Chapter 9 Flashcards
categorical logic
logic based on the relations of inclusion and exclusion among classes (or “categories”) as stated in categorical terms
-useful in clarifying and analyzing deductive arguments
category
In logic, a c ategory is a group or a class or a population; any bunch of things can serve as a category for our purposes
terms
Terms are noun phrases, like “dogs,” “cats,”
“Christians,” “Arabs,” “people who read logic books,” and so on. These terms are
labels for categories or classes.
-NOUNS
subject term (S)
subject of the conclusion
-goes in the first blank (All __ are __.)
predicate term (P)
predicate of the conclusion
-goes in the second blank (All __ are __.)
A-claims
ALL metals are conductors
- general law
- affirmative claim
E-claims
NO metals are conductors
- general law
- negative claim
I-claims
SOME metals are conductors
(this metal is a conductor)
-observational law
-affirmative claim
O-claims
SOME metals are NOT conductors
(this metal is not an conductor)
-observational law
-negative claim
venn diagramms
illustrations of categorical claims
- p.246 in critical thinking (newest edition)
equivalent claims
if, and only if, they would be true in all and exactly the same circumstances— that is, under no circumstances could one of them be true and the other false
“only”
The word “only,” used by itself, introduces the predicate term of an
A-claim.
The phrase “the only” introduces the subject term of an A-claim.
“only matinees are half-price shows” as well as “matinees are the only half-price shows” can be translated to:
“all half-price shows are matinees”
square of opposition
shows the logical relationships of categorical claims that correspond with each other
p. 254
- very interesting, I
corresponding claims
claims who have the same subjects and the same predicates.
NOTE: S = S and P = P is CORRECT/CORRESPONDING
S=P and P=S is NOT CORRECT/NOT CORRESPONDING
contrary claims
can both be false but not both be true
-A and E claims (general laws) are contrary
subcontrary claims
can both be true but cannot not both be false
-I and O claims (observational laws) are subcontrary
contradictory claims
never have the same truth value
- A and O are contradictory, as well as E and I
e. g. if A is true, O cannot be true
conversion
switching S and P
E and I claims are equivalent to their converses
A and O claims are not
universe of discourse
a closed system in which a discourse takes place
e.g. if the professor says: “everybody passed the exam” the universe of discourse are all the students. Your mother didn’t pass the exam.
complementary class/term
contains everything in the universe of discourse that is not part of the first class
-e.g. student and nonstudents
an obversion
- change a claim from affirmative to negative or vice versa
- change the predicate term to its complementary
- all categorical claims are equal to their obversion
All pairs of terms (S and P) of a true e-claim are complementary terms.
True or false?
False, but all complementary terms are pairs of terms of a true e-claim
contra-positive
- converse the claim (switch S and P)
- replace both terms with complementary terms
All A- and O-claims, but not E- and I-claims, are equivalent to their contrapositives
Some mosquitoes carry West Nile virus. So it must be
that there are some that don’t.
Is this necessarily true?
No, The only way to get an I-claim from an O-claim is by obverting the O-claim.
a syllogism
two-premise deductive arguement
a categorical syllogism
A categorical syllogism (in standard form) is a syllogism whose every claim is a standard-form categorical claim and in which three terms each occur exactly twice and in exactly two of the claims.
major term (P)
the term that occurs as the predicate term of the syllogism’s conclusion
minor term (S)
the term that occurs as the subject term of the syllogism’s conclusion
middle term (M)
the term that occurs in both of the premises but not at all in the conclusion
validity
An argument is valid if, and only if, it is not possible for its premises to be true while its conclusion is false.
This is just another way of saying that, were the premises of a valid argument true (whether or not they are in fact true), then the truth of the conclusion would be guaranteed.
a sound argument
If the premises of a valid argument are in fact true, then that argument is said to be sound, and its conclusion has been proven to be true
The venn diagramm method of testing validity
a syllogism is valid only if diagramming the premises automatically produces a correct diagram of the conclusion. The exact process can’t be displayed here.
Handy rules to remember:
1. When one premise is an A- or E-premise and the other is an I- or O-premise, diagram the A or E-premise first
2. An X that can go in either of two areas goes on the line separating the areas
3. If any circle has only one area remaining uncolored,
an X should be put in that area
Categorical Syllogisms with Unstated Premises
In everyday life, people often give syllogisms, but they leave out one premise, probably because they think it is too obvious to state. It might be sensible to question such unstated premises
popular syllogisms
P1: All As are Bs; P2: All Bs are Cs; C: All As are Cs
or
P1: All As are Bs; P2: No Bs are Cs; C: No As are Cs
affirmative and negative
A and I are affirmative; E and O are negative
distribution
a claim is distributed if it says something about every member of a class A claim: only S is distributed E claim: both S and P are distributed I claim: neither S nor P are distributed O claim: only P is distributed
The three rules to test validity
- The number of negative claims in the premises must be the same as the number of negative claims in the conclusion. (Because the conclusion is always one claim, this implies that no valid syllogism has two negative premises.)
- At least one premise must distribute the middle term.
- Any term that is distributed in the conclusion of the syllogism must be distributed in its premises.
