PHIL 4160: Symbolic Logic
Dr. Robert Lane
Lecture Notes: Tuesday January 21, 2014

 

[2.11.] “Material Conditionals.”

 

Our fourth connective symbolizes the English language phrase “If ... then .....”. E.g.,

 

“If Murphy has a bone, then Murphy is happy.”

 

A sentence of this form is called a conditional (it is also sometimes called a hypothetical): “A compound sentence that expresses an ‘If ... then’ relationship between its component sentences.” (p.53)

 

In a conditional, the component sentence between the “if” and the “then” is the antecedent; the component sentence after the “then” is called the consequent. In the example, “Murphy has a bone” is the antecedent and “Murphy is happy” is the consequent.

 

Important: conditionals are frequently expressed in English without using the word “then”:

§  “If Murphy has a bone, he is happy.”

and without using the words “if” or “then”:

§  “As long as Murphy has a bone, he is happy.”

§  “Murphy is happy, assuming that he has a bone.”

 

The symbol for the conditional is the horseshoe:   É

 

So to represent the sentence about Murphy in our system, we write:   B É H

 

(where “B” is a sentence constant representing “Murphy has a bone” and “H” is a sentence constant representing “Murphy is happy.”)

 

In our system of sentential logic, the conditional is truth-functional: the truth-value of a given material conditional is always completely determined by the truth-values of the atomic sentences of which it is composed (the antecedent and the consequent):

 

p

q

p É q

T

T

T

T

F

F

F

T

T

F

F

T

 

In our system, a conditional is false exactly when its antecedent is true and its consequent is false; in all other cases, the conditional is true.

 

While the connectives we have looked at so far (the dot, the tilde and the wedge) pretty closely capture the meanings of their English-language counterparts (“and,” “not” and “or”), the horseshoe captures only some of the meaning of the words “if...then...”

 

This is because the conditional in sentential logic is truth-functional, but many conditionals in ordinary language are not truth-functional. In other words, for many ordinary-language “if-then” sentences, you can’t always tell the truth-value of the entire sentence just from knowing the truth value of its antecedent and consequent.

 

For instance, consider “If it rains tonight, then I will stay home.” (Compare the Notre Dame vs. Miami example on p.36):

 

It rains tonight.

I will stay home.

If it rains tonight, then I will stay home.

T

T

T -- if it actually does rain, and I actually do stay home, then the conditional sentence I uttered is true.

T

F

F -- if it actually does rain, but I go out, then the conditional sentence I uttered is false.

F

T

??? -- if it doesn’t rain tonight and I stay home, it’s not clear whether the conditional is true or false (since it’s unknown what would have happened had it rained)

F

F

??? -- if it doesn’t rain tonight and I go out, it’s not clear whether the conditional is true or false (since it’s unknown what would have happened had it rained)

 

So we can’t always tell whether a conditional in ordinary English is true or false, even if we know the truth values of its antecedent and consequent. This is what is meant by saying that conditional sentences in ordinary English are not always truth-functional.

 

But in sentential logic, all connectives must be truth-functional. We must be able to assign a “T” or an “F” to every conditional, based on its component sentences. So in the two problem cases above, where the antecedent is false, we have to assign either a “T” or an “F”.

 

Although it is not obviously correct to say that the conditional in this example is true in the case that it doesn’t rain, it seems less accurate to say that it is false. It is this sort of thinking that grounds the assignment of “T” to conditionals with false antecedents:

 

p

q

p É q

T

T

T

T

F

F

F

T

T

F

F

T

 

Because there is this difference in meaning between (frequently non truth-functional) conditionals in ordinary language and (always truth-functional) conditionals in sentential logic, conditionals in sentential logic are not just called conditionals; they are given a more specific name: “material conditionals”. The truth-function expressed by a material conditional is called material implication.

 

 

truth-function

symbol

name of symbol

English equivalent(s)

conjunction

·

Dot

and

negation

~

Tilde

not ; it is not the case that

disjunction

Ú

Wedge (a.k.a. vel)

or ; either...or...

material implication

É

Horseshoe

if …, then …

 

EXAMPLES p.40 -- We won’t go through these in class. Look at them closely to be sure you understand that all of them express the same conditional and can all be symbolized the same way.

 

 

[2.11.1.] Oddities of Material Implication.

 

Because material implication is truth-functional, it results in some oddities:

 

Any material conditional with a false antecedent or a true consequent is true, even if the antecedent and the consequent have nothing to do with one another: e.g.,

·         “If Joe Biden is Vice President [T], then DNA takes the form of a double helix [T].”

·         “If Miley Cyrus is an evil cyborg from the future [F], then Atlanta is the capitol of Georgia [T].”

·         “If the Nazis won WWII [F], then Cuba is a representative democracy [F].”

All three of these conditionals are true, if they are interpreted as material conditionals.

 

However, this rarely has an effect on our translations of actual, ordinary language arguments into symbolization. This is because people rarely assert conditionals the antecedent and consequent of which have nothing to do with each other, and they rarely assert conditionals with antecedents they know to be false.

 

 

[2.12.] “Material Biconditionals.”

