Today’s Lecture 4/1/10

9.1

Symbolizations continued

Announcements

Homework:

--Ex 9.1 pgs. 432-433 Part D (1-25 All). (I will post answers to these shortly).

-- Read 9.3 pgs. 444-457

--Have a great Spring Break!

General tips on symbolizing categorical statements (and those that resemble them)

1. Memorize the form (e.g. All S are P) for each of the four types of categorical statements as well as the stylistic variants for each type. For Univ. Affirmatives utilize the (x) and the !; for Univ. Negatives, utilize the (x), !, and the ~ in the consequent; for Particular Affirmatives, utilize the ("x) and the •; for Particular Negatives, utilize the ("x), •, and the ~. Note that our four types of categorical statements can be negated.

General tips on symbolizing categorical statements (and those that resemble them)

2. Remember the “truth conditions” argument (see pg. 425) for why the • is not equivalent to the !; More specifically, why Particular type statements should not be symbolized using the !.

General tips on symbolizing categorical statements (and those that resemble them)

3. Remember that there are some statements that resemble our four types of categorical statements (e.g. ‘Everything is red’). See book and slides for the standard way to translate these.

General tips on symbolizing categorical statements (and those that resemble them)

4. Take note of the predicates in the scheme of abbreviation. The English statement will declare that these predicates apply to a thing or things. If the statement declares that the predicates apply to a particular named thing or things (e.g. Obama, Los Angeles--these are proper nouns/names), simply connect the predicate letter(s) with the individual constant(s).

General tips on symbolizing categorical statements (and those that resemble them)

5. Continuing from 4: the English statement most often will not declare that the predicates apply to proper nouns/names. If so, the statement will declare that the predicates apply to things in general -- non-named things, in which case we will capture this fact symbolically by utilizing individual variables (e.g. lower case x and y). In our translation, the predicate letter(s) will be connected (applied) to the variable.

General tips on symbolizing categorical statements (and those that resemble them)

6. Continuing from 5: if the English statement declares that the predicates apply to non-named things, it will also declare (sometimes implicitly) the number of said non-named things. Pay special attention to this. If the statement claims (or implies) that the predicates apply to nothing (‘no thing’), or all, every, or each thing, then you’re dealing with either a Univ. Affirmative or a Univ. Negatives and you’ll want to capture this fact symbolically by utilizing, at the least, the universal quantifier (x).

General tips on symbolizing categorical statements (and those that resemble them)

7. Continuing from 6: If the statement claims (or implies) that the predicates apply to at least one non-named thing (or something), then you’re dealing with a Particular Affirmative and you’ll want to capture this fact symbolically by utilizing, at the least, the existential quantifier ("x). The statement may declare that the predicates (oftentimes one of the predicates) don’t apply to at least one non-named thing (or something). If so, you’re dealing with a Particular Negative and you’ll want to utilize, at the least, the ("x).

General tips on symbolizing categorical statements (and those that resemble them)

8. Note that you can still come across statements that are conditionals, conjunctions, negations, etc. The main operators will of course be the !, •, ~ respectively. For example, you could have (x)(Hx ! Mx). This can be called a quantified statement (All humans are mortal); it’s not a conditional. The (x) ‘governs’ the ! and is thus the main operator. But contrast this with:

(x)(Hx ! Mx) ! Ma (If all humans are mortal, then Aristotle is mortal). This is a conditional.

General tips on symbolizing categorical statements (and those that resemble them)

9. Determining whether a statement is a quantified statement that is a Universal or a genuine conditional can be tricky at times; after all, much of the meaning of a Universal statement is captured with the ! (a Universal statement is a conditional that is governed by a universal quantifier). As a general tip, look to the consequent of the conditional. If there is a pronoun (e.g. it/he) that is not attached to a quantifier also in the consequent, chances are the pronoun ‘refers back to’ (is governed by) a quantifier (that is sometimes implicit) that also governs the content of the antecedent. If so, you’re dealing with a Universal and not a genuine conditional. Contrast #s 11 and 20 in part C for an example.

