TEST #2 Flashcards
logical operators and argument forms (ill-formed argument, invalid argument, non-cogent, cogency, conjunction, contraposition, universal modus ponens, biconditional, hypothetical syllogism) ( + original arguments using them)
what are invalid forms of arguments
the truth of the premises don’t guarantee the truth of the conclusion
denying the antecedent
affirming the consequent
truth vs rational strenght of premises
- evaluating the truth- value of a premise and conclusion is different from evaluating the rational strength of arguments
- it’s one thing to ask whether statements are true or not, but its another to ask: if those statements were true, what would that mean for the conclusion?
standard form
STANDARD FORM: arguments written out as consecutively numbered premises and conclusions, with justifications for each line in the argument stated
- provides a clear reconstruction of the argument (interpreting and rewriting an argument), which is essential for evaluating the argument (deciding whether or not the argument is a good argument)
benefits of standard form:
1) helps exclude any extra non-argument material
2) helps include unstated/missing premises
3) helps encourage clearer and precise formulation of premises and conclusion
4) helps make discussion of argument more convenient since each element is numbered
components:
- individually numbered premises and conclusion
- only one premise or conclusion per line
- the word “therefore” or equivalent symbol before the conclusion
- horizontal line between premises and conclusion
- brackets of the conclusion to show which premise the conclusion is supported by
deductive argument
DEDUCTIVE ARGUMENT: aims to provide a logically conclusive support for the conclusion
- it is impossible for the premises to be true, and the conclusion false
- shows that if the premises are true, the conclusion has to be true
well-formed argument
WELL-FORMED ARGUMENT: an argument is well-formed if and only if the argument is valid or cogent
- arguments whose conclusion does logically follow from its premises
- valid and cogent arguments
validity / a valid argument
- validity is a feature an argument either has, or doesn’t have
- in logic, validity applies only to arguments and the logical relationship between the premises and conclusions, not the premises or conclusion individually or their truth value
- a valid argument can have false premises and a false conclusion
- a valid argument is considered to be truth-preserving ( if premises are true, conclusion is true)
VALIDITY: an argument is deductively valid if and only if it is impossible for all the premises to be true, and the conclusion false
- A1) IF the premises are true, then the conclusion must be true as well
- A2) the conclusion logically follows the premise
- A3) in a world where the premises are all true, the conclusion is GUARANTEED to be true as well
invalid argument
INVALID ARGUMENT: it is possible for the premises to be true, and the conclusion to be false
- the premises dont guarantee the truth if the conclusion
validity test
VALIDITY TEST: the way in which we test if an argument is valid or not
1) Assume that the premises are all true. Would the conclusion have to be true as well?
2) If the answer is “yes” then the argument is valid. If the answer is “no” then the answer is invalid
is validity related to truth-value?
- No. a valid argument can have false conclusion and false premises.
- No. We don’t even have to know the truth-value of the premises and conclusion. as long as we evaluate the logical relationship between the premises and conclusion
-the point of validity is not whether each premise/conclusion is true or false, but rather that if the premises were true, would thy guarantee the truth of the conclusion?
sentential connectives + 5 types (dont explain)
SENTENTIAL CONNECTIVES: logical system in which sentences are not broken down into smaller units
- the letters stand for whole sentences
- use of UPPERCASE LETTERS
- the whole conclusion is found inside the premise
COMPOUND STATEMENTS: combination of two or more simpler sentences (we abbreviate them by using A and B)
1) conjunction
2) disjunction
3) negation
4) conditional
5) biconditional
predicate logic
opposite of sentential connectives
PREDICATE LOGIC: logical system in which sentences are broken down in subunits such as subjects and predicates (descriptive phrases)
SMALL LOWERCASE LETTERS (x): particular person or thing
BIG UPPERCASE LETTERS (A, B, C): descriptive phrases/characteristics
conjunction (sentential connective)
1) CONJUNCTION
- P and/& Q
- they are compound statements composed of two parts call the conjuncts
- ex. Today is tuesday and I am in class.”
