3 Deduction

inference by pattern or form

Deduction is inferring a conclusion from premises by their pattern or form. If the premises are true, the conclusion must be true. For example: if objects have mass, then they attract one another; you and the Earth have mass; therefore, you and the Earth attract one another.

Goals

• Assess an argument as valid or sound
• Identify the forms of common effective arguments
• Distinguish the forms of syllogisms and dilemmas
• Apply transformation rules to construct proofs

Deductive Argument

Logical form makes the structure of a statement clearer.

• 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.

Form is shown with connectives between statements (represented by capital letters).

• Connectives: not, and, or, if-then, is equivalent to.
• Parentheses are grouping indicators.

To reduce parentheses, connectives have an order of priority.

• From highest to lowest: negation (not), conjunction (and), disjunction (or), implication (if-then), equivalence (if and only if). 
• So if P or Q and not R, then S means if (P or (Q and (not R))), then S.

An argument is valid if it has one of the deductive forms.

• ‘The light is not on’ and ‘the baby is not a boy’ both have the form ‘not A.’ 
• From ‘A or B’ and ‘A,’ one can conclude ‘B.’ So the logical form of the argument is: A or B; not A; ∴ B. This form is one of the deductive forms of an argument. 

If an argument can be expressed in one of the deductive forms, then the argument is valid.

• Assuming the premises are true, the conclusion must be true.
• 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.

An argument is sound if it is valid and has true premises.

• This argument is valid (it has a deductive form we will explore in the next topic), but isn’t sound. If every animal can fly, then pigs can fly; a pig is an animal; therefore, a pig can fly. 
• The following is sound. If every animal has DNA, then a pig has DNA; a pig is an animal; therefore, a pig has DNA.

What do you think?

1. If an argument is valid, what does it matter if it is sound?
2. Deduction is based on the form of an argument. Is induction based on the content of an argument?

Modus Ponens, Modus Tollens

In affirm the antecedent, that the antecedent is true is reason to believe its consequent is true.

• As rain beings to fall, the umpire looks to the sky and shouts “well, there goes the game”. This is an enthymeme in which the umpire states only the conclusion: if it is raining, then the game is cancelled; it is raining, so ‘there goes the game’.
• If an artifact evokes an emotion, that’s art. I kicked over my nasty neighbour’s garbage can and he reacted with plenty of emotion. I explained it is just art. That’s valid, but not likely sound.

Affirming the antecedent means: if the antecedent is true, the consequent is true.

• If A is true, then B is true; A is true; therefore, B is true.
• Affirm the Antecedent (a.k.a. “modus ponens”) uses an if-then conditional statement. The antecedent is the ‘if’ part. The consequent is the ‘then’ part. 

In deny the consequent, a false consequent is reason to not believe its antecedent.

• If there is fire, there must be oxygen; the sun has no oxygen; therefore the sun is not on fire.
• If you love me, you would not leave me; you did leave me; therefore, you do not love me.

Deny the consequent means: if the consequent isn’t true, the antecedent can’t be true.

• If A is true, then B is true; but B is not true; so, A is not true.
• Deny the Consequent (a.k.a. “modus tollens”) is based on a conditional. If the consequent is false, then the antecedent is false.

What do you think?

1. Is affirm the antecedent basically the reverse of deny the consequent? What does that mean, being the reverse?
2. A sci-fi movie has the tagline: “nobody can hear you scream in space.” That’s an enthymeme for a deny the consequent argument. If there is sound, then there must be a medium; there is no medium in outer space; therefore, in space, nobody can hear you scream. Expand some other message from a movie or commercial into an argument.

Hypothetical & Disjunctive

In hypothetical syllogism, a true first-antecedent is reason to believe the last-consequent.

• If you sleep in, you’ll miss the bus, then you’ll have to walk. So — sleep in, you walk.
• 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. So, if the valve is closed, no dinner.

In a hypothetical syllogism 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.
• In Hypothetical Syllogism (or “conditional syllogism”), if the first antecedent in a series of overlapping conditionals is true, then the final consequent is true. 

