Train” consists of refuting an argument by taking it to extremes.
1 Smith, a wealthy business person, has argued that government should get off the back of the business. Obviously, Smith wants to abolish government altogether. Yet without government there would be no defence, no judicial system, no pensions, no health and safety regulations. None of us wants to forgo these benefits. Thus we can see that Smith’s argument is absurd.
2 The opposition wants to lower the highway speed limit by 10% in order to save lives. But why stop there; why not lower it 50% or more? Obviously saving lives is not the real agenda of the opposition.
Suggestion
Suggestion is a non-inference idea or plan put forward for consideration. It introduces a thought.
1 I suggest we open these windows to let some fresh air inside.
2 Might I recommend the fresh lobster entrée, madam. It is exquisite.
Suppressed Evidence
Suppressed Evidence is a type of fallacy that leaves out information that would lead to a different conclusion. It uses only the facts that support the conclusion, disregarding any other pertinent facts. This can occur by inadvertent omission. It can also occur by deliberate deception, as is the case where those making judgment consider only one side of the evidence.
1 Let’s get a bulldog puppy for the kids. They are ugly-cute and nobody around here already has one. [This is a one-sided assessment since bulldogs are not especially good around children, difficult to train, and have habits of drooling and breaking wind.]
2 This ad says that we can buy a Brand X cell phone for only $99. That sounds like a great bargain. For less than $100 we can make all the phone calls we want!
In the one-sided assessment of Suppressed Evidence, evidence to the contrary is suppressed. In a balanced assessment, by comparison, the preponderance of evidence is reason for a belief or action. This is a process of inferring a conclusion by weighing reasons for (pro) and against (con) to reach a decision based on the greater body of evidence.
Syllogism
A syllogism is a form of reasoning that has a minor premise (A is B), major premise (B is C), and a conclusion that consists of the subject from the minor premise and the predicate from the major premise (therefore, A is C). It leaves out the middle term connecting the two premises.
1 Spot is a dog; a dog is a mammal; thus, Spot is a mammal [the conclusion leaves out middle term ‘dog’].
2 Dragonflies are insects that eat mosquitoes. No insects that eat mosquitoes should be harmed. So, no dragonflies should be harmed [the conclusion leaves out the middle term ‘eats mosquitoes’]
Tautology
Tautology is a rule of equivalence that a statement is equivalent to multiple statements of itself joined by “and” or by “or”. Thus, A ≡ A & A. Likewise A ≡ A | A. Often a redundant expression is a tautology in that it repeats without adding information.
1 You will receive a “free gift” book.
2 “Each and every” victim was “dead or deceased”.
3 If it’s in stock, we have it.
Taxonomy
A taxonomy is a classification system, such as a library uses to organize books or a biologist uses to classify organisms. Bloom’s taxonomy is a classification of thinking skills. The taxonomy of the mental skills was developed in the 1950s and revised in 2001.
Transposition
Transposition (or contrapositive) is a rule of equivalence that if the consequent of a conditional is false, then the antecedent must also be false. A ⊃ B ≡ ~B ⊃ ~A.
1 Rain requires clouds, so the absence of clouds indicates no rain
2 The Big Bang was silent since sound must have a medium to travel through; no medium, no sound
True and False
True and False are values indicating whether a statement accurately represents that to which it refers. “A unicorn has a horn” is true (by definition), even if not factual. For someone with phantom limb sensations, “my toes are itchy” can be false yet “it feels like my toes are itchy” is true.
Tu Quoque
Tu Quoque (or You Too) is a type of fallacy that rejects a claim because the person making it is not acting in a manner consistent with that claim. In effect, I reject your claim since you are being a hypocrite. Tu Quoque is fallacious because the claim may be valid regardless of the claimant’s behaviour.
1 Dad, I don’t see how you can ask me to not smoke. You and mom both said that you smoked when you were young. Maybe I will quite later on, like you did.
2 The ambassador’s complaint about poverty in our nation is ridiculous since there are people below the poverty level in his country as well.
Two Wrongs Make a Right
Two Wrongs Make a Right is a fallacy that concludes one wrong action can cancel out another. The conduct of others is taken as sufficient reason for belief or action, justifying an action as what others have or would have done. More specifically, the fallacy runs, if one mistake is made, another can cancel it out.
1 She pinched me, so I pulled her hair.
2 The library keeps calling me to donate to their fund-raising campaign. I’ve asked them repeatedly to stop. If they don’t stop, I will rip pages out of books before I return them. I’ll make sure that they’re the pages with important plot points.
This fallacy does not include preventative action, however, such as punching a mugger or a naval blockade to prevent nuclear hostilities.
Understanding
Understanding is knowing the meaning (use, significance, implications) of content.
1 You need to understand [be aware of the consequences] that once you leave, you can’t return.
2 I know you’re talking, but I can’t understand [make sense of] what you are saying.
3 I am supposed to give tell you ‘the canary sleeps,’ but I don’t understand [know the meaning of] what that means.
Undistributed Middle
Undistributed Middle is a fallacy that concludes the subjects in a syllogism are the same because the subjects have the same predicate. This is a formal fallacy: A is B, C is B, therefore A is C. The fallacy occurs when there is no middle term connecting the two premises. . Undistributed Middle is fallacious because the subjects can be separate groups even though they have a common property.
1 The mayor is a person. I am a person. Therefore I am the mayor.
2 All living things can reproduce. A virus can reproduce, so a virus is alive.
Weak Analogy
Weak Analogy is a fallacy in which the conclusion depends on a similarity that is not relevant to the claim. An analogy is a comparison: A is like B, so if A has a certain property, B likely has a similar property. In a weak analogy the items compared are significantly different or don’t have many relevant similarities.
1 A laxative capsule looks like a jelly bean, so it will be just as tasty.
2 When water is poured on the top of a pile of rocks, it trickles down to the rocks on the bottom. Similarly, when rich people make lots of money, this money will trickle down to the poor.
Wish
Wish is a type of non-inference expressing the feeling of a hope or desire for something to happen or to be true.
1 This lottery ticket has to be the winner; we really need the money.
2 The donut franchise is bound to succeed. You’ll get a huge return on your investment.