Rules of Inference

Arguments

An argument is a sequence of statements that starts with one or more premises (statements that are assumed to be true) and ends with a conclusion. We say that the argument is valid if the conclusion follows logically from the premises, i.e. if the conclusion must be true given that the premises are true.

In declaring an argument valid, we do NOT question the truth of the premises. Therefore, the following argument is valid, even though its premise is false.

Suppose I have ten million dollars. Then I can retire.

Argument Forms

To help us identify arguments as valid or invalid, we will identify valid argument forms. An argument form is an abstraction of an argument, where the particulars of the situation have been stripped away and only the logical structure remains. Basic valid argument forms are known as rules of inference.

Let us begin with the following example argument: When the sun shines, I’m happy. The sun is shining. Therefore, I’m happy.

Let’s introduce the following propositional variables:

p = “the sun shines/is shining.” q = “I’m happy.”

The argument above can than be rewritten as

It is customary to separate the conclusion with a horizontal bar. The triple dot ∴ symbol is read as “therefore”. This particular argument form is called the modus ponens.

𝑝 → 𝑞𝑝__________∴ 𝑞

Proving a Rule of Inference

To formally demonstrate that a an argument form is valid, we must show that the conditional

if (conjunction of all premises) then conclusion

is true if all the premises are true. Since that conditional is true by default if the premises are not all true, it must always be true (be a tautology) if the argument form is to be valid.

Let us use this formal definition of validity to show that the modus ponens is valid. To do that, we must establish that

𝑝 → 𝑞 ∧ 𝑝 → 𝑞Is a tautology. We could use a truth table to do that, but it is more elegant to use logical equivalences:

𝑝 → 𝑞 ∧ 𝑝 → 𝑞

≡ ¬ ¬𝑝 ∨ 𝑞 ∧ 𝑝 ∨ 𝑞

≡ ¬ ¬𝑝 ∨ 𝑞 ∨ ¬𝑝 ∨ 𝑞

≡ 𝑝 ∧ ¬𝑞 ∨ ¬𝑝 ∨ 𝑞

≡ 𝑝 ∨ ¬𝑝 ∧ ¬𝑞 ∨ ¬𝑝 ∨ 𝑞≡ ¬𝑞 ∨ ¬𝑝 ∨ 𝑞 ≡ ¬𝑞 ∨ 𝑞 ∨ ¬𝑝

≡ 𝑇 ∨ ¬𝑝 ≡ 𝑇

Modus Tollens

We have learned that a conditional is equivalent to its contrapositive. Therefore, if we know 𝑝 → 𝑞, then we also know ¬𝑞 → ¬𝑝. If we then also know ¬𝑞, we can conclude ¬𝑝. This argument form is known as the modus tollens:

To prove the validity of the modus tollens, we have to show that the following statement is a tautology:

𝑝 → 𝑞 ∧ ¬𝑞 → ¬𝑝

This is left as an exercise for the student.

𝑝 → 𝑞¬𝑞__________∴ ¬𝑝

Other Valid Rules of InferenceArgument Form Name Corresponding Tautology

𝑝 → 𝑞𝑞 → 𝑟∴ 𝑝 → 𝑟

HypotheticalSyllogism

𝑝 → 𝑞 ∧ 𝑞 → 𝑟 → (𝑝 → 𝑟)

𝑝 ∨ 𝑞¬𝑝∴ 𝑞

Disjunctive Syllogism

𝑝 ∨ 𝑞 ∧ ¬𝑝 → 𝑞

𝑝∴ 𝑝 ∨ 𝑞

Addition 𝑝 → 𝑝 ∨ 𝑞

𝑝 ∧ 𝑞∴ 𝑝

Simplification 𝑝 ∧ 𝑞 → 𝑝

𝑝𝑞

∴ 𝑝 ∧ 𝑞

Conjunction 𝑝 ∧ 𝑞 → 𝑝 ∧ 𝑞

𝑝 ∨ 𝑞¬𝑝 ∨ 𝑟

∴ 𝑞 ∨ 𝑟

Resolution 𝑝 ∨ 𝑞 ∧ ¬𝑝 ∨ 𝑟 → (𝑞 ∨ 𝑟)

Some Real-Life Examples of the use of Rules of Inference

Hypothetical Syllogism: “If we had faster than light travel, we could travel to other star systems. If we could travel to other star systems, we would meet aliens. Therefore, if we had faster than light travel, we would meet aliens.”

Disjunctive Syllogism: “the accused is either innocent, or he is lying. He is not lying. Therefore, he is innocent.”

Addition: at a certain state park, you can get free admission if you are a student or a senior. You are a student. Therefore, you satisfy the condition of being a student or a senior.