disjunction (sentential connective)
2) DISJUNCTION
- P or/v Q
- they are compound statements composed of two parts called the disjuncts
- ex. either the picnic was cancelled or it was sunny
negation (sentential connective)
3) NEGATION
- Not P / ~ P
- saying the statement P is false/not the case
- ex. it is not sunny
conditional (sentential connective)
4) CONDITIONAL
- If P, then Q / If P -> Q
- if P happens, Q is guaranteed to happen (Q depends on P) (P guarantees Q)
- conditionals do not assert that either the antecedent or the consequent is true. they merely state a logical relationship between P and Q
- they are compound statements composed of two parts:
- ANTECEDENT: what follows the “if”
- CONSEQUENT: what follows the “then”
- ex. if it rains, then the picnic will be cancelled
conditionals are not always expressed in their logical form
- ex. since your lease expired, the landlord is free to raise the rent
- ex. being a teenager means you have lots of problems
- ex. anyone who likes logical is a fool
- ex. the truth of evolution implies the falsity of the Bible
- ex. whenever i drink coffee, i get antsy
“IF” introduces the antecedent, no matter where it occurs in a statement
- ex. “I’ll find the material difficult IF i skip a class”
- “ONLY IF” introduces the consequent, no matter where it occurs in a statement
- ex. “I will buy the giant TV ONLY IF the price drops”
- this should be written as Q -> P
biconditional (sentential connective)
5) BICONDITIONAL
- P if and only if Q
- if P then Q, if Q then P
- ex. you can enter the club if and only if you have legitamate ID
11 valid argument patterns (just list)
VALID ARGUMENT PATTERNS: common patterns shared by many deductive arguments
- help determine if an argument is deductive
- help determine if an argument is valid or invalid
- this is a means to an end: evaluating an argument
1) argument by elimination
2) conjunction
3) simplification
4) affirming the antecedent (modus ponens)
5) denying the consequent (modus tollens)
6) hypothetical syllogism
7) contraposition
8) universal modus ponens
9) universal modus tollens
10) universal hypothetical syllogism
11) universal ruling out
1) argument by elimination
- P v Q
- ~ P
:. 3. Q
(from 1, 2) - P v Q
- ~ Q
:. 3. P (from 1, 2)
ex. 1. either it will rain today or it will snow today
2. it will not snow today
therefore,
.: 3. it will rain today
2) conjunction
- P
- Q
:. 3. P & Q
ex, 1. I have a dog
2. I have a cat
therefore,
:. 3. I have a dog and a cat
3) simplification
- P & Q
.: 2. P (from 1) - P & Q
.: 2. Q (from 1)
ex. 1. i have a dog and a cat
therefore,
.: 2. i have a dog
4) affirming the antecedent (modus ponens)
- If P, then Q
- P
therefore,
.: 3. Q (from 1, 2)
ex. 1. if tmu is a good university, then many students apply there
2. tmu is a good university
therefore,
:. 3.many students apply there
5) denying the consequent (modus tollens)
- if P, then Q
- ~ Q
therefore,
.: 3. ~ P
ex. 1. if Jim ate a burger, then he wore red pants
2. Jim did not wear red pants
3. Jim did not commit the murder
6) hypothetical syllogism
- If A, then B
- If B, then C
.: 3. If A, then C (from 1, 2)
ex. 1.
2.
.: 3.
