Deductive Arguments and Logical Symbols

Types of Deductive Arguments and Symbols

• This lecture covers valid and invalid deductive arguments
• The use of symbols to express deductive arguments is explored
• Validity is a property relating to arguments, not statements
• For an argument to be valid, the premise must be true

Symbols Used in Logic

• Implication (If-then) is represented by ⊃ (or an arrow or >)
• Conclusion (therefore) is represented by ∴
• Negation (not) is represented by ~
• Disjunction (or) is represented by v
• Conjunction (and) is represented by *
• Inclusion (brackets) is represented by ( )

Symbols and Conditionals

• A claim in the form "If P then Q" is a conditional statement
• P is the antecedent of the conditional
• Q is the consequent of the conditional

Variant Ways of Expressing Conditionals

• "If P then Q" can also be expressed as:
  • P entails Q
  • P implies Q
  • P -> Q
  • P is sufficient for Q
  • If you've got P you must have Q
  • A necessary condition for having P is that you have Q
  • Q is necessary for having P
  • It’s only the case that P if it's also the case that Q
  • P only if Q.

Sufficient and Necessary Conditions

• A sufficient condition guarantees that if the first fact obtains, the second fact also obtains
• Example: Having ten children is sufficient to be a parent
• A necessary condition requires that for the second fact to be true, the first fact must also be true
• Example: Being male is necessary to be a biological father
• If P entails Q:
  • P is a sufficient condition for Q
  • Q is a necessary condition for P

Entailment

• While P entails Q, Q does not necessarily entail P
• Sometimes P entails Q and Q entails P can occur
• Variant ways of expressing when entailment occurs:
  • P if and only if Q
  • P if Q
  • P just in case Q
  • P <-> Q
  • if P then Q, and if Q then P
  • P is both sufficient and necessary for Q
  • P is a necessary and sufficient condition for Q

Example of Necessary and Sufficient Condition

• Being a male parent is both necessary and sufficient for being a biological father
• If you're a biological father, it's necessary for you to be a male parent
• if you're a male parent, you’re considered biological father
• Someone is a biological father if and only if he's a male parent

Deductive Argument Forms

• Modus Ponens (& Fallacy of affirming the consequent)
• Modus Tollens (& Fallacy of denying the antecedent)
• Disjunctive Syllogism
• Hypothetical Syllogism
• Constructive Dilemma

Modus Ponens

• Modus ponens affirms the antecedent
• It is the simplest form of a valid argument
• Example:
  • P1: If Spot is a dog, then he has a tail and legs
  • P2: Spot is a dog
  • C: Therefore, he has a tail and legs
• This involves an implicit conditional or if-then structure

Symbolic Representation of Modus Ponens

• Logicians express "if-then" structure using symbols
• "if-then" is expressed by the symbol: ⊃
• "if p then q" is expressed as: p ⊃ q
• the conclusion utilizes the symbol: ∴
• The valid form (modus ponens) uses the symbols:
  • p ⊃ q
  • p
  • ∴ q

Fallacy of Affirming the Consequent

• The invalid form of Modus Ponens is:
  • P1: If Spot is a dog, then has a tail and legs
  • P2: Spot has a tail and legs
  • C: Therefore, he is a dog

Symbolic Representation of the Fallacy of Affirming the Consequent

• p ⊃ q
• q
• ∴ p

Abortion Example of Modus Ponens

• P1: If abortion is murder, then it is morally false
• P2: Abortion is murder
• C: Therefore abortion is morally false
• Symbolic representation:
  • p ⊃ q
  • p
  • ∴ q

Validity and Soundness

• Arguments may be valid and unsound
• A valid argument is not necessarily sound
• Example:
  • P1: If abortion is murder, then it is morally false
  • P2: Abortion is murder
  • C: Therefore abortion is morally false
• The valid argument above is not necessarily sound, because 'abortion is murder' is questionable

Modus tollens

• Modus tollens is the method of denying the consequent
• Conclusion is formed validly
• Example:
  • If Spot is a dog, then he has a tail and legs
  • Spot has no tail and legs
  • Therefore, he is not a dog

Symbolic Representation of Modus Tollens

• Denial is expressed by the symbol ~
• Using symbols, modus tollens can be represented as:
  • p ⊃ q
  • ~q
  • ∴ ~p
• Example 2:
  • If I am hungry, I am confused and angry
  • I am not confused and angry
  • Thus I am not hungry

