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Graham Priest - 6. Paradoxes

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you [Music]

hi this is Graham priest again and this

is the last of the short lectures we've

been recording on Logic and what I want

to do today is talk about some paradoxes

last week we talked about Dolly thism

the view that some contradictions are

true and I gave you some examples of

things that some philosophers at least

have thought were daily thick these

concern the law and they concern the

limits of thought but I want to give you

some more examples today of where you

might think that dynasties occur and

these concerned exactly paradoxes the

paradox is often phrased as an argument

which ends in a contradiction and I died

early the solution is essentially one

that accepts the contradiction I'm going

to give you three examples of paradoxes

and point out what the diocese would say

about them and the first one concerns

some paradoxes of emotion

now the paradