Conditional Conditional Statements and Statements and

Material Material ImplicationImplication

The Conditional: The Conditional: The Fourth ConnectiveThe Fourth Connective

Conditional statementConditional statement: when two : when two statements are combined by placing the statements are combined by placing the word “if” before the first and “then” before word “if” before the first and “then” before the secondthe second ““If…then…”If…then…” We use the arrow → or horseshoe to We use the arrow → or horseshoe to

represent the “if-then” phrase represent the “if-then” phrase Also called a Also called a hypotheticalhypothetical, an , an implicationimplication, or an , or an

implicativeimplicative statement statement The component statement that follows the The component statement that follows the

“if” is called the “if” is called the antecedentantecedent The component statement that follows the The component statement that follows the

“then” is called the “then” is called the consequentconsequent If (antecedent), then (consequent)If (antecedent), then (consequent)

∩∩

The ConditionalThe Conditional

A conditional statement asserts that A conditional statement asserts that (in any case) (in any case) ifif its antecedent is true, its antecedent is true, then its consequent is also truethen its consequent is also true

But as in disjunction, there are a few But as in disjunction, there are a few different senses in which a different senses in which a conditional can be interpretedconditional can be interpreted

The Four Types of The Four Types of ImplicationImplication

1. If all humans are mortal and Socrates is a 1. If all humans are mortal and Socrates is a human, then Socrates is mortal.human, then Socrates is mortal.

• Logical ImplicationLogical Implication: the consequent follows : the consequent follows logically from its antecedentlogically from its antecedent

2. If Leslie is a bachelor, then Leslie is 2. If Leslie is a bachelor, then Leslie is unmarried.unmarried.

• Definitional ImplicationDefinitional Implication: the consequent follows : the consequent follows the antecedent by definitionthe antecedent by definition

3. If I put X in acid, then X will turn red.3. If I put X in acid, then X will turn red.• Causal ImplicationCausal Implication: The connection between : The connection between

antecedent and consequent is discovered antecedent and consequent is discovered empiricallyempirically

4. Is we lose the game, then I’ll eat my hat.4. Is we lose the game, then I’ll eat my hat.• Decisional ImplicationDecisional Implication: no logical connection nor : no logical connection nor

one by definition between the consequent and one by definition between the consequent and antecedent. This is a decision of the speaker to antecedent. This is a decision of the speaker to behave in the specified way under the specified behave in the specified way under the specified circumstances.circumstances.

Which Sense of Implication Do Which Sense of Implication Do We Use?We Use?

We must try to find a sense that is at least a part of We must try to find a sense that is at least a part of the meaning of the meaning of allall four different types of implication four different types of implication No matter what type of implication is asserted by a No matter what type of implication is asserted by a

conditional statement, part of its meaning is the negation conditional statement, part of its meaning is the negation of the conjunction of its antecedent with the negation of of the conjunction of its antecedent with the negation of its consequentits consequent

For a conditional to be true (e.g. “If p then q”), ~(p For a conditional to be true (e.g. “If p then q”), ~(p • • ~q) must be ~q) must be true:true:

Think p=“A piece of blue litmus paper is placed in that solution.”Think p=“A piece of blue litmus paper is placed in that solution.” q=“The piece of blue litmus paper will turn red.”q=“The piece of blue litmus paper will turn red.”

““If p then q” = false if paper is placed in solution, but doesn’t turn red If p then q” = false if paper is placed in solution, but doesn’t turn red The horseshoe symbol does not stand, therefore, for The horseshoe symbol does not stand, therefore, for

all all the meanings of “if-then” – there are several the meanings of “if-then” – there are several meaningsmeanings

p q abbreviates ~(p p q abbreviates ~(p • ~q), whose meaning is • ~q), whose meaning is included in the meanings of each kind of implicationincluded in the meanings of each kind of implication∩∩

What this means…What this means…

pp qq ~q~q p p • ~q• ~q ~(p ~(p • • ~q)~q)

TT TT FF FF TT

TT FF TT TT FF

FF TT FF FF TT

FF FF TT FF TT

*Abbreviated Truth Table *Abbreviated Truth Table for the Conditionalfor the Conditional

pp qq p qp q

TT TT TT

TT FF FF

FF TT TT

FF FF TT

∩∩

The only time a conditional is FALSE is when the antecedent is true

and the consequent is false.

Material ImplicationMaterial Implication represents the represents the material implicationmaterial implication

A fifth type of implicationA fifth type of implication E.g. “If Hitler was a military genius, then I’m a E.g. “If Hitler was a military genius, then I’m a

monkey’s uncle.”monkey’s uncle.” No real connection between antecedent and consequentNo real connection between antecedent and consequent

This kind of relationship is what is meant by material This kind of relationship is what is meant by material implicationimplication

It just asserts that it is It just asserts that it is notnot the case that the antecedent is the case that the antecedent is true when the consequent is false.true when the consequent is false.

