Formal logic The part of logic that deals with arguments with forms

Types of Propositions in a formal argument There are five types of propositions in a formal argument. The knowledge of these will help in understanding what is done in formal argument The propositions are: 1: Negation 2. Conjunction 3. Disjunction 4 Conditional statement 5. Bi-Conditional statement

negation A negation is a statement in which something is denied. The word usually employed for this purpose is NOT Example: It is NOT raining Nigeria is NOT in America Ibadan is not the capital of Ogun State

Negation (Contd) There are, however, other ways of expressing negation. Example: He is unfriendly He is ungrateful

Logical symbol of negation The logical symbol for a negation is ~ It is called a tilde The statement “it is not raining” will be symbolised as ~P

Conjunction This is a statement involving the use of “AND” EXAMPLE: It is raining and the sun is shining

Logical symbol of Conjunction The logical symbol of conjunction is. It is called “dot”

In the example “It is raining and the sun is shining” The statement will be symbolised as “P.Q” P= It is raining Q= the sun is shining and =.

Parts of a conjunction There are two parts to a conjunction. The parts of a conjunction are called “conjuncts.” The part before the “and” is referred to as the first conjunct. While the part after the “and” is the second conjunct. In the example: “It is raining and the sun is shining,” “it is raining” is the first conjunct; while “the sun is shining” is the second conjunct.

Disjunction A disjunction is a compound statement in which two statements are joined together by the use of “OR” EXAMPLE: It is raining or the sun is shining

Parts of a Disjunction There are two parts to a disjunction- the part before the OR and the part after the OR. The part before the OR is called the first disjunct, while the part after the or is called the second disjunct.

Logical connective of a disjunction The logical connective of a disjunction is “ V ” It is called a vee or wedge The statement “it is raining or the sun is shining” will be symbolised as “PvQ”

Conditional statement A conditional statement is a proposition involving the use of the connective “if…then…” EXAMPLE If it is raining then the ground is wet

Conditional statement A conditional statement is any statement of the form “if…then…” E.g. If (it rains) then (the ground is wet) There are two parts of a conditional statement- the antecedent and the consequent The antecedent is the part after the if but before the then- it rains The consequent is the part after the then

Parts of a conditional statement A conditional statement consists of two parts- the antecedent and the consequent. The antecedent of a conditional statement is the part of the proposition after the “if” but before the “then;” while the consequent of the conditional statement is the part after the “then.”

The logical connective of a conditional The logical connective of a conditional statement is called “horse shoe” The logical connective of a conditional statement is “ ⊃ ”

Bi-conditional A bi-conditional is a compound statement involving the use of “…if and only if…” Kunle will come to the wedding if and only if Janet will accompany him.

FORMS OF VALID DEDUCTIVE ARGUMENTS There are Nine forms of valid deductive arguments. They are Modus Ponens Modus Tollens Hypothetical Syllogism Disjunctive Syllogism Addition Conjunction Simplification Constructive Dilemma Destructive Dilemma

Schemas of the Nine Valid Deductive Arguments 1. Modus Ponens p ⊃ q p ∴ q 2. Modus Tollens p ⊃ q ~q ∴ ~p

3. Hypothetical Syllogism p ⊃ q q ⊃ r ∴ p ⊃ r 4. Disjunctive Syllogism p v q ~p ∴ q

5. Addition p ∴ p v q 6. Conjunction p q ∴ p q

7. Simplification p q ∴ p 8. Constructive Dilemma (p ⊃ q) (r ⊃ s) p ∨ r ∴ q ∨ s

9. Destructive Dilemma (p ⊃ q) (r ⊃ s) ~q v ~s ∴ ~p v ~r

Modus Ponens In the Modus Ponens there are two premises. The first is a conditional statement. The second is the affirmation of the antecedent of the conditional statement in the first premise. The conclusion is the affirmation of the consequent of the conditional statement in the first premise p ⊃ q p ∴ q

Example of Modus Ponens If it rains then the ground is wet. It rains. Therefore, the ground is wet Premsie1: If it rains then the ground is wet. Premise 2: It rains. Conclusion: the ground is wet

Modus Ponens If he is the best in the team then he scores three goals tonight. He is the best in the team. Therefore, he scores three goals tonight. P= he is the best in the team Q= he scores three goals

Invalid Form of Modus Ponens p ⊃ q q ∴ p

Invalid Form of Modus Ponens Description: In the invalid form of Modus Ponens, which commits the fallacy of affirming the consequent, the first premise is a conditional statement. The second premise is the affirmation of the consequent of the conditional statement in premise one, while the conclusion is the affirmation of the antecedent of the conditional statement in the first premise.

Formal fallacy committed by the invalid form of MP The invalid form commits the fallacy of affirming the consequent. This name derives from the fact that what we have in premise 2 is the affirmation of the consequent of the conditional statement in premise one instead of the affirmation of the antecedent.

