Deductive Arguments

Arguments An argument is a set of claims put forward as reasons to believe some statement.

Arguments An argument is a set of claims put forward as reasons to believe some statement. The reasons are given in the premisses The statement they support is the conclusion

Arguments An argument is a set of claims put forward as reasons to believe some statement. The reasons are given in the premisses The statement they support is the conclusion An argument If children like ice-cream, and Bob is a child, then Bob likes ice-cream

Arguments An argument is a set of claims put forward as reasons to believe some statement. The reasons are given in the premisses The statement they support is the conclusion An argument in standard form P1 Children like ice-cream P2 Bob is a child C Bob likes ice-cream

Arguments Two kinds of arguments Deductive - conclusion doesn’t tell us more about the world than the premisses

Arguments Two kinds of arguments Deductive - conclusion doesn’t tell us more about the world than the premisses Inductive – does claim to tell us more

Arguments Two kinds of arguments Deductive - conclusion doesn’t tell us more about the world than the premisses If children like ice-cream, and Bob is a child, then Bob likes ice-cream Inductive – does claim to tell us more

Arguments Two kinds of arguments Deductive - conclusion doesn’t tell us more about the world than the premisses If children like ice-cream, and Bob is a child, then Bob likes ice-cream Inductive – does claim to tell us more All the swans I have seen are black. Therefore all swans are black

Deductive Arguments Validity If the premisses are true then the conclusion must be true

Deductive Arguments Validity If the premisses are true then the conclusion must be true All men are mortal Socrates is a man Socrates is mortal

Deductive Arguments Validity If the premisses are true then the conclusion must be true Note: if the premisses are false then the conclusion may be false

Deductive Arguments Validity If the premisses are true then the conclusion must be true Note: if the premisses are false then the conclusion may be false All men are turtles Socrates is a man Socrates is a turtle

Deductive Arguments Validity Invalid: If the premisses are true then the conclusion must be true Note: if the premisses are false then the conclusion may be false Invalid: Even if the premisses are true the conclusion may be false

Deductive Arguments Validity Soundness If the premisses are true then the conclusion must be true Note: if the premisses are false then the conclusion may be false Soundness The argument is valid and the premisses are true Note: the conclusion must be true

Deductive Arguments Logic Some arguments are valid just because of their form All men are mortal Socrates is a man Socrates is mortal

Deductive Arguments Logic Some arguments are valid just because of their form All men are mortal All A are B Socrates is a man C is an A Socrates is mortal C is B

Deductive Arguments Logic Some arguments are valid just because of their form All men are mortal All A are B Socrates is a man C is an A Socrates is mortal C is B These are Formally Valid

Deductive Arguments Not Logic Some arguments are not valid just because of their form Socrates is a bachelor Socrates is unmarried.

Deductive Arguments Not Logic Some arguments are not valid just because of their form Socrates is a bachelor A is a B Socrates is unmarried A is C Not much to say about these sorts of arguments

Logical Arguments Propositional logic Syllogistic The logic of sentences and connectives Syllogistic The logic of categories

Propositional Logic Propositional Connectives Disjunction ‘… or …’ ‘Grass is green or snow is white’ is true if ‘grass is green’ is true or ‘snow is white is true’

Propositional Logic Propositional Connectives Conjunction ‘… and …’ ‘Grass is green and snow is white’ is true if ‘grass is green’ is true and ‘snow is white is true’

Propositional Logic Propositional Connectives Negation ‘not …’ ‘Grass is not green’ is true if ‘grass is green’ is not true

Propositional Logic Propositional Connectives Implication ‘if … then …’ ‘If grass is green then snow is white’ is true if in any case that ‘grass is green’ is true it is also the case that ‘snow is white is true’

Truth Tables Negation A                not A T                  F F                  T

Truth Tables Disjunction A                B                A or B T                T                  T T                F                   T F                T                  T F                F                   F

Truth Tables Conjunction A                B                A and B T                T                  T T                F                   F F                T                  F F                F                   F

Truth Tables Implication A                B                A and B T                T                  T T                F                   F F                T                  T F                F                   T

Conditional Statements ‘if P then Q’ is a Conditional Statement. P is the Antecedent Q is the Consequent

