Logical Argumentation Final Exam 2015

Major Term

"P" term (predicate). Major term is always the predicate term of the conclusion.

Disjunction

"p v q" is true, if "p" is true, or "q" is true, But if both "p" is false and "q" is false then "p v q" is false. In other words, at least one statement has to be true in order to not be false.

Commutation

( p & q) = (q & p) (p v q) = (q v p) _______________________________________ similiar to math; X times Y = Y times X. - One of the Equivalence Rule in Logic

Biconditional

( p <=> q) = [ (p > q) & (q > p) ] = [(p & q) v (~p & ~q) _________________________ biconditional is "p if and only if q". A biconditional is true only if both components have the same truth values. Normally have to replace the biconditional with another equivalence rules.

Contraposition

( p > q ) = ( ~q > ~p ) > = the horseshoe ______________________ We use contraposition in proofs when it is easier to work with the contrapositive of the statement than with the statement itself ____________________________________ Example: - If it rained last night, the ground is wet R > W - if the ground is not wet, it did not rain last night ~W > ~R

Implication

( p > q ) = (~p v q ) = ~(p & ~q) ______________________________ The conditional cannot be true if its antecedent is true and its consequent is false. _________________________________________ A) If I stay home for dinner, I'll have to listen to my uncle doing his Elvis impressions S > L B) Either I don't stay for dinner or I'll have to listen to my uncle doing his Elvis impression ~S V L (page 336) C) I cannot stay for dinner and not listen to my uncle doing his Elvis impression. ~(S . ~L )

Consistent

(On Truth Tables) 2 Truths in the same row of 2 different propositions. Making them both possible to be true at the same time. Reference columns are set up normally as T T F F and T F T F.

Equivalent

(On Truth Tables) Identical rows in the final column. Propositions have the exact same T's and F's. Reference columns are set up normally as T T F F and T F T F.

Inconsistent

(On Truth Tables) Never true in same rows of 2 different propositions. Reference columns set up normally as T T F F and T F T F.

Reference Column

(On truth tables) always set up as p: T T F F , q: T F T F at the beginning of your truth tables. ALWAYS REQUIRED.

Rules of Proofs

- Each step must be a valid inference -All steps can be proven valid by truth-tables

Figure

1 , 2 , 3, 4

Categorical Proposition

An assertion about relations amongst classes. "Animal is a genus, mammals are animals, whales are mammals."

Appeal to Ignorance

Asserting that we should believe a claim because it has not been proven.

Diversion

Changing the issue in the middle of the argument.

The Ten Equivalence Rules

Commutation (Comm.), Association (Assoc.), Tautology (Taut), Double Negation (DN), De Morgan's Law (DM), Distribution (Dist.) , Contraposition (Contra) , Implication (Imp) , Biconditional (Bicon) , Exportation (Exp) .

Four Formal Relationships

Consistent , Equivalent , Contradiction , Inconsistent. Used for two propositions in a truth table.

Disjunctive Syllogism

From a disjunction and the negation of one disjunct, the other disjunct can be derived. (Or switch the -p with a -q)

Particular Statements

Have existential imports. Their truths does not depend on existence of S's. categories (I and O).

Simplification

Highlights the fact that all conjunctions are composed of singular atomic fats. If the conjunction is true, the conjuncts must BOTH be true. We can thus pull out one conjunct from any conjunction.

Valid argument

If premises are true, then conclusion is true.

Inductive Argument

Inductive reasoning (as opposed to deductive reasoning or abductive reasoning) is reasoning in which the premises seek to supply strong evidence for (not absolute proof of) the truth of the conclusion. Strong and/or Weak argument.

Sound argument

Is an argument that is not only valid, but begins with premises that are actually true (facts).

Consequence

Is represented by "q" generally in logical problems. It is the statement that comes after the antecedent. Example: "if p then q" or "if I don't study then I won't pass". The consequence here is "I won't pass".

Antecedent

Is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause. Example: "if p then q" . It is represented by "p" generally.

Universal Statements

Lack existential imports. Their truths do not depend on existence of S's. categories (A and E)

Self-Contradictions Truth Table

Logically false. Using the example problem "p & ~(q > p)" the truth table will look like..."p": T T F F , "&": F F F F , "~": F F T F , "(q": T F T F , ">": T T F T , "p)": T T F F . Another example of self-contradiction is "p & ~p" or "q & ~q" .

The Nine Basic Inference Rules

Modus Ponens (MP) , Modus Tollens (MT) , Hypothetical Syllogism (HS) , Disjunctive Syllogism (DS) , Simplification (Simp) , Conjunction (Conj.) , Addition (Add) , Constructive Dilemma (CD) , Destructive Dilemma (DD) .

Subject

S (Some S are P, All S are P, etc.)

Affirmative

Saying some is definitely existent

Negative

Saying they are not

Minor Term

The "S" term (subject). "Minor leagues before the Major leagues". Minor premise has to come before the major premise.

Negated Truth Table

The "~p" is always the opposite of "P". When a negation is in a Truth Table make sure you have the non-negated form too. For example, if the table has "~( p v q)" , make sure you have "( p v q)" as a column too.

Constructive Dilemma

The disjunction of the antecedents of any two conditionals allows for the derivation of the disjunction of their consequent. This one is tricky. 1) Remember MP (Modus Ponens), which tells us that we have a conditional as one premise (p > q) and the antecedent of the same conditional as another premise (p) we are able to derive the consequent (q). 2) Here, we have two conditionals ( p > q and r > s) AND one of the two antecedents (p v r). We should be able to run MP on at least one of the conditionals, given that we know at least one of the antecedents. Thus, at least one of the two consequents (q v s) will follow.

Destructive Dilemma

The disjunction of the negations of the consequents of the two conditionals allows for the derivation of the disjunction of the negations of their antecedents.

Modus Ponens

The rule of logic stating that if a conditional statement ("if p then q ") is accepted, and the antecedent ( p ) holds, then the consequent ( q ) may be inferred.

Division

The same as composition, but going the opposite way. Can't assume the whole attributes to the parts.

Singular Terms / Singular Proposition

The subject term is a name, pronoun, or phase standing for a single subject.

Middle Term

The term that appears twice in both premises. M never should appear in conclusion. Example: No S are M Some M are P _________________ Some S are not P

Tautology Truth Table

There are no F's in the final column.

Truth Tables

Truth-tables methods demonstrate an argument's validity. This method first tries to show that an argument is invalid. If it fails to show it is invalid, we conclude that it is valid.

Hypothetical Syllogism

We can derive a conditional from the two we already have, provided the antecedent of one of the conditionals is the same as the consequent of another. (Also called a chain argument).

Conjunction

We can put two lines of deduction together to form a conjunction. This works together with rule #7 addition. "p & q" is true if both "p" and "q" are true. p q _____ p & q OR q p ______ q & p

Composition

When a feature of the part(s) of something are erroneously attributed to the whole.

Conditional Truth Table

When the antecedent is True and the consequence is False, it is Invalid. "p": T T F F , "q" : T F T F , "p > q": T F T T . DO NOT CONFUSE WITH BICONDITIONAL TRUTH TABLE!

Equivocation

When words switch meaning in the argument. When a word expresses one concept in one premise and expresses a different concept in another premise or in the conclusion.

Exportation

[ (p & q) > r ] = [ p > (q > r) ] _______________________________ Allows us to isolate p as an antecedent; if we have already derived "p", we can then infer "q > r". Conversely, if we have the statement of the form "p > (p > r)" , this rule allows us to isolate "r" as a consequent.

Association

[ p & (q & r)] = [ (p & q) & r] AND [ p v ( q v r ) = [ (p v q) v r] _____________________________ Whereas the Comm. rule says that the ORDER of conjuncts and disjuncts makes no difference in truth value. The Association rule says GROUPING makes no difference. Equivalence Rule in Logic.

Distribution

[ p . (q v r) ] = [(p . r) v (p . r) OR [p v (q . r)] = [(p v q) . (p v r) ______________________________ Distribute the common letter outside of the ( ) to the letters inside the parantheses. Can be applied to conjunction and disjunction.

Tautology

p = (p & p) p = (p v p) ______________________________ Example: Imagine a child given the choice of "You can clean your room, or you can clean your room." <<< p = (p v p). Equivalence Rule in Logic.

Double Negation

p = ~~p ______________________ The two signs on the right cancel out. You can use Disjunctive Syllogism and Double negation together. Equivalence Rule in Logic.

Disjunction Truth Table

p: T T F F , q: T F T F , p v q: T T T F .

Truth Tables for two variables (reference table)

p: T T F F , q: T F T F . Always put at beginning of truth tables. (put T's and F's going down a row)

Conjunction Truth Table

p: T T F F , q: T F T F, p & q: T F F F

Hypothetical Syllogism Truth Tables

p: T T T T F F F F , q: T T F F T T F F , r: T F T F T F T F , (premise 1) "p > q": T T F F T T T T , (premise 2) "q > r": T F T T T F T T , (conclusion) "p > r": T F T F T T T T .

Truth Tables for Three variables

p: T T T T F F F F , q: T T F F T T F F , r: T F T F T F T F .

Modus Tollens

the rule of logic stating that if a conditional statement ("if p then q ") is accepted, and the consequent does not hold ( not-q ), then the negation of the antecedent ( not-p ) can be inferred.

De Morgan's Law

~ (p & q) = ~p v ~q ~ (p v q) = ~p & ~q ___________________ DM's law is useful because it allows you to get rid of negated compound statement in a proof. __________________________________________ ~ (p v q) = Cheating is NEITHER honest NOR smart. (English example) ~p . ~q = Cheating is NOT honest, AND cheating is NOT smart *** Equivalence Rule ***

Connectives

•Compound statements are formed by using connectives to combine simple statements. •Compound statements include disjunction, conjunction, conditionals, and negation . . .

Mood

A , E , I , O

Deductive Argument

A deductive argument is an argument that is intended by the arguer to be (deductively) valid, that is, to provide a guarantee of the truth of the conclusion provided that the argument's premises (assumptions) are true. Valid and Sound Argument.

Existential Import

A statement only has existential imports only if the truth depends on the existence of things in certain categories.

Traditional Square of Opposition

A: All S are P E: No S is P I: Some S are P O: Some S are not P

Addition

Addition tells us that p entails p v q, and q entails p v q. No matter what claims p and q might stand for, if p is true, then (either p or q) must be true. The truth of one disjunct is all it takes to make the whole disjunction true. p ____ p v q OR q ____ p v q