Simplification: you apply for a scholarship. You mention on your resume that you have very good grades, and an outstanding collection of baseball cards. The selection committee makes a note that you have very good grades.

Making Sense of the Resolution Rule

Resolution, as stated, may seem somewhat mysterious:𝑝 ∨ 𝑞

¬𝑝 ∨ 𝑟∴ 𝑞 ∨ 𝑟

The rule makes much more intuitive sense if we rewrite the two premises as conditionals and switch their order:

𝑝 → 𝑟¬𝑝 → 𝑞∴ 𝑞 ∨ 𝑟

It’s certain that either 𝑝 or¬𝑝 is the case. Since the first case guarantees 𝑟, and the second case guarantees 𝑞, then 𝑞 or 𝑟 must be the case.

Example: During the day time, Chris works. At night, he sleeps. Therefore, Chris always works or sleeps.

FallaciesA fallacy is an invalid argument form. Two common fallacies are:

The fallacy of affirming the conclusion:

This fallacy confuses the conditional premise with its converse. We have learned that a conditional and its converse are not logically equivalent.

Example: if it rained, then the street is wet. The street is wet. Therefore, it rained. [The streets may be wet for a different reason. Perhaps someone sprayed water on the streets.]

The fallacy of denying the hypothesis:

This fallacy confuses the conditional premise with its inverse, which we know is not logically equivalent.

Example: if you stop breathing, you will die. You don’t stop breathing. Therefore, you won’t die. [There are other causes of death.]

𝑝 → 𝑞𝑞∴ 𝑝

𝑝 → 𝑞¬𝑝∴ ¬𝑞

Analyzing an Argument (1)

Show that the following is a valid argument:

If it rains, I don’t go to work. If I don’t go to work, I get fired. I’m not getting fired today. Therefore, it is not raining.

To prove the validity of the argument, we will introduce assign appropriate propositional variables, rewrite the argument in terms of those variables and identify valid rules for inference that lead from the premises to the conclusion:

1. 𝑝 → ¬𝑞 (premise)2.¬𝑞 → 𝑟 (premise)3.¬𝑟 (premise)4. 𝑝 → 𝑟 (hypothetical syllogism from 1, 2)5. ∴ ¬𝑝 (modus tollens from 3, 4)

Needed Propositions

𝑝 = "it rains"

𝑞 = "I go to work"

𝑟 = "I get fired"

Instantiation and Generalization

This slide discusses a set of four basic rules of inference involving the quantifiers.

1. If a statement is true about all objects, then it is true about any specific, given object. This is called universal instantiation.

∀𝑥𝑃 𝑥∴ 𝑃 𝑐 for any arbitrary c

2. If a statement is true about every single object, then it is true about all objects. This is called universal generalization.

𝑃 𝑐 for any arbitrary c∴ ∀𝑥𝑃 𝑥

3. If an object exists that makes a statement true, then it is possible to produce such an object. This is called existential instantiation.

∃𝑥𝑃 𝑥∴ 𝑃 𝑐 for some particular c

4. If we have an object with a property, then we know that objects with this property exist. This is called existential generalization.

𝑃 𝑐 for some particular c∴ ∃𝑥𝑃 𝑥

Analyzing an Argument (2)Show that the following is a valid argument:

Every kid loves ice cream. Joey doesn’t love ice cream. Therefore, Joey is not a kid.

To answer the question, we will introduce appropriate predicates, rewrite the argument using those predicates and identify the valid argument forms that lead from the premises to the conclusion.

1. ∀𝑥 𝐾 𝑥 → 𝐿 𝑥 (premise)2.¬𝐿 Joey (premise)3. 𝐾 Joey → 𝐿 Joey (universal instantiation of 1.)4. ∴ ¬𝐾 Joey (modus Tollens from 2 and 3) (conclusion)

Needed Predicates

K 𝑥 = "𝑥 is a kid"

L 𝑥 = "𝑥 loves ice cream"

Two Common Mistakes

A very common mistake students make in formally analyzing arguments is to forget the quantifiers. For example, they would give 𝐾 𝑥 → 𝐿 𝑥 as the first premise of the previous argument. But this “premise” is not even a statement because 𝑥 is a free variable. Without quantification, the “statement” is meaningless.

A second common mistake is to confuse conditional statements with conjunctions. Let us consider the example on the previous slide.