Example 2

• If Spot is a dog, then he has a tail and legs
• Spot is not a dog
• Therefore he doesn’t have a tail and legs
• In symbols:
  • p ⊃ q
  • ~p
  • ∴ ~q

Fallacy of Denying the Antecedent

• The invalid form of Modus Tollens is the fallacy of denying the antecedent

Disjunctive Syllogism

• A disjunction or alternation is formed by inserting "or" between two statements
  • e.g. either p or q, Something is either blue or red, The blind prisoner has a red hat or a white hat
• Two component statements combined using "or" are called "disjuncts" or alternatives
• In symbols: p v q
• A syllogism is a deductive argument consisting of two premises and a conclusion
• A disjunctive syllogism is a deductive argument where at least one of its premises' disjuncts is true

Example of Disjunctive Syllogism

• The blind prisoner has a red hat or a white hat
• The blind prisoner does not have a red hat
• Therefore the blind prisoner has a white hat
• In symbols:
  • p v q
  • ~p
  • ∴ q
• Another example:
  • Colours are either properties of objects or they are subjective sensations
  • Colours do not belong to properties
  • Thus colours are subjective.

Hypothetical syllogism

• States that if one implies another, and that other implies a third, then the first implies the third
• Example is:
  • If God created the universe then the universe will be perfect
  • If the universe is perfect then there will be no evil
  • So if God created the universe there will be no evil
• If God created the universe, then the universe will be perfect symbolic representation:
  • p ⊃ q
  • If the universe is perfect then there will be no evil symbolic representation: q ⊃ r
  • So if God created the universe there will be no evil symbolic representation: ∴ p ⊃ r
• Example 2:
  • If I do not wake up, then I cannot go to work
  • If I cannot go to work, then I will not get paid
  • Therefore, if I do not wake up, then I will not get paid

Constructive dilemma

• If two conditionals are true and at least one of their antecedents is, then at least one of their consequents must be too.
• Example:
  • Either you will go to heaven or you will go to hell
  • If you go to heaven you will have eternal bliss
  • If you go to hell you will have eternal sweating
  • Therefore you will either have eternal bliss or eternal sweating
• Either you will go to heaven or you will go to hell Symbolic representation: p v q
• If you go to heaven you will have eternal bliss Symbolic representation: p ⊃ r
• If you go to hell you will have eternal sweating Symbolic representation: q ⊃ s
• Therefore you will either have eternal bliss or eternal sweating Symbolic representation: ∴ r v s

Identification of Standard Argument Forms

• Argument 1:
  • If bacteria is in the egg salad, then the guests will become ill
  • The guests have not become ill
  • Hence, bacteria is not in the egg salad
  • If P then Q, Not Q, Therefore not P
  • Answer: Modus Tollens
• Argument 2:
  • Andiswa is either a first year or a second year
  • Andiswa is not a first year
  • Hence, Andiswa is a second year
  • Either P or Q, Not P, Therefore Q
  • Answer: Disjunctive Syllogism
• Argument 3:
  • If Malwande winks at Tammy, then Malwande is interested in dating her
  • Malwande winks at Tammy
  • Hence, Malwande is interested in dating her
  • If P then Q, P, Therefore Q
  • Answer: Modus Ponens
• Argument 4:
  • The universe is the result of chance, or it is the result of design
  • The evidence suggests the universe is not the result of chance
  • Hence, it is likely that the universe is the result of design
  • Either P or Q, Not P, Therefore Q
  • Answer: Disjunctive Syllogism
• Argument 5:
  • If there are virtuous leaders, then there will be virtuous citizens
  • There are no virtuous leaders
  • Hence, there are no virtuous citizens
  • If P then Q, Not P, Therefore not Q
  • Answer: Fallacy of Denying the Antecedent
  • (If spot is a dog, he will have four legs, Spot is not a dog, Therefore he does not have four legs)
• Argument 6:
  • Simone is bored or she is sick
  • She is not sick
  • So she must be bored
  • Either P or Q, Not Q, Therefore P
  • Answer: Disjunctive Syllogism
• Argument 7:
  • If Talitha studies for the test, then she will pass
  • She does not study for the test
  • Hence, she will not pass
  • If P then Q, Not P, Therefore Not Q
  • Answer: Fallacy of Denying the Antecedent
• Argument 8:
  • If we spend enough money, then poverty will end
  • Poverty has not ended
  • Therefore, we haven't spent enough money
  • If P then Q, Not Q, Therefore Not P
  • Answer: Modus Tollens