(logic) symbols and translations (6.1)

key words for disjunctions

"or," "either-or," "unless," "Unless," "or"

aspirin relieves aches and ibuprofen relieves pains, or burn cream relieves burns

(A * I) v B

if either (avarice is a vice) or (temperance is a virtue), then neither (fortune favors the foolish) nor (the glass is half full)

(A v T) > ~(F v G)

either burn cream relieves burns and aspirin relives aches or ibuprofen relieves pains

(B * A) v I

either burn cream relieves burns and ibuprofen relieves pains or aspirin relieves aches

(B * I) v A

M if C

C > M

if D, then B

D > B

L is a necessary condition for E

E > L

only if L, F happens

F > L

translate "p only if q" as

P > Q

should translate the statement "today is W if tomorrow is T" to

T > W

B, provided that V

V > B

only if H, W happens

W > H

disjunction

an either-or statement; should translate this from ordinary language into propositional logic notation using the wedge; "either" is often optional in ordinary language disjunctions - but may serve as a punctuational role in compound statements containing disjunctions; assumed to be inclusive - meaning that a disjunction is true if either/both of the two statements being connected are true

>, or horseshoe

an implication, meaning "if... then," "only if," or "implies;" in front of sentences it connects

main operator

connects the largest statement components in a propositional logic sentence

conditional

contains a hypothetical supposition; translated from ordinary language into a propositional logic notation with the horseshoe; the statement to the left of the horseshoe is the antecedent and the statement to the right is the consequent

compound statement

contains at least one simple statement as a component

compound statements containing one-or-more negations

correctly translating these is dependent upon correct placement of the tilde/s within the compound expression (a misplaced tilde can change the meaning of a proposition in logical notation so that it no longer conveys the meaning of the ordinary language statement it is intended to represent)

simple statement

does not contain any other statement as a component

(Z) * ~(R v F)

dot is the main operator, because it governs the components (Z) and ~(R v F); the tilde isn't, because it only governs that portion of the statement within the preferences (R v F)

it is not the case that both (temperance is a virtue) and either (love is eternal) or (avarice is a vice)

~[T * (L v A)]

examples of WWFs

~[~(K>R)=~(D>E)]; ~S>(Sv~S); ~{~[~(~E>~R)=~E]v(U=S)}; [~Lv(L*S)]=~W

if burn cream's relieving burns implies that aspirin relieves aches, then ibuprofen relieves pains

(B > A) > I

if burn cream's relieving burns implies that ibuprofen relieves pains, then aspirin relieves aches

(B > I) > A

jerry is a student, or jerry is a professor

(j) v (p), where (j) represents "jerry is a student" and (p) represents "jerry is a professor"

provided that A, you are T

A > T

aspirin relieves aches or both ibuprofen relieves pains and burn cream relieves burns

A v (I * B)

C is a sufficient and necessary condition for O

C = O

given that D, F is happening

D > F

(love is eternal) if and only if neither (the glass is half full) nor (temperance is a virtue)

L = ~ (G v T)

statement

a sentence/sentence-component with a truth value - meaning its either true or false

conjunction

a statement that claims two statements are both true; should translate ordinary language into propositional logic notation with the dot * operator; should always place the dot BETWEEN two statements that the dot * is connecting; can connect simple statements/negations/compound statements inside parenthesis

compound statement contains at least

one simple statement as a component

negation

statement that claims some statement is false

antecedent of a conditional represents

sufficient condition

B is a sufficient and necessary condition for U

B = U

example of a WFF

~ (S > ~I) * [(X = ~X) * O)

today is neither w nor t

~ (W v T)

ibuprofen relives pains and burn cream relives burns, or aspirin relives aches

(I * B) v A

(love's being eternal) is sufficient condition for (the glass's being half full), only if (fortune favoring the foolish) is a necessary condition for (space's being the final frontier)

(L > G) > (S > F)

if space's being the final frontier implies that fortune favors the foolish, then the glass is half full

(S > F) > G

space's being the final frontier is a sufficient and necessary condition for fortune's not favoring the foolish

(S > ~F) * (~F > S)

the entire statement is comprised of

(the main operator) and (the component statement/s)

if aspirin relieves aches, then if burn cream relieves burns, then ibuprofen relieves pains