In disjunctive syllogism, that one disjunct is false is reason to believe its alternative is true.

• We know that she paid Pete or Paul. Our sources show that she didn’t pay Pete, so she must have paid Paul.
• 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.

In Disjunctive Syllogism (or “excluded middle”), 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; therefore, B is true.
• A disjunction is an exclusive statement of alternatives: this or that — not both, not something else. 

What do you think?

1. Which kind(s) of syllogism are these arguments? Squares are rectangles; rectangles are quadrilaterals; so, squares are quadrilaterals. My cat is not male; cats are male or female; thus, my cat is female.
2. Hypothetical syllogism is like a row of falling dominoes. What visual metaphor describes disjunctive syllogism?

Constructive & Destructive Dilemmas

In a constructive dilemma, if at least one antecedent is true, then at least one consequent is true.

• 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.
• 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.

Constructive dilemma starts with 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.

In destructive dilemma, if at least one consequent is false, then at least one antecedent is false.

• If we are going paint the deck, then we need to buy brushes; but if we are to stay within budget, then 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.
• If the model car isn’t 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.

Destructive dilemma also starts with two or more true conditionals. 

• If one of the consequents is false, then one of the antecedents must also be false.
• If A is true, then B is true and if X is true, then Y is true; B or Y is false; so, A or X is false.

What do you think?

1. Is the following dilemma constructive or destructive? We made a deal: if I win the lottery, I will donate it to the animal hospital; if you win, you’ll donate it to the orphanage. One of us has the winning ticket. Therefore, either the animal hospital or the orphanage will get a donation.
2. Symptoms suggest a problem with the heart or liver. The first will show up on x-ray and the second on a blood text, but neither showed up. So, it isn’t a heart or liver problem. This seems to be a destructive dilemma, but is it?

Simplification & Addition

A conjunction is a statement made joining other statements (called conjuncts) with ‘and.’ 

• The sky is blue; the grass is green; so it is true that the sky is blue and the grass is green.
• The rain stopped; the sun is shining; so it is true that the rain stopped and the sun is shining.

By conjunctive addition, if this is true and that is true, then “this and that” is true.

• The ‘and’ may be implied or in another form, such as ‘yet’ or ‘but.’
• A conjunction has at least two, but can have many conjuncts.

What this means is, any two true statements can be joined to form a true conjunction. 

• A is true; B is true; so, A and B is true.
• If two or more statements are true on their own, then they are true together. 

By simplification, if two statements are true together, then each statement is true on its own.

• In other words: if a conjunction is true, then each conjunct on its own is true.
• A and B is true; so, A alone is true. (Also, B alone is true.)

Just as conjunctive addition puts together; conjunctive simplification takes apart.

• 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.
• If the movie is short yet funny, then it is true that the movie is short and also that is funny

By disjunctive addition: if a statement is true, then any statement in which it is a disjunct is also true.

• A is true; so, A or B is true — even if B is false or unrelated to A.
• A 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.

• Any mass has inertia, so it is true that “any mass has inertia or the Earth is flat.”
• Since 2+2=4, it is true that “2+2=4 or I am exactly 6 cm tall.”

What do you think?

1. Does the order of conjuncts matter? Is, A and B equivalent to B and A and A or B equivalent to B or A?
2. Why is it that a true statement plus any nonsense statement produces a true statement? (Careful with the word ‘plus’ in the previous sentence.)

Conditional & Indirect Proof

In a conditional proof, if A is assumed to be true and B results, then “if A, then B” is true.

• 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.
• 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.

A conditional proof doesn’t say the antecedent is actually true. Only that if it is, then the consequent will also be true.

• A conditional proof proves a condition. It proves there is an if-then.
• If I assume A is true and, applying rules of logic, B results; therefore, if A, then Bis true.

In indirect proof, statement A is true if not A results in a contradiction.

• Statement A is true if not A reduces to an absurdity, meaning it leads to a contradiction. 
• If not A leads to a contradiction, it has to be false, so indirectly statement A has to be true. 

Indirect proof presumes A and not A are the only options. If one is false by leading to a contradiction, then the other is true.

