Philosophy: Intro to Logic

Weak; Uncogent (true premise; not very likely conclusion)

A few of my grandparent's pets are dogs. Therefore, the next pet they buy will probably be a dog.

the truth value of a biconditional is true when both premises have the same truth value example: Emma Watson is a soccer player if and only if Barney is a blue bird. E ≡ B > F ≡ F > T Emma Watson is an actress if and only if Barney is a blue bird. E ≡ B > T ≡ F > F

Biconditional Truth Table Rules: When is a biconditional true? False?

the truth value of a conditional is only false when the first premise is true and the second premise is false example: If Nicole Kidman is an actress, then so is George Bush. N ⊃ G > T ⊃ F > F If Bill Maher is a show host, then so is Jimmy Fallon. B ⊃ J > T ⊃ T > T

Conditional Truth Table Rules: When is a conditional true? False?

the truth value of a conjunction is only true when both premises are true

Conjunction Truth Table Rules: When is a conjunction true? False?

the truth value of a disjunction is only false when both premises are false example: Either humans are dogs, or dogs are humans. H v D > F v F > F Either Tim Duncan or Kawhi Leonard is a basketball player. T v K > T v T > T

Disjunction Truth Table Rules: When is a disjunction true? False?

Strong; Uncogent (false premise; semi-likely conclusion) *strong because assuming the validity of the premise, the conclusion is highly likely

Every grown adult is married. Therefore, it is probable that when I am a grown adult I will be married.

Answer: L = 2^n = 2^2 = 4 (A v ~ B) ⊃ B T F t T T T f F F F t T F T f F check for lines with true premises and a false conclusion. invalid (line 2)

How do you determine the validity of a proposition using truth tables? use this example:(A v ~ B) ⊃ B

the tilde makes the truth value of the premise opposite example: if p is false, then ~p is true

Negation Truth Table Rules: When is a negation true? False?

B ≡ R

Rewrite using operators B if and only if R.

D • C

Rewrite using operators D and C

P v E

Rewrite using operators Either P or E

N ⊃ F

Rewrite using operators F if N. OR If N then F.

~A

Rewrite using operators It is not the case that A.

Strong; Cogent (true premises; highly likely conclusion)

Strong or weak? Cogent or uncogent? Because every former U.S. president was older than 40, it is probable that the next U.S. president will be older than 40.

Invalid; Unsound (true premises; false conclusion)

Valid or invalid? Sound or unsound? All dogs are animals. All cats are animals. Therefore, all dogs are cats.

Valid; Sound (true premises; true conclusion)

Valid or invalid? Sound or unsound? Since San Antonio is a city in Texas, and Texas is part of the United States, it follows that San Antonio is a city in the United States.

Valid; Unsound (one false premise; true conclusion) *valid because assuming the validity of the premises, the conclusion is true

Valid or invalid? Sound or unsound? The Mississippi River flows through California and California is a part of the United States. Therefore, the Mississippi River is a river in the United States.

Answer: procedure found on pg. 335 1. [ ~ ( X v Z ) • ( B v ~ A ) ] ⊃ ~ ( Z ⊃ ~ C ) 2. F F T T F T 3. F F T F t F F t 4. f F f t T f f T f 5. T f T F t 6. t T t F 6.2. what's left: T ⊃ F = False

What are the steps for computing the truth value of a longer (or complex) proposition? use this example: A, B, and C are true; X and Z are false) [~(X v Z) • (B v ~ A)] ⊃ ~ (Z ⊃ ~ C)

to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own

What is the aim, or purpose, of logic?

Dot (•)

What operator is this? Logical function: conjunction Used to translate: and, also, moreover, but, however, yet, still, although, nevertheless

Wedge (v) *it's a down arrow, not really a v

What operator is this? Logical function: disjunction Used to translate: or, unless

Triple Bar (≡)

What operator is this? Logical function: equivalence Used to translate: if and only if, is a sufficient and necessary condition for

Horseshoe (⊃)

What operator is this? Logical function: implication Used to translate: if...then..., only if

Tilde (~)

What operator is this? Logical function: negation Used to translate: not, it is not the case that

Valid Deductive Argument

an argument in which it is impossible for the conclusion to be false given that the premises are true

Strong Inductive Argument

an argument in which it is improbable that the conclusion be false given that the premises are true

Invalid Deductive Argument

an argument in which it is possible for the conclusion to be false given that the premises are true

Weak Inductive Argument

an argument in which the conclusion does not follow probably from the premises, even though it is claimed to