A > (B > I)

if aspirin relieves aches, then ibuprofen relieves pains, then burn cream relieves burns

A > (I > B)

burn cream relieves burns and either aspirin relieves aches or ibuprofen relieves pains

B * (A v I)

burn cream relives burns and aspirin or ibuprofen relieves aches

B * (A v I)

burn cream relieves burns and either ibuprofen relieves pains or aspirin relieves aches

B * (I v A)

a B only if G

B > G

B only if O

B > O

if C, then U

C > U

M if D

D > M

if the glass is half full, then if fortune favors the foolish, then love's being eternal implies that space is the final frontier

G > [F > (L > S)]

ibuprofen relieves pains, and aspirin relieves aches or burn cream relieves burns

I * (A v B)

ibuprofen relieves pains and burn cream or aspirin relieves aches

I * (B v A)

ibuprofen relieves pains, and burn cream relieves burns or aspirin relieves aches

I * (B v A)

ibuprofen relieves pains or both burn cream relieves burns and aspirin relieves aches

I v (B * A)

(the glass NOT being half full) is a necessary condition for (love's being eternal)

L > ~G

love is eternal, unless avarice is not a vice and temperance is not a virtue

L v (~A * ~T)

p if q

Q > P

T, given that Y

Y > T

if (avarice's being a vice) and (temperance's being a virtue) are sufficient and necessary conditions for (love's being eternal), then neither (the glass is half full) nor (space is the final frontier)

[(A * T) = L] > ~(G v S)

avarice's being a vice implies that the glass is half full, given that love's being eternal implies that neither fortune favors the foolish nor temperance is a virtue

[L > ~(F v T)] > (A > G)

*, or "dot"

a conjunction, meaning "and," "also," "moreover," "however," or "but;" in front of sentences it connects

v, or "wedge"

a disjunction, meaning "or" or "unless;" in front of sentences it connects

consequent of a conditional represents

a necessary condition

~, or "tilde"

a negation, meaning "not" or "it is not the case that;" always in front of a statement that it negates; only one that can immediately follow another operator

ket words for *

and/but/however/moreover/although/nevertheless/yet/and-are

ordinary language negation words

apply only to the unit that follows them

noth both b and c is equivalent to

either not b or not x

=, or "triple bar"

equivalence, meaning "if and only if;" in front of sentences it connects

tilde is always placed

in front of a sufficient condition it replaces

dot operator expresses

logical function of conjunction

biconditional

statements that use if-and-only-if

well-formed formulas (WFFs, pronounced "woofs")

syntactically correct arrangements of logical symbols and proposition letters in propositional logic

either or statements are best translated with

the wedge operator

parenthesis

use f more than two instances of sentence letters are needed to translate a compound statement from ordinary language into propositional logic; used to indicate the range of the operators; if you need parenthesis within parenthesis, use brackets or braces to keep nested parentheses clear visually

operators/connectives

used to combine simple whole statements into compound statements

propositional logic

whole statements/propositions, which are represented by capital letters

it is not the case that both temperance's being a virtue implies that avarice is a vice and space's being the final frontier implies that fortune favors the foolish

~ [(T > A) * (S > F)]

not both (avarice is a vice) and (temperance is a virtue)

~(A * T)

neither biology nor psychology is reducible to physics

~(B v P) === ~B * ~P

both michael and elliot are not bald

~(M v B) === ~M * ~E

not either romeo or belvolio is a capulet

~(R v B) === ~R * ~B

avarice is a vice, given that not both temperance is a virtue and the glass is half full

~(T * G) > A

alabama is not a western state, but utah is

~A * U

either china is not a socialist country or north korea is

~C v N

not both donald trump and bill gates are real estate moguls

~D v ~B === ~(D * B)

either the glass is not half full or love is not eternal

~G v ~L

either immanuel kant or democritus was not a pre-socratic philosopher

~I v ~D === ~(I * D)

space is not the final frontier; however, if the glass is half full, then either fortune favors the foolish or love is eternal

~S * [G > (F V L)]

either space is not the final frontier or if the glass is half full then fortune favors the foolish

~S v (G > F)

it is not the case that either (avarice is a vice) and (temperance is a virtue), or (the glass is half full) and (fortune favors the foolish)

~[(A * T) v (G * F)