7) contraposition
- If P, then Q
.: 2. If ~ Q, then ~ P (from 1)
ex. 1. if Donald Trump loses the election, then Kamala Harris wins
.: 2. if Kamala Harris doesn’t win, then Donald Trump doesn’t lose the election (from 1 by contraposition)
8) universal modus ponens
- All A’s are B’s
- x is an A
- x is a B
ex. 1. all students are smart
2. Omar is a student
.: 3. Omar is smart (from 1 and 2 by universal modus ponens)
9) universal modus tollens
- All As are Ba
- x is not a B
- x is not an A
ex. 1. All students are hard-working
2. Omar is not hard-working
.: 3. Omar is not a student
10) universal hypothetical syllogism
- All A’s are B’s
- All B’s are C’s
.: 3. All A’s are C’s
ex. 1. All whales are mammals
2. All mammals are black
.: 3. All whales are black (from 1 and 2 by universal hypothetical syllogism)
11) universal ruling out
- No As are Bs
- x is an A
.: 3. x is not a B
ex. 1. no children are well-behaved
2. Jacob is a child
.: 3. Jacob is not well-behaved
invalid argument patterns (argument of elimination)
- P or Q
- P
- Q
- P or Q
- Q
- P
(left vs righty ex.)
- P v Q
- P
- ~ Q
- P v Q
- Q
- ~ P
(left or righty committed the crime example)
(we cannot exclude that (ex.) righty didnt help commit the crime)
in logic we don’t use exclusive “or”, but the inclusive “or”
invalid argument pattern (denying the antecedent)
- if P, then Q
- ~ P
.: 3. ~ Q
ex. 1. if Einstein is smart, then he has white hair
2. Einstein is not smart
.: 3. Einstein does not have white hair
invalid argument pattern (affirming the consequent)
- If P, then Q
- Q
.: 3. P
ex. 1. If Einstein invented the computer, then he’s a genius
2. Einstein is a genius
.: 3. Einstein invented the computer
(the premises don’t guarantee the truth of the conclusions)
invalid argument patterns (hypothetical syllogism)
- If A, then B
- If C, then B
.: 3. If A, then C - If A, then B
- If B, then C
.: 3. If B, then C - If B, then A
- If C, then B
.: 3. If B, then A - If C, then B
.: 3. If A, then C - If A, then B
- If D, then C
.: 3. If A, then C
invalid argument patterns (universal ruling out)
- All A’s are B’s
- x is not an A
- x is not a B
ex. 1. All students are kind
2. Jacob is not a student
.: 3. Jacob is not kind
- All A’s are B’s
- x is a B
.: 3. x is an A
ex. 1. All students are kind
2. Jacob is kind
3. Jacob is a student
inductive arguments
INDUCTIVE ARGUMENTS: aims to give probable, not conclusive, support for the conclusion
- shows that if the premises are true, the conclusion is likely true
cogency
COGENCY: an argument is cogent if and only if it is invalid, but all the premises give good/probable reason for the conclusion
- requiring no background information/assumptions
- if the premises are true, it is likely/probable for the conclusion to be false
- cogent arguments dont have to have true premises or true conclusions; what matters is their logical relationship
- an argument is cogent if and only if it is invalid, but if all the premises are true, the conclusion is likely to be true
- a cogent argument is invalid because even if the premises are true, they don’t guarantee the truth of the conclusion
cogent argument
ex. 1. Quitting smoking tens to improve one’s health
2. Mary has quit smoking
.: 3. Mary’s health will improve
(it is not guaranteed, but probable that her health will improve)
ex. 1. Boris is a student at NYU
2. Most NYU students voted
.: 3. Boris voted
ex. 1. most chairs have 10 legs
2. Justin Trudeau is a chair
therefore, probably,
.: 3. Justin Trudeau has 10 legs
this is an invalid argument because it is possible for the premises to be true, and the conclusion to be false
this is a cogent argument because the premises make it likely for the conclusion to be true
non-cogent argument
even if the premises are true, they don’t make it likely for conclusion to be true
ex. 1. a few police officers are corrupt
2. Jim is a police officer
3. Jim is corrupt
(Jim could be a part of the rest of the police officers who aren’t corrupt)
this is an invalid argument because it is possible for the premises to be true and the conclusion false.
this is a non-cogent argument because even if the premises were true, they don’t make it likely for the conclusion to be true.
the cogency test
1) imagine or suppose that the premises are all true. Is the conclusion likely to be true as well?