Conditional Statements ‘if P then Q’ is a Conditional Statement. P is the Antecedent Q is the Consequent The conditional ‘if P then Q’ can be drawn as

Conditional Statements ‘if P then Q’ is a Conditional Statement. P is a sufficient condition for Q Q is a necessary condition for P If P is a necessary and sufficient condition for Q then Q is also a sufficient and necessary condition for P.                   If P then Q and if Q then P. This is a biconditional.   P if and only if Q

Valid Arguments in PL Modus Ponens: If P then Q P Q If you are English then you like fish and chips. You are English. So, you like fish and chips.

Valid Arguments in PL Modus Tollens: If P then Q not Q not P If you are English then you like fish and chips You don’t like fish and chips So, you are not English

Formal Fallacies in PL Affirming the Consequent: If P then Q Q P If you are English then you like fish and chips. You like fish and chips. So, you are English.

Valid Arguments in PL Denying the Antecedent: If P then Q not P not Q If you are English then you like fish and chips You are not English So, you don’t like fish and chips

Syllogistic Categorical Propositions All S are P Universal Affirmative A No S are P Universal Negative E Some S are P Particular Affirmative I Some S are not P Particular Negative O

Syllogistic Sample Arguments No G are H All F are G No F are H   No men are perfect All Greeks are men No Greeks are perfect

Syllogistic Sample Arguments No G are H Some F are G Some F are not H   No philosophers are wicked Some Greeks are philosophers Some Greeks are not wicked

Euler Diagrams Universal Affirmative All S are P

Euler Diagrams Universal Negative No S are P

Euler Diagrams Particular Affirmative Some S are P

Euler Diagrams Particular Negative Some S are not P

Euler Diagrams Sample Arguments No G are H                             All F are G                             No F are H

Euler Diagrams Sample Arguments No G are H Some F are G Some F are not H

Testing Arguments Disproving Validity Method 1 — The Counterexample Method

Deductive Arguments Disproving Validity Method 1 — The Counterexample Method Determine the pattern of the argument to be criticised

Deductive Arguments Disproving Validity Method 1 — The Counterexample Method Determine the pattern of the argument to be criticised Construct a new argument with: (a) the same pattern (b) obviously true premises; and (c) an obviously false conclusion.

Deductive Arguments Disproving Validity Method 1 — The Counterexample Method Example If God created the universe then the theory of evolution is wrong The theory of evolution is wrong God created the universe

Deductive Arguments Disproving Validity Method 1 — The Counterexample Method Example If A then B B A

Deductive Arguments Disproving Validity Method 1 — The Counterexample Method Example If Stephen is a wombat then Stephen is a mammal T Stephen is a mammal T Stephen is a wombat F !

Deductive Arguments Disproving Validity Method 2 — Invalidating Possible Situations

Deductive Arguments Disproving Validity Method 2 — Invalidating Possible Situations Describe a possible situation in which the premises are obviously true and the conclusion is obviously false

Deductive Arguments Disproving Validity Method 2 — Invalidating Possible Situations Example (Fallacy of Affirming the Consequent) If my car is out of fuel it won’t start My car won’t start My car is out of fuel

Deductive Arguments Disproving Validity Method 2 — Invalidating Possible Situations Example (Fallacy of Affirming the Consequent) My car will indeed not start without fuel (it is a fuel-driven car) and the electrical system needed to start the car has been taken out for repairs (so it won't start). Yet the car has a full tank of petrol.

Deductive Arguments Disproving Validity Method 2 — Invalidating Possible Situations Example (Fallacy of Denying the Antecedent) If the committee addresses wilderness values it must address naturalness It will not address wilderness values It need not address naturalness

Deductive Arguments Disproving Validity Method 2 — Invalidating Possible Situations Example (Fallacy of Denying the Antecedent) Wilderness value involves, amongst other things, naturalness (Federal legislation actually defines 'wilderness value' this way). Moreover, the Committee's terms of reference do not include consideration of wilderness value (so it won't address it). Yet the Committee is explicitly formed to consider naturalness (to feed their findings into those of other Committees, so that a joint finding can be made regarding wilderness values)