Form and Function
This is a hypothetical syllogism.
If p, then q.
If q, then r.
Therefore, if p, then r.
This is the definition for a valid argument.
If all of the premises are true, then the conclusion must also be true.
This translates to: Logic is fun, but it is not easy.
P = logic is fun
Q = logic is easyP . ~ Q
This is a standard argument for: If I study for the test, then I will not be able to sleep. But without sleep, my brain won't work when I take the test. Then again, if I don't study, then I won't be prepared for the test. So, I damned if I do and I'm damned if I don't.
P = study
Q = not sleep
R = not prepared for test
If P, then Q
If Q, then R
Therefore, if P, then R
This comes down to this hypothetical syllogism.
But it is really an extended argument in which the above conclusion is the premise in a disjunction of sorts:If P, then R. (If study, then be unprepared)
If ~P, then R. (if no study, then be unprepared)
P or ~P (I must study or not study)
Therefore, R. (I will be unprepared)This is why the statement ends with "So, I am damned if I do and I'm damned if I don't."
This may be a valid form, and it may even be sound. But more than anything else, it reflects why cramming (waiting until the last minute to study) is a bad plan.
That is a venn diagram for:
Some valid arguments have false conclusions, but not all syllogisms are valid. So, some syllogisms have true conclusions.
S = Valid Arguments
M = Arguments with false conclusions
P = Syllogisms
Some S are M.
Some P and S.
Therefore, some P are not M.
The Venn Diagram will contain floating X's in the middle spaces between S/M and S/P, which makes the conclusion invalid.
This is a conditional.
If p, then q.
p
Therefore, q.
(What is the name of this valid argument?)This is an explanation of a false dilemma and its relationship to black and white thinking.
What is a dilemma in which the two choices presented are treated like the only two valid choices, thus obscuring the possibility of an alternative outcome, which is like the fallacy of false alternatives because there is at least a third choice that is ignored.
Example:
If I become a lawyer, I will be rich and unhappy.
If I become a teacher, I will poor and happy.
I must be a lawyer or a teacher.
Therefore I must either be rich and unhappy or poor and happy.
This translates to: If logic is easy, then it is fun.
P = logic is fun
Q = logic is easy
q -> p
Genetic diversity in plants protects against blight. Weeds have held on to their diversity and thus to their defenses.
1. Plants with genetic diversity
2. Plants that are protected against blight.
3. Weeds
All plants with genetic diversity are plants that are protected against blight.
Weeds are plants that have genetic diversity.
Therefore, weeds are protected against blight.
That is a truth table for:
p and q
p
not p or not q
Therefore, q
P ~P Q ~Q P.Q ~P v ~Q
T F T F T F
T F F T F T
F T T F F T
F T F T F T
You can now check to see if this is valid or not
This is a disjunctive syllogism.
p or q
not p
Therefore, q
This is the difference between a contrary and a contradiction.
Contrary = can't both be true, but can both be false.
Contradiction = one true, and one false
Logic is fun if and only if it is easy.
P = logic is fun
Q = logic is easy
q <-> p
If a ruler who wants always to act honorably is surrounded by many unscrupulous men, his downfall is inevitable. Therefore, a ruler who wishes to maintain his power must be prepared to act immorally when this becomes necessary (i.e. when surrounded by unscrupulous men.)
P = desire to act honorably amongst unscrupulous men
Q = can maintain his power
P -> ~Q
Therefore, ~P -> Q
Is this valid? You can check with a truth table that might look like this:
P ~P Q ~Q P -> ~Q ~P -> Q
T F T F F T
T F F T T T
F T T F T T
F T F T T F
Since we are concerned if we see T premises and a F conclusion, we are really looking at the last two rows. Low and behold, in the last line we see that the premise is T but the conclusion is F, so the argument form is invalid. (Way to go, Machiavelli.)
That is a Venn diagram for: None but the rich can afford to be idle. Anyone who can afford to be idle is a threat to the Puritan work ethic, so some rich persons threaten the Puritan work ethic.
All who can afford to be idle are rich.
All you can afford to be idle are a threat to Puritan work ethic.
Therefore, some rich threaten the puritan work ethic.Or
All S are M.
All S are P.
Therefore, some M are P.
Now, you fill in the circles to check the validity.
This is a destructive dilemma.
if r, then s.
not q or not s
therefore, not p or not r
(What form does this mimic?)
This is an explanation for the fallacy of the undistributed middle.
What is the problem when the middle term (the one that does not appear in the conclusion) of a syllogism is not found in the predicate of one premise and the subject of the other.
Fallacy Example:
All S are M.
All P and M.
Therefore, all S are P.In this example, the middle term "M" is only found the in the predicate position of both premises. But, it must be distributed between the predicate and subject position of the premises.
Example of good distribution:
All S are M.
All M are P.
Therefore, all S are P.
Logic is fun only if it is easy.
P = logic is fun
Q = logic is easy
P -> Q
Austin must be rich since he drives a Porsche.
Categories:
1. Austin
2. Rich people
3. People who drive Porsches.
Austin drives a Porsche.
(All people who drive Porsches are rich.)
Therefore, Austin must be rich.
Truth table for: All forms of pantheism which involves the belief that man is a part of God must be rejected, because if man is actually a part of God, the evil in man is also in God, but there is not evil in God.
P = man is a part of God
Q = Man evil is a part of God
P -> Q
~Q
Therefore, ~P
This is denying the consequent. Is this valid or not? Can you see this when you create the truth table?
This is an indirect argument or proof by contradiction.
p
if p, then q.
not q.
Therefore, not p.
This is the explanation of how you can represent both a sufficient and necessary cause using conditionals.
P = Studying is challenging.
Q = Studying is productive.
If studying is challenging, then it is productive.
(P -> Q)
This shows that studying being challenging is a sufficient condition for it being productive. But, it is not necessary because there could be other types of studying that are productive. But it is to say that if the studying is challenging then you are guaranteed that it is also productive.
If I want to express challenge as a necessary condition of studying, I can write this in a couple of ways:
If studying is productive, then it is challenging OR Studying is productive only if it is challenging.
(Q -> P)
Now I am saying that any time studying is productive, it is also challenging. There is no productivity without challenge.If I want to express that each are a necessary and sufficient condition of each other, then I would say:
Studying is productive if and only if it is challenging. ( Q<-> P)
Which is the same as saying:
Studying is challenging if and only if it is productive. (P <-> Q)
Logic is fun when it is easy, but not always easy when it is fun.
P = logic is fun
Q = logic is easy
Q -> P and ~(P -> Q)All reporters are aggressive. Only reporters have daily deadlines. So no non-aggressive persons have daily deadlines.
Categories:
1. Reporters
2. Aggressive People
3. People with Daily Deadlines
All reporters are aggressive people.
All people with deadlines are reporters.
Therefore, all persons who have daily deadlines are aggressive. (Through conversion and obversion of: No non-aggressive persons have daily deadlines.)We did not cover obversion and conversion together, so it will not be on the test. But they are ways to translate sentences into other forms while preserving their original meaning.
Venn diagram for: Some moral beliefs transcend the cultural context in which they arise. Beliefs that transcend their cultural contexts are not relative to the culture in which they arise. Therefore, some moral beliefs are not relative to the culture in which they arise.
S = Moral beliefs
M = Beliefs that transcend
P = Not relative
Some S are M.
All M are P.
Therefore, some S are P.
We did this in class! Refer to your notes for the correct circles.