 

The final truth-functional connective in sentential logic is the material biconditional (a.k.a. biconditional), symbolized by the tribar: º

 

p

q

p º q

T

T

T

T

F

F

F

T

F

F

F

T

 

The truth-function expressed by a material biconditional is called material equivalence:

 

material equivalence (df.): two sentences are materially equivalent if they have the same truth value; if they do not have the same truth value, then they are not materially equivalent.

 

Notice that in the truth-table definition for “º”, “p º q” is true when both p and q are true and when both p and q are false. In other words, a material biconditional is true when its component sentences have the same truth-value; it is false exactly when they have different truth values.

 

p º q” is called a biconditional because it is equivalent to the conjunction of two conditionals:

 

p º q                             is equivalent to           (p É q) · (q É p)

 

 

This explains why we read “p º q” as “p if and only if q”. This is because...

 

p if and only if q

or p º q

is equivalent to the conjunction of

 

p if q

 

which is equivalent to

 

“if q then p

or q É p

and

p only if q

 

which is equivalent to

 

“if p then q

or p É q

 

So “Anne is an actress if and only if Bert is a bachelor” is equivalent to “If Anne is an actress, then Bert is a bachelor, and if Bert is a bachelor, then Anne is an actress.”

 

So “A º B” is equivalent to “(A É B) · (B É A)”.

 

Notice that the phrase “just in case” expresses material equivalence. So an accurate translation of “Murphy is happy just in case he has a bone” is “H º B”.

 

--

 

You have now been introduced to all five connectives of sentential logic. You MUST master these connectives. In particular, you should be able to reproduce the truth table definitions for all of them.  In summary, they are:

 

function

negation

conjunction

disjunction

material implication

material equivalence

operator

~

·

Ú

Ì

º

English term

not

and

or

if...then

if and only if; just in case

number of inputs

1

2

2

2

2

 

 

[2.13.] “‘Only if’ and ‘Unless’”.

 

[2.13.1.] “Only if.”

 

p only if q        =          if p, then q                   =          p É q

 

Example:

 

“Romney is President only if he won Florida.” This means--

 

“If Romney is President, then he won Florida.” In other words, there is no way that Romney will be President without winning Florida. Romney winning Florida is a necessary condition of Romney becoming President. So if Romney does in fact become President, that guarantees that he has actually won Florida.

 

So “Romney is President only if he won Florida” should be symbolized “R É F” (where “R” symbolizes “Romney is President” and “F” symbolizes “Romney won Florida”).

 

 

[2.13.2] “Unless.”

 

The most intuitive translation of “unless” into an English form that can be easily translated into our system is: if not.

 

p unless q   =   p if not-q

 

Once you replace “unless” with “if not”, you can then shift the “if” portion to the beginning of the sentence…

 

p if not-q  =   if not-q, then p

 

From here, the symbolization is easy:

 

if not-q, then p   =   ~q É p

 

Notice that the symbolization (which basically means “if q is not true, then p is true”) is equivalent to “either q is true or p is true”: q Ú p … which is equivalent to p Ú q. So

 

p unless q   =   p if not-q   =   if not-q, then p   =   ~q É p    =   q Ú p    =   p Ú q

 

Example:

 

“Romney will win unless Palin speaks.” This means--

 

“If Palin does not speak, then Romney will win.” In other words, the only condition in which Romney does not win is if Palin speaks. Palin not speaking is a sufficient condition of Romney winning. So if Palin does not speak, that is a guarantee that Romney will win.

 

So “Romney wins unless Palin speaks” should be symbolized  “~P É R” (where “P” symbolizes “Palin speaks” and “R” symbolizes “Romney wins”).

 

“Unless” can also be directly translated with the wedge (Ú), although this might seem less obviously correct than translating it with the horseshoe. “Romney wins unless Palin speaks” is equivalent to “Either Romney wins or Palin speaks” and so can also be symbolized as “R Ú P”.

 

Note that sometimes “unless” and “only if,” when used in everyday English, are intended by their speakers to express a biconditional, rather than a conditional. As your textbook notes, you should translate these phrases as expressing conditionals according to the above guidelines, unless there is (i.e., if there is not) some clear indication in what you are translating that the speaker or writer intends to express a biconditional.

 

EXAMPLES (p.43)

 

 

EXERCISE 2-5 (p.43)

·         Complete all of these and check your answers to the even-numbered problems to the answers in the back of the book. We will review the odd-numbered problems at the beginning of the next class.

 

Exercise 2-6 (pp.47-48)

·         Do all of these exercises. Check your even-numbered answers against the book. We will go through the odd-numbered problems next time.

 

Exercises 2-7 (pp.48-49) and 2-8 (pp.49-50)

·         Do all of these exercises. Check your even-numbered answers against the book. We will go through the odd-numbered problems next time.

 

 

 

Stopping point for Tuesday January 21. For next time:

·         complete exercises 2-5, 2-6, 2-7, and 2-8

·         read all of chapter 3

·         Heads up: your first test is still scheduled for Thursday January 30. I will decide after our next class whether to move the test to Tuesday February 4.

 

 

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