Some would translate the statement every kid loves ice cream as ∀𝑥 𝐾 𝑥 ∧ 𝐿 𝑥 . But this means that everyone (every person) is a kid and loves ice cream, quite a different statement. The correct symbolic representation of every kid loves ice cream is

∀𝑥 𝐾 𝑥 → 𝐿 𝑥

This is actually old news- in a previous presentation, we already learned that domain restricted universal quantification of a statement is equivalent to the unrestricted universal quantification of a conditional, where the membership in the restricted domain is the premise of the conditional, and the original statement is the conclusion. Therefore, every kid loves ice cream is equivalent to it is true for every person that if the person is a kid, they love ice cream.

Analyzing an Argument (3)Show that the following is a valid argument:

When dogs play, they get dirty. Dogs that don’t play sleep. Sleeping dogs do not chase cats. Billy is not a dirty dog. Therefore, Billy does not chase cats.

To answer the question, we will introduce appropriate predicates, rewrite the argument using those predicates and identify the valid argument forms that lead from the premises to the conclusion.

1. ∀𝑥(𝑃 𝑥 → 𝐷 𝑥 ) (premise)2. ∀𝑥(¬𝑃 𝑥 → 𝑆 𝑥 ) (premise)3. ∀𝑥(𝑆 𝑥 → ¬𝐶 𝑥 ) (premise)4. ¬𝐷 Billy (premise)5. 𝑃 Billy → 𝐷 Billy (universal instantiation of 1.)6. ¬ 𝑃 Billy (modus tollens using 4. and 5.)7. ¬𝑃 Billy → 𝑆 Billy (universal instantiation of 2.)8. 𝑆 Billy (modus ponens using 6. and 7.)9. 𝑆 Billy → ¬𝐶 Billy (universal instantiation of 3.)10. ∴ ¬𝐶 Billy (modus ponens using 8. and 9.)

There is more than one correct solution. For example, we could have combined 7 and 9 using hypothetical syllogism to get . ¬𝑃 Billy → ¬𝐶 Billy and then applied the modus ponens to that and to 6 to get the conclusion.

Needed Predicates

𝑃 𝑥 = "𝑥 plays"

S 𝑥 = "𝑥 sleeps"

𝐶 𝑥 = "𝑥 chases cats"

D 𝑥 = "𝑥 is dirty"

More Rules of Inference for Quantifiers

In the previous example, we repeatedly instantiated universally quantified conditionals and then used a modus ponens or tollens. It is convenient to adopt these argument patterns as new, named rules of inference:

Universal Modus Ponens:∀𝑥 𝑃 𝑥 → 𝑄 𝑥𝑃(𝑐)∴ 𝑄 𝑐

Universal Modus Tollens:

∀𝑥 𝑃 𝑥 → 𝑄 𝑥

¬𝑄(𝑐)

∴ ¬𝑃 𝑐

A Common Type of Real-Life Fallacy: False Dichotomies involving Universal Generalization (1)

Some people simplify the world through generalization, i.e. universal statements of the form ∀x∈X (Q(X)) where X is a set of objects or people and Q(x) is some statement about x.

They may react to an example that breaks this rule by jumping to the conclusion that the critic is proposing ∀x∈X (¬Q(X)). Examples:

Earthling: All Klingons are warlike and violent.Vulcan: I know Klingons who are not.Earthling: Oh I see. So all Klingons love peace, don’t they? Don’t be so naive!

Jack: I have never done anything wrong. All my mistakes are someone else’s fault.Mike: That thing you did yesterday was definitely your fault.Jack: Oh, so everything is my fault. I get it.

The thinking underlying this fallacy is the unquestioned assumption that all elements of X are essentially the same. They are all either one thing, or the opposite. The possibility that some of them satisfy Q(x) and others don’t is not considered. Correctly, ¬∀x∈X (Q(X)) only means ∃x∈X (¬ Q(X)), nothing more. It doesn’t mean ∀x∈X (¬Q(X)).

A Common Type of Real-Life Fallacy: False Dichotomies involving Universal Generalization (2)

We know that restricted universal statement can be expressed as an unrestricted universal conditional: all Swedes are tall is equivalent to if someone is Swedish, they are tall.

Therefore, another form of the fallacy ¬∀x∈X (Q(X)) ≡ ∀x∈X (¬Q(X)) is

¬∀x (P(x) → Q(X)) ≡ ∀x (P(x) → ¬ Q(X)) where P(x)=(x∈X).

The following exchange illustrates this form of the fallacy and may feel familiar:

Pundit 1: Joe Jackson still has a chance to win the election.Pundit 2: Joe Jackson is running way behind in the polls. He'll lose.Pundit 1: Mike Johnson was running way behind in the polls in 1996 and he ended up winning.Pundit 2: Yes but not everyone who is running way behind in the polls ends up winning.

