Symbolizing & Truth Tables
The four types of arguments for conditional arguments
What are:
affirming the antecedent (valid, modus ponens)
denying the antecedent (invalid)
affirming the consequent (invalid)
denying the consequent (valid, modus tollens)
Symbolize the following, and name the type of argument and whether it's a valid argument form:
If it's cold, then I turn the heat on. I have the heat on. Therefore, it's cold.
P -> Q
Q
-------
P
Affirming the consequent (invalid)
Explain contradictory, contrary, and subcontrary
Contradictory: one sentence must be false if the other is true
"Every S is P" and "Some S are not P"
Contrary: Both can be false, but both cannot be true
"Every S is P" and "No S is P"
Subcontrary: Both can be true, but both cannot be false
"Some S is P" and "Some S is not P"
The hypothetico-deductive method
What is "an important method in science and everyday reasoning, in which a hypothesis is tested by deducing from it an observational prediction, and then determining whether or not that observational prediction is borne out"?
Main symbols: H & O
The three ways a sentence can relate to transitivity
What are transitive (a>b, b>c, a>c), intransitive (a>b, b>c, IS NOT POSSIBLE THAT a>c), nontransitive (a>b, b>c, idk)
The two other kinds of properties of relation: symmetry (symmetric, asymmetric, and nonsymmetric) and reflex (reflexive, irreflexive, and nonreflexive)
For symmetry, think same relationship to another thing
For reflex, think relationship to oneself (like reflexive verbs in language learning)
P->Q
Q->R
-------
P->R
Is this kind of argument
What is a hypothetical syllogism?
Truth table for:
(PvQ)->(P&Q)
P Q (PvQ) -> (P&Q)
-----------------------
T T T T T
T F T F F
F T T F F
F F F T F
The rules of distribution
Rule 1: Is the middle term distributed once?
Rule 2: Are the end terms in the conclusion distributed only if they are distributed in the premise?
Rule 3: Does the number (0 or 1) of negative conclusions equal the number of negative premises?
Deducing an observational prediction from a hypothesis requires the use of this
What are auxiliary hypotheses?
Symbol: A
Ex: proper testing conditions and theoretical background knowledge
The symbolic form of an O-type sentence
What is (∃x)(Sx & ~Px)?
Similarly:
I-type: (∃x)(Sx & Px)
A-type: (x)(Sx->Px)
E-type: (x)(Sx->~Px)
Put in standard form:
"I'll graduate only if I do my work"
What is "If I graduate then I've done my work"?
Remember: "only if" means "then." Suggests a necessary condition (which we know is the consequent). Could also be said that doing my work is a necessary condition for graduating.
Symbolize the following, and name the type of argument and whether it's a valid argument form:
"The game has been sold out unless it has been canceled. If it has been sold out, I won’t be able to see it, and if it has been canceled, I won’t be able to see it. Either way, I won’t be able to see the game." (8.9.2 #1)
PvQ
P->R
Q->R
--------
R
Valid: Constructive Dilemma
How you remove terms if a syllogism contains more than 3
What is find synonyms, obversion (changing quality and changing the predicate to its complement), and conversion (switch subject and predicate terms)?
Obverse ex: Every flower is pretty -> No flower is not pretty
Conversion ex: Some games are fun -> Some fun things are games
When using this method, you also need to consider this (NOT implicit assumptions)
What are alternative hypotheses?
Symbol: H'
Ex: If the original hypothesis is that there's snow outside because nighttime is lighter than usual, the alternative hypothesis could be that the street lights are brighter than usual.
Symbolize this statement:
Mary washes her hair
What is mWm?
This is reflexive! (x)xRx
Irreflexive: (x)~xRx
Nonreflextive: ~(x)(xRx) & ~(x)(~xRx)
Explain tautologies, self-contradictions, and contingent sentences
Tautologies: P->P
Self-Contradictions: P&~P
Contingent Sentence: (all else)
The two types of valid dilemmas
What are "constructive dilemma" and "destructive dilemma"?
Constructive dilemma: similar to affirming the antecedent
Destructive dilemma: similar to denying the consequent
Both need a disjunction in the premises and need 3 statements as premises.
Daily Double!!!!!!!
What is it called when you have a 2 premise argument with 1 premise being a disjunction and the other premise denying the truth of one of the disjuncts?
Make a Venn Diagram for:
"All dogs are cute. Some dogs are not quiet. Therefore, some cute things are not quiet"
Also note validity
See Board
The four rules of confirming an argument using the H-D method
What are:
1.) The hypothesis is initially plausible.
2.) If the hypothesis and auxiliary hypotheses are true, then the observational prediction is true.
3.) The observational prediction is in fact true.
4.) No alternative hypothesis has as high of a prior probability as the one being tested.
If these are met, then we can conclude a hypothesis is confirmed true
Symbolic express the property of symmetry
What is (x)(y)(xRy->yRx)?
Asymmetric: (x)(y)(xRy->~yRx)
Nonsymmetic: ~(x)(y)(xRy->yRx) & ~(x)(y)(xRy->~yRx)
These four rules
"1. The negation of a conjunction is logically equivalent to the disjunction of the negations of the conjuncts.
2. The negation of a disjunction is logically equivalent to the conjunction of the negations of the disjuncts.
3. The conjunction of two sentences is logically equivalent to the negation of the disjunction of their negations.
4. The disjunction of two sentences is logically equivalent to the negation of the conjunction of their negations."
Make up THIS
What is De Morgan's Laws
The symbols:
1. “~(p • q)” is logically equivalent to “~p v ~q.”
2. “,(p v q)” is logically equivalent to “~p • ~q.”
3. “p • q” is logically equivalent to “~(~p v ~q).”
4. “p v q” is logically equivalent to “~(~p • ~q).”
Truth table for:
P&Q
P
~Pv~Q
----------
Q
(8.9.1 #1)
p q ~p ~q ~p v ~q p & q
T T F F F T
T F F T T F
F T T F T F
F F T T T F
Valid
Use the distributive method to find the validity of:
All baristas support tipping, but some customers do not support tipping. Hence, some baristas are not themselves customers.
Conclusion means Baristas = S, Customers = P, Supports tipping = M
All S(d) is M
Some P is not M(d)
--------------------
Some S is not P(d)
(Remember how we place the distribution marker! A-type = distribute the S. E-type = distribute BOTH. I-type = nothing. O-type = distribute the P)
Rule 1: Yes, good
Rule 2: No, bad!
Rule 3: Yes, good
INVALID
Since Popper thought the scientific method was deductive rather than inductive, his H-D method looks different from the standard one. It looks like this.
Falsifiability
More specifically, he thought you can confirm pretty much anything (dubious) and irrefutability is a bad thing (means you're forcing it to be true). Speak in terms of corroboration not confirmation. Introducing ad hoc assumptions or reinterpretation in an ad hoc way if the original hypothesis is found to be false will save the theory BUT will "lower its scientific status"
Symbolize this argument:
Chris has more money than Susan. Nancy has more money than Chris. Therefore, Nancy has more money than Susan.
What is
(x)(y)(z)((xMy & zMx)->zMy)
(cMs & nMc)->nMs
cMs & nMc
---------------
nMs
This argument is a transitive one!
Intransitive: (x)(y)(z)((xRy & yRz)->~xRz)