Many arguments contain conditional Many arguments contain conditional statements of various kinds of implication, but statements of various kinds of implication, but the validity of all valid arguments (of the the validity of all valid arguments (of the general type with which we will be concerned) general type with which we will be concerned) is preserved, even if the additional meanings of is preserved, even if the additional meanings of their conditional statements are ignored.their conditional statements are ignored.

∩∩

Some “If” Indicator Some “If” Indicator WordsWords

““If…” can be replaced by such phrases If…” can be replaced by such phrases as:as: ““in case…”in case…” ““provided that…”provided that…” ““given that…”given that…” ““on condition that…”on condition that…”

Some indicator words for “then…” Some indicator words for “then…” include:include: ““implies...”implies...” ““entails…”entails…”

Necessary ConditionsNecessary Conditions For a normal car to run, it is necessary For a normal car to run, it is necessary

that there is fuel in its tank, its spark that there is fuel in its tank, its spark plugs properly adjusted, its oil pump plugs properly adjusted, its oil pump working, etc.working, etc. If the car runs, then every one of the If the car runs, then every one of the

conditions necessary for its occurrence must conditions necessary for its occurrence must be fulfilledbe fulfilled““That there is fuel in its tank is a necessary That there is fuel in its tank is a necessary

condition for the car to run” condition for the car to run” = “The car runs only if there is fuel in its tank” = “The car runs only if there is fuel in its tank” = “If the car runs then there is fuel in its tank”= “If the car runs then there is fuel in its tank”

All these are “R F” All these are “R F” ∩∩

HintsHints

““p is a necessary condition for q”p is a necessary condition for q”= q p= q p

“…“…only if…” =only if…” = ““p only if q” = p q p only if q” = p q

∩∩

∩∩∩∩

Sufficient ConditionsSufficient Conditions

For a purse to contain over a dollar, it For a purse to contain over a dollar, it would be sufficient for it to contain 101 would be sufficient for it to contain 101 pennies, 21 nickels, 11 dimes, 5 quarters, pennies, 21 nickels, 11 dimes, 5 quarters, etc.etc. If any one of these circumstances obtains, the If any one of these circumstances obtains, the

specified situation will be realizedspecified situation will be realized ““That a purse contains 5 quarters is a That a purse contains 5 quarters is a

sufficient condition for it to contain over a sufficient condition for it to contain over a dollar” dollar” = “If the purse contains 5 quarters then it = “If the purse contains 5 quarters then it

contains over a dollar”contains over a dollar”

HintsHints ““p is a sufficient condition for q”p is a sufficient condition for q”

= p q (Note: here q is a necessary condition for p)= p q (Note: here q is a necessary condition for p)

Compare: “p is a Compare: “p is a necessarynecessary condition for q” condition for q”= q p (Note: here q is a sufficient condition for p)= q p (Note: here q is a sufficient condition for p)

““Formula” to help you remember, given any Formula” to help you remember, given any sentence saying something is a necessary or sentence saying something is a necessary or sufficient condition:sufficient condition: p qp q

∩∩∩∩

∩∩

Necessary conditionSufficient condition

ExampleExample

A, B, C are true statementsA, B, C are true statementsX, Y, Z are false statementsX, Y, Z are false statements

Determine whether the following is Determine whether the following is true or false:true or false:

(X Y) Z(X Y) Z

∩∩∩∩

SolutionSolution (X Y) Z(X Y) Z

1. Main connective is the second horseshoe 1. Main connective is the second horseshoe (conditional)(conditional)

2. Look at antecedent: (X Y):2. Look at antecedent: (X Y):

-X and Y are false, so this makes this conditional -X and Y are false, so this makes this conditional truetrue

-we know this by using our knowledge of the -we know this by using our knowledge of the conditional (i.e. truth table for the material conditional (i.e. truth table for the material implication)implication)

3. We know Z (the consequent) is also false3. We know Z (the consequent) is also false

4. Therefore, a conditional with a true antecedent 4. Therefore, a conditional with a true antecedent and false consequent is and false consequent is falsefalse

∩∩∩∩∩∩

ExampleExample

A, B, C are true statementsA, B, C are true statementsX, Y, Z are false statementsX, Y, Z are false statements

Determine whether the following is Determine whether the following is true or false:true or false:

[(A • X) v (~A X)] [(A X) • (X A)][(A • X) v (~A X)] [(A X) • (X A)]∩∩ ∩∩ ∩∩∩∩