Propositional Logic

Question 1

Propositional Logic is also known as

Answer 1

• Modern logic
• Symbolic logic
• Boolean Logic

Question 2

what was the main early logician and head of stoic academy

Answer 2

Chrysippos

Question 3

describe Zeno the Stoic Philosopher

Answer 3

Zeno ff301BC
* Zeno Famous for logical paradoxes
-“This statement is false”
-First examples of “reductio ad absurdum” (reduce to absurity) Arguments
-realized infinity is a complex concept: arrow going to target has distances it goes before hitting target
-Zeno taught Philosophy under the “Stoia Poikile” (Painted Porch) giving the school the name its name

Question 4

what was the fate of the stoics

Answer 4

• This school of philosphy persisted until 529 AD when the Christian Emperor Justinian ordered all philosophy schools closed (bc he thought they were asking too many questions)
• We know that several hundred Texts were written by stoic logicians, none survive

Question 5

describe stoic Philosphers

Answer 5

Were adamant that reason, rather than passion was the correct attitude for someone seeking wisdom (and happiness)
-“freedom is secured not by the fulfilling of one’s desires, but by the removal of desire” (Epictus)
* thus in modern English, we call someone Stoic who is dispassionate, especially in the face of hardship (had to be good at logic to use as their reasoning)

Question 6

How does Propositional logic vs Predicate logic differ in who dictated them?

Answer 6

Propositional: Stoics
Predicate: Aristotle

Question 7

How does Propositional logic vs Predicate logic differ in allowable expressions?

Answer 7

Propositonal - more variety allowable expressions
Predicate - very limited expressions

Question 8

How does Propositional logic vs Predicate logic differ in forms?

Answer 8

Propositional -Arguments have variable forms
Predicate - arguments have fixed form (3 line syllogisms)

Question 9

How does Propositional logic vs Predicate logic differ in operators?

Answer 9

Propositional -allows more than one operator
Predicate -allows only one operator set ‘to be’

Question 10

How does Propositional logic vs Predicate logic differ in proofs?

Answer 10

Propositional -less simple proofs
Predicate -very simple proofs for validity

Question 11

Propositional and predicate logic were both:

Answer 11

• Scorned and burned by the early christians
• Rediscovered

Question 12

How does Propositional logic vs Predicate logic differ in when they were rediscovered?

Answer 12

Propositional -rediscovered in late 19th Century (copies originally written in arabic were found and translated)
Predicate - rediscovered in early medieval period

Question 13

what are atomic propositions

Answer 13

The simplest possible propositions (bc scientists gave the name atom to what they thought was the smallest thing to exist)
ex: I am walking = Atomic
-I am walking and texting = not atomic
-I am walking and I am texting = two propositions conjoined
* a statement is atomic if no logical connectives are in the statement: words like “and, or, if -then, but, not”, any logical words = NOT atomic

Question 14

what is the relationship for the propositional operator “and” and its symbol

Answer 14

• conjunction
• &

Question 15

what is the relationship for the propositional operator “or” and its symbol

Answer 15

• disjunction
• v

Question 16

what is the relationship for the propositional operator “if… then” and its symbol

Answer 16

• conditional
• ->

Question 17

what is the name for the propositional operator “not” and its symbol

Answer 17

• negation
• ~

Question 18

Truth Values

Answer 18

• Every statement is either true or false
• In ordinary language we represent a statement as true by merely writing, speaking etc
• by convention, “it is raining” means:
  -it is true that it is raining, if no negation > true is assumed
  -to deny, we negate: “it is not raining”

Question 19

describe propositional notation

Answer 19

any atomic statement can be represented by a single letter (usually uppercase)
ex: The statement Phil is a Philosophy prof can be represented by P
-to say that it is true that Phil is a Philosophy Prof, we merely write “P”: read as “P is true”
-to deny that phil is a philosophy prof we write not-P or ~P
-read as “P is false”

Question 20

what are the logical words to look for that must separate letters in propositional notation

Answer 20

And, or, but, not, if -then, therefore

Question 21

Truth tables

Answer 21

• Using a truth table we can represent all the possible truth values for any combination of statements
• One row represents one possible arrangement of truth values
• for one statement, there would be 2 rows
• for two statements (2 variables) this would double to 4 rows
• for three statements (3 variables) this would double to 8 rows
• the rightmost variable is alternated singly, the 2nd rightmost is alternated doubly and so on
• ex: if you are evaluating the statement “P > (P and Q)