Pundit 2 may seem to be the 'winner' of this debate, but only to illogical minds. Pundit 1 never claimed that Joe Jackson will certainly win because he is behind in the polls, merely that it is not certain that he will lose because he is behind in the polls, as Pundit 2 thinks. His example of Mike Johnson’s surprise win correctly refutes Pundit 2’s claim that being behind in the polls guarantees electoral loss. Pundit 2's final response demolishes a straw man he himself set up - that Pundit 1 claimed that being behind guarantees a win.

A Common Type of Real-Life Fallacy: False Dichotomies involving Universal Generalization (3)

Let us dissect another example of this fallacy in every technical detail.

Two scientists are having a professional argument.

Jane: Have you heard of Jackson's new theory? It's very promising.

Mary: I don't think so. All my colleagues think it's rubbish.

Jane: Many scientists first thought that about Wegener's theory of continental drift.

Mary: Yes but not every theory that is rejected at first is accepted later.

It may seem to the casual observer that Mary prevailed in this debate, even though her final argument is based on a logical fallacy. Jane had never argued that initial rejection of a theory implies that it must be true. She merely observed that initial rejection of a theory does not imply that it is false, which refutes Mary's implied argument.

A Common Type of Real-Life Fallacy: False Dichotomies involving Universal Generalization (4)

Given an idea x, let R(x) be the statement that x was initially rejected by the relevant experts. Let A(x) be the statement that the consensus eventually accepted x. Let us also use JT for Jackson's theory, and WT for Wegener's theory of continental drift.

The debate between Jane and Mary concerns the truth of A(JT). Jane expresses her belief that A(JT) is true in her opening statement. Mary expresses her belief that A(JT) is false in her first sentence, and justifies it in her second sentence, based on the following logic:

1. R(JT) (Mary's stated premise)2. MIP = ∀x (R(x) → ¬A(X)) (Mary's implied premise)3. R(JT) → ¬A(JT) (universal instantiation of the previous rule)4. ∴ ¬A(JT) (modus ponens using 1. and 3.)

But Jane notices that MIP is not true. A statement for all x is false if you can find one x for which it is false. Jane observes in her final statement that R(WT) is true. She takes A(WT) to be generally accepted knowledge. Her implied argument is that therefore, the conditional

R(WT) → ¬A(WT)

is false since its premise is true but the conclusion is false. This means that therefore, MIP is false as well and Mary’s belief in ¬A(JT) is unjustified.

A Common Type of Real-Life Fallacy: False Dichotomies involving Universal Generalization (5)

An alternative way of understanding Jane's argument for ¬MIP is to observe that

¬MIP = ¬∀x (R(x) → ¬A(X)) = ∃x ¬(R(x) → ¬A(X)) = ∃x ¬(¬R(x) ∨ A(x))= ∃x (R(x) ∧ ¬A(x)).

We used three rules here we have learned so far: 1. The negation of a universal quantification is the existential quantification of the

negation. 2. The conditional can be expressed as a disjunction. 3. One of De Morgan’s laws.

Since (R(WT) ∧ ¬ A(WT)) is true, ¬MIP is true.

Mary responds to Jane’s refutation with a misunderstanding. She thinks that Jane is claiming

JPC = ∀x (R(x) → A(x)) (Jane's presumed claim).

A Common Type of Real-Life Fallacy: False Dichotomies involving Universal Generalization (6)

JPC is a false statement because its negation ¬JPC = ∃x (R(x) ∧ ¬A(x)) is true. There are many ideas that were rejected by experts at first, and remain rejected. Velikovsky's catastrophism is one example.

Let us juxtapose JPC and ¬MIP:

JPC = ∀x (R(x) → A(x)) ¬MIP = ¬∀x (R(x) → ¬A(X)).

Mary appears to believe that these statements are logically equivalent.

Her misunderstanding seems to be two-fold: she seems to think that negation of a universal quantification is the universal quantification of the negation, and that the negation of a conditional is simply the conditional with the same premise but negated conclusion. Therefore, she believes that¬∀x (P(x) → Q(X)) is equivalent to ∀x (P(x) → ¬Q(X) .

We know based on what we have already learned that these rules are incorrect. In fact, JPC and ¬MIP are not logically equivalent.

A Common Type of Real-Life Fallacy: False Dichotomies involving Universal Generalization (7)

The fallacy is subtle. Mary's final statement ¬JPC is true, so many people will erroneously assume that she 'won' the debate. They overlook that ¬JPC only refutes JPC, an argument that Jane never made in the first place. Jane never said that initial rejection always implies eventual acceptance (JPC). She stated correctly that initial rejection does not always imply eventual non-acceptance (¬MIP). Mary demolished a straw man in her final statement.