Example of invalid MP If it rains then the ground is wet The ground is wet Therefore, it rains

Modus Tollens (M.T.) p ⊃ q ~q ∴ ~p Description: In the Modus Tollens, the first premise is a conditional statement. The second premise is the negation of the consequent of the conditional statement in the first premise, while the conclusion is the negation of the antecedent in the first premise.

Example of Modus Tollens Premise 1:If it rains then the ground is wet Premise 2: The ground is not wet Conclusion: It did not rain P= it rains Q= the ground is wet

Example 2 of MT If Christiana is a from Ghana then Christiana is a Black Christiana is not a Black Therefore, Christiana is not from Ghana

Invalid form of MT p ⊃ q ~p ∴ ~q

Description of Invalid form of MT Description: in the invalid form of Modus Tollens, the first premise is a conditional statement. The second premise is the erroneous denial of antecedent of the conditional statement in premise one, while the conclusion is the wrong denial of the consequent of the conditional statement in premise one. This variant of Modus Tollens commits the fallacy of denying the antecedent. In the valid form of Modus Tollens, it is the consequent of the conditional statement in Premise 1 that is denied in Premise 2. But in the invalid form, it is the antecedent, rather than the consequent, that is being denied in Premise 2, while the consequent is erroneously denied in the conclusion.

Example of the invalid form of MT If it rains then the ground wet it did not rain Therefore, the ground is not wet

Hypothetical Syllogism (H.S.) p ⊃ q q ⊃ r ∴ p ⊃ r Description: Hypothetical Syllogism contains two premises and a conclusion and all three are conditional statements. In the second premise, the consequent of the conditional statement in Premise 1 implies another consequent, while in the conclusion the antecedent of the first premise implies the consequent of the second premise.

Example of Hypothetical Syllogism If it rains then the ground is wet If the ground is wet then the ground is slippery Therefore, if it rains then the ground is slippery.

Example 2 If Kunbi is serious then he studies hard for his exams If Kunbi studies hard for his exams then he passes his exams Therefore, if Kunbi is serious then he passes his exams

Disjunctive Syllogism (D.S.) p ∨ q ~p ∴ q Description: Disjunctive Syllogism is a valid deductive argument with two premises and a conclusion. The first premise is a disjunction. The second premise is the denial of one of the conjuncts in the first premise, while the conclusion is the affirmation of the conjunct that is not denied in the first premise.

Example of Disjunctive Syllogism Either it rains or the ground is dry It did not rain Therefore, the ground is dry

Example 2 of Disjunctive Syllogism Either he wins the election or he loses the election He did not win the election Therefore, he lost the election

Addition (Add) p ∴ p v q Description: Addition has only one premise, and a conclusion in which a proposition is added to the one in the premise with the use of “or” such that the conclusion becomes a disjunction

Example of Addition Nigeria’s Federal Capital Territory is Abuja Nigeria’s Federal Capital Territory is Abuja or Ibadan is in the South West

Example 2 of Addition It is raining therefore, it is raining or the sun is shining

Conjunction (Conj.) p q ∴ p q Description: A conjunction has two premises, and a conclusion in which the proposition in premise one is joined to the one in premise two with the use of “and” such that the conclusion is a conjunction.

Example of conjunction It is raining The ground is wet Therefore, it is raining and the ground is wet.

Example 2 of Conjunction Dr. Ebele Goodluck Jonathan is the president of Nigeria Olusegun Obasanjo is a former military Head of State Therefore, Dr. Ebele Goodluck Jonathan is the president of Nigeria and Olusegun Obasanjo is a former military Head of State

Simplification (Simp.) p q ∴ p Description: A simplification has just a premise which is a conjunction, and a conclusion in which one of the conjuncts in the premise (usually the first) is affirmed in the conclusion.

Example of simplification It is raining and the sun is shining Therefore, it is raining

Constructive Dilemma (CD) (p ⊃ q) (r ⊃ s) p ∨ r ∴ q ∨ s Description: In Constructive Dilemma, the first premise is a conjunction in which the first and second conjuncts are conditional statements. The second premise is a disjunction in which the antecedents of the conditional statements in the first premise form the disjuncts, while the conclusion is a disjunction in which the consequents of the conditional statement in the first premise are the disjuncts.

Example If it is raining then the ground is wet and if the sun is shining then the weather is hot Either it is raining or the sun is shining Therefore, either the ground is wet or the weather is hot

Destructive Dilemma (D.D) (p ⊃ q) (r ⊃ s) ~q ∨ ~s ∴ ~p ∨ ~r Description: Destructive Dilemma has two premises. The first premise is a conjunction in which the first and second conjuncts are conditional statements. The second premise is a disjunction in which the consequents of the conditional statements in the first premise are negated. The conclusion is a disjunction in which the antecedents of the conditional statement in premise one are negated.

Example If it is raining then the ground is wet and if the sun is shining then the weather is hot Either the ground is not wet or the weather is not hot Therefore, either it is not raining or the sun is not shining