2) if the answer is “yes” its cogent
if the answer is “no” its non-cogent
validity vs cogency vs non-cogency (key words) (questions to ask) (valid or invalid)
validity:
- “always” “all”
- if the premises were true, is it guaranteed that the conclusion is true as well?
- always valid
- both validity and cogency are a feature that an argument either has or doesnt have. they both depend on the logical relationship between the premises and the conclusion.
cogency:
- “most of,” “a lot of,”
- if all the premises are true, would that make the conclusion probable?
non-cogency:
- “few”
- the truth of the premises neither guarantees the truth of the conclusion nor makes the conclusion probable
- invalid
valid arguments can have:
false premises, false conclusion
false premises, true conclusion
true premises, true conclusion
(remember, in valid arguments it is impossible for the premises to be true and the conclusion false)
cogent arguments can have:
true premises, true conclusion
true premises, false conclusion
one or more false premises, true conclusion
one or more true premises, false conclusion
(remember, in cogent arguments, it is possible for the premises to be true and the conclusion false)
common patterns of cogent arguments
- most As are Bs
- x is an A
therefore, probably,
.: 3. x is a B
ex. 1. most professors have PHDs
2. Kraay is a professor
therefore, probably,
.: 3. Kraay has a PHD
- x is an A
- x is a B
- most ABS are Cs
therefore, probabaly,
.: 4. x is a C (from 1, 2, 3)
ex. 1. Kraay is right-handed
2. Kraay is a professor
3. Most right-handed professors are nice.
therefore, probabaly,
.: 4. Kraay is nice
common patterns of non-cogent arguments
- most As are Bs
- x is not an A
therefore, probabaly,
.: 3. x is not a B
ex. 1. most professors are nice
2. Kraay is not a professor
therefore, probably,
.: 3. Kraay is not nice
- most As are Bs
- x is a B
therefore, probably,
.: 3. x is an A
ex. 1. most professors have beards
2. Kraay has a beard
therefore, probabaly,
.: 3. Kraay is a professor
ill-formed arguments
ILL-FORMED ARGUMENTS: arguments that are neither valid not cogent (they are invalid and not cogent)
- the premises do not guarantee that the conclusion is true or even probable
- ill-formed arguments are usually based on “assumptions”
- ex. 1. My teacher’s name is John
2. My teacher is a male
patterns of ill-formed arguments
- Most As are Bs
- x is not an A
- x is not a B
- most police officers are funny
- Jared is not a police officer
- Jared is not funn
- Most As are Bs
- x is a B
- x is an A
- Most robins can fly
- Tweet can fly
- Tweet is a robin
incomplete arguments
INCOMPLETE ARGUMENTS: arguments that are ill-formed but can be made cogent or valid by adding one obvious premise
summary notes about cogency and validity
validity:
- impossible for the premises to be true and the conclusion false
- the truth of the premises guarantee the truth of the conclusion
cogency:
- possible for the premises to be true and the conclusion false
- the truth of the premises make it probable that the conclusion is true
reminder
WELL-FORMED ARGUMENTS: an argument is well-formed if and only if the argument is valid or cogent
ILL-FORMED ARGUMENTS:
- (the truth of the premises neither guarantees nor makes probable the truth of the conclusion)
degrees of validity vs degrees of cogency
validity does not come in degrees
- an argument is either 100% valid, or invalid
- like a light switch
- an argument cannot be more valid than another
- if all the premises are true, then there’s no possibility for the conclusion to be false, and degrees require levels of possibility
- truth does not come in degrees, rather, cogency and possibility does
cogency (probability) comes in degrees
- an argument can be more or less cogent than another argument
- the premises of one cogent argument support the conclusion more strongly than do the premises of another argument
- like a fluid light switch
- ex. 40% rain vs 90% rain
- ex. “most students voted” is more cogent than “some students voted”