Deductive Argument

an argument incorporating the claim that it is impossible for the conclusion to be false given that the premises are true

Inductive Argument

an argument incorporating the claim that it is improbable that the conclusion be false given that the premises are true

Argument from Authority

an argument that concludes something is true because a presumed expert or witness has said that it is

Argument from Analogy

an argument that depends on the existence of an analogy, or similarity, between two things or states of affairs

Casual Inference

an argument that proceeds from knowledge of a cause to a claim about an effect, or, conversely, from knowledge of an effect to a claim about a cause

Prediction

an argument that proceeds from our knowledge of the past to claim about the future

Generalization

an argument that proceeds from the knowledge of a selected sample to some claim about the whole group

Argument Based on Signs

an argument that proceeds from the knowledge of a sign to a claim about the thing or situation that the sign symbolizes

Statement Form

an arrangement of statement variables and operators such that the uniform substitution of statements in places of the variables results in a statement

Truth Table

an arrangement of truth values that shows in every possible case how the truth value of compound proposition is determined by the truth values of its simple components

Cogent Argument

an inductive argument that is strong and has all true premises

Uncogent Argument

an inductive argument that is weak, has one or more false premises, fails to meet the total evidence requirement, or any combination of these

Truth Function

any compound proposition whose truth value is completely determined by the truth values of its components

L = 2^n where L is the number of lines and n is the number of different simple propositions (capital letters)

formula used to determine how many lines a truth table will consist of

Statement Variables

lowercase letters (p, q, r, s) that can stand for any statement

SUN (S ⊃ N)

mnemonic device used to remember that whatever is given as a sufficient condition goes in front of the horseshoe, and whatever is given as a necessary condition goes after.

Operators (Connectives)

special symbols that connect two or more statements in a grammatically valid way

Truth Values

the characteristic a statement possesses as to whether it is true or false

Consequent

the component of a conditional statement that is the "...then _____" factor

Antecedent

the component of a conditional statement that is the "if ____..." factor

Conjuncts

the components of a conjunctive statement

Disjuncts

the components of a disjunctive statement

Biconditional Statement (Biconditional)

the logical function of the statement "B ≡ R"

Conjunctive Statement (Conjunction)

the logical function of the statement "D • C"

Conditional Statement (Conditional)

the logical function of the statement "N ⊃ F"

Disjunctive Statement (Disjunction)

the logical function of the statement "P v E"

Negation

the logical function of the statement "~A"

Proposition

the meaning or information content of a statement; interchangeable with "statement"

Main Operator

the operator that has as its scope everything else in the statement

Logic

the organized body of knowledge, or science, that evaluate arguments

Inference

the reasoning process expressed by an argument

Material Equivalence

the relation of which a biconditional statement expresses

Material Implication

the relation of which a conditional statement expresses

Conclusion

the statement that is claimed to follow from the premises

Premises

the statements that set forth the reasons or evidence

Logically Equivalent Propositions

these propositions necessarily have the same truth value

D ⊃ T

using the mnemonic device SUN, rewrite this argument with operants. Getting Tucker's shots is a necessary condition for his stay at doggie-daycare.

B ⊃ A

using the mnemonic device SUN, rewrite this argument with operants. Getting a B on the final is sufficient for Jerry to receive an A in the class.

Sufficient Condition

when the event A is all that is required for the occurrence of event B; i.e. having the flu (A) is a sufficient condition for feeling miserable (B), however there are many other ways to feel miserable. Event A is just sufficient enough to validate B.

Necessary Condition

when the event B cannot occur without the occurrence of even A; i.e. having air to breathe (A) is a necessary condition for survival (B). In this case, without event A, event B is not possible.

Unsound Argument

a deductive argument that is invalid, has one or more false premises, or both

Sound Argument

a deductive argument that is valid and has all true premises

Propositional Logic

a form of logic in which the fundamental elements are whole statements (or propositions)

Argument

a group of statements, one or more of which (the premises) are claimed to provide support for, or reasons to believe, one of the others (the conclusion)

Syllogistic Logic

a kind of logic in which the fundamental elements are terms, and arguments are evaluated as good or bad depending on how the terms are arranged in the argument

Modal Logic

a kind of logic that involved such concepts as possibility, necessity, belief, and doubt

Statement

a sentence that is either true or false

Simple Statement

a statement that does not contain any other statement as a component

Compound Statement

a statements that contains at least one simple statement as a component

Categorical Syllogism

a syllogism in which each statement begins with one of the words "all," "no," or "some."

Well-Formed Formulas (WFFs)

a syntactically correct arrangement of symbols