Question 22

Uses for truth tables

Answer 22

• define operators
• analyze statements -statements have different possible truth values
• analyze arguments - valid arguments never have a false conclusion with all true premises

Question 23

what do you do for a truth table for negation

Answer 23

A negated statement has the opposite truth value to the un-negated statement
ex: a row with P at the top would be the possible values, a row with ~P would be the result (with opposite values)

Question 24

what do you do for a truth table for conjunction

Answer 24

• a conjunction is true (P & Q) when both conjuncts are true, otherwise false
• in other words the only time P & Q is true is when both P & Q rows are separately true.
• symbol for conjunction is the ampersand: &

Question 25

what do you do for a truth table for disjunction?

Answer 25

* A disjunction is true when either or both disjuncts are true, otherwise false (at least one must be true for P v Q to be true) * symbol is v

Question 26

How do you determine whether or in a statement means: one or the other, or both (inclusive sense of or), or one or the other but not both (exclusive sense)

Answer 26

we use the principle of charity to pick the more inclusive sense that its one or the other or both (more reasonable interpretation)

Question 27

describe conditional statements

Answer 27

* statement form: if antecedent then consequent * interpretation = "material conditonal" -don't assume any actual causal relationship

Question 28

what do you do for truth tables for conditionals?

Answer 28

* A conditional is true in every case except when the antecedent is true and the consequent is false * symbol is ->

Question 29

describe the katniss analogy for conditionals

Answer 29

* Consider betting on katniss your bet is expressed as a conditional * if the arrow hits the target you win the prize * when is the deal broken? -only when the arrow hits and you don't get the prize, you can say the bet is broken -if you get the prize for no good reason or for some other bet, when katniss doesn't hit the target, it's still true that you would get a prize if she did, bc its unrelated -if the arrow misses and you don't get the prize the bet is still true bc if she did hit it you still would get a prize

Question 30

how can categorical A form statements: All A's are B's be converted to conditional statements?

Answer 30

* If A (member of A) then B (member of B)

Question 31

How can categorical E form statements: No A's are B's be converted to conditional statements

Answer 31

Not (A then B) ~(A -> B)

Question 32

describe how biconditional statements: "If and only if" work in propositional logic

Answer 32

* A if and only if B means (If A then B) and (If B then A) * recall how "only" worked in categorical logic: -Only A's are B's = All B's are A's -the bi-conditional is the same as equivalence symbolized by the tribar: ≡

Question 33

describe the relationship and symbol for "if and only if"

Answer 33

* Equivalence * ≡

Question 34

How do you do a truth table for equivalence (A if and only if B)?

Answer 34

* (P -> Q) & (Q ->) * Work from inside brackets to out treat (P->Q) and (Q-> P) as separate if thens, only false when P is true and Q is false * then between the results of those two below the & in the top of the table, use conjunction/and rule, only true when both are true * then in last column (P≡ Q) column, copy results of &? -fact check this, -rows with same truth values are equivalent

Question 35

compound statements can be analyzed for their possible truth values, there's 3 possible outcomes, what are they?

Answer 35

* some statements can never be true: -contradictions (P & ~P) * Some statements can never be false: -Tautologies (P v ~P) * Some statements can either be true or false: -contingent (P v Q)

Question 36

contradictions (P & ~P)

Answer 36

can never be true, (outcomes always false)

Question 37

Tautologies (P v ~P)

Answer 37

Can never be false

Question 38

Contigent (P v Q)

Answer 38

can either be true or false

Question 39

Propositional arguments

Answer 39

Arguments are any number of statements intended to support a conclusion * Notational convention -each premise gets its own line -each premise is numbered -conclusion added to end of last line after "/ and 3 dots" (means therefore)

Question 40

statements vs arguments

Answer 40

* Statements and arguments are inter-changeable * any argument is a statement formed by a conjunction of the premises being the antecedent of a conditional with the conclusion being the consequent. (Must have if then or therefore)

Question 41

a valid argument presented as a single statement will be a

Answer 41

* tautology (a statement that can never be false) * (final column all T)