• Beachcombers are asked to leave pebbles on the beach, so people don’t spoil the ecosystem. [Leave pebbles because not leaving them leads to an unnatural natural environment.]
• Earth must be a sphere; otherwise, you’d fall off the edge. [Sphere or flat; if flat, then fall off the edge; ships don’t fall off, therefore, not flat; therefore sphere.]

What do you think?

1. If I have a good day whenever I carry my lucky charm, why doesn’t that conclude the conditional “if carry charm, then good day”?
2. In a country cottage on a well, if you run the dishwasher, there won’t be enough water for my shower and I always wait until my shower before doing the laundry. So, if you run the dishwasher, the laundry will have to wait. Create another country cottage scene using a conditional proof (e.g.: mice in the attic or low tide along the beach).

Transformation

Some logical forms can be changed to other forms that are equivalent.

• Transformation does not make a new argument. It puts one statement in another form. 
• In the following rules of replacement, one expression can be replaced with the other. That is, they tell how to rephrase one statement into an equivalent.

Association: to switch the grouping of statements joined by “and” or by “or” does not change whether the overall expression is true or false.

• (A and B) and C is equivalent to A and (B and C). 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 or B) or C is equivalent to A or (B or C). “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.”

Commutation: switching the order of statements joined by “and” or by “or” does not change whether the overall expression is true or false.

• A and B is equivalent to B and A. A dollar and a dime have the same value as a dime and a dollar.
• A or B is equivalent to B or A. Whether you put on the left glove first or right glove first, the result will be the same.

Transposition: if the consequent of a conditional is false, then the antecedent must also be false.

• If A, then B is equivalent to if not B, then not A. Rain requires clouds, so the absence of clouds indicates no rain.
• The Big Bang was silent since sound must have a medium to travel through; no medium, no sound.

DeMorgan’s Law: a negative can be distributed to a conjunction or disjunction. 

• Not (A and B) is equivalent to not A or not B. “He isn’t tall, tanned, and handsome” means he is not tall or not tanned or not handsome.
• Not (A or B) is equivalent to not A and not B. “The subway does not run north or south” means the subway does not run north and it doesn’t run south.

Distribution: disjunction is distributive over conjunction, and conjunction is distributive over disjunction.

• A and (B or C) is equivalent to (A and B) or (A and C). Ice cream on cake or pie is the same as ice cream on cake or ice cream on pie
• A or (B and C) is equivalent to (A or B) and (A or C). “Jogging or sitting and reading” is the same as “jogging or sitting and jogging or reading”

Double negation: a double negative is equivalent to a positive. 

That is, not not A is equivalent to A.

• I wouldn’t ask you if this wasn’t so important = I ask since this is important.
• There isn’t a day when I don’t think about it = I think about it every day.

Exportation: a series of antecedents is equivalent to their conjunction. 

If A, then (if B, then C) is equivalent to if (A and B), then C.

• 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. [This example uses ‘then’ as producing a result.]
• If you put water in the tray and put the tray in the freezer, then there are ice cubes = if you put water in the tray, then put the tray in the freezer, then there are ice cubes. [This example uses ‘then’ as later in time.]

Material Equivalence: two items are equivalent when the imply one another. 

A is equivalent to B is the same as (if A, then B) and (if B, then A) and also equivalent to (A and B) or (not A and not B).

• 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.
• Hesperus (the evening star) is Phosphorus (the morning star), since both are the planet Venus.

Material Implication: a conditional is the same as saying the “antecedent is false or the consequent is true”. 

If A, then B is equivalent to not A or B and also equivalent to not (A and not B).

• 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.
• 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.

Tautology: a statement is equivalent to multiple statements of itself joined by “and” or by “or”. 

• A is equivalent to A and A. You will receive a “free gift” book.
• Likewise A is equivalent to A or A. “Each and every” victim was “dead or deceased.”

What do you think?

1. Suppose you were given the task of stating the rules of replacement more simply. For instance, “Association means AND is true in any order.” Completing that task is an example of using which transformation rule?
2. Are the following examples of double negation? It is not true that the performance was boring. I don’t think the building is in total darkness.