Side Note on Terminology

!! A Universal Statement -- one that is either a Universal Affirmative or a Universal Negative (see pg. 420) -- is a conditional that is governed by the (x). Thus a Universal Statement is a universally quantified statement. However not all universally quantified statements are conditionals governed by the (x). We could have (x)(Fx • Sx). This could say: Everything is such that it is funny and smart. I.e. All things are funny and smart. This is a universally quantified statement but it not a Universal Affirmative nor a Universal Negative statement.

Some Answers to HW

Ex 9.1 pgs. 431-432 Part C (11-25):

#11

A thing is a logician only if it is rational

(Univ. Affirmative)

Recall that 'only if' introduces a consequent. Thus this could read: 'if a thing is a logician, then it is rational.

The statement makes a claim about anything. Every x is such that if x is a logician, then x is rational.

(x)(Lx ! Rx)

#12

All trees are non-animals

(Universal Negative)

Everything is such that, if it is a tree, then it’s not an animal.

(x)(Tx ! ~Ax)

#13

Some people are good and some people are not good

(Conjunction of a Particular Affirmative and a Particular Negative)

There exists at least one thing that is both a person and good; and there exists at least one thing that is a person but not good).

There is an x such that x is a person and x is good; and there is an x such that x is a person but x is not good.

("x)(Px • Gx) • ("x)(Px • ~Gx)

#14

Something is both good and evil

(Particular Affirmative)

There’s at least one x such that x is good and x is evil

("x)(Gx • Ex)

#15

There exists a person who is good.

(Particular Affirmative)

There exists a thing that is both a person and good.

There is an x such x is a person and x is good

("x)(Px • Gx)

#16

Only blue things are sky blue.

(Universal Affirmative)

If anything is not blue, then it’s not sky blue.

Equivalently: If anything is sky blue, then it’s blue.

Every x is such that if x is sky blue, then x is blue.

(x)(Sx ! Bx)

#17

If Socrates is not a philosopher, then Aristotle is not a philosopher

~Ps ! ~Pa

#18

Not all animals are rational

(Negation of a Univ Affirmative; or a Particular Negative)

Not everything is such that if it is an animal, then it’s rational

Not every x is such that if x is an animal, then x is rational.

~(x)(Ax ! Rx) or ("x)(Ax • ~Rx)

#19

There exists an animal that has a soul

(Particular Affirmative)

There is a thing such that it is an animal and it has a soul

There is an x such that x is an animal and x has a soul

("x)(Ax • Sx)

#20

If all bats are mammals, then some mammals have wings

(A conditional w/ Univ. Affirm as antecedent and a Partic. Affirm as the consequent)

If (every x is such that if x is a bat, then x is a mammal), then there is an x such that x is a mammal and x has wings)

If (x)(Bx ! Mx), then ("x)(Mx • Wx)

(x)(Bx ! Mx) ! ("x)(Mx • Wx)

#21

All birds except penguins can fly

(Univ. Affirmative)

Everything is such that if it is a bird and not a penguin, then it can fly

Every x is such that if x is a bird and x is not a penguin, then x can fly

(x)[(Bx • ~Px) ! Fx]

#22

All and only circles are perfect

(Univ. Affirmative*)

All circles are perfect and only circles are perfect.

(Every x is such that if x is a circle, then x is perfect) and (every x is such that if x is perfect, then x is a circle)

(x)(Cx ! Px) • (x)(Px ! Cx) or (x)(Cx # Px)

#23

One fails the course when blowing off the final exam

(Univ. Affirmative)

Any person who blows off the final exam will the fail the course in question

Every x is such that if x is a person and x blows off the final exam, then x will fail the course.

(x)[(Px • Bx) ! Fx]

#24

If any explorer discovers gold, then he or she will become famous

(Univ. Affirmative)

All explorers who discover gold will become famous

Everything is such that if it is both an explorer and discoverer of gold, then it will become famous

(x)[(Ex • Dx) !Fx]

#25

Humans are featherless bi-peds

(Univ. Affirmative)

(x)[Hx ! (Fx • Bx)]

Ex 9.1 pgs. 432-433 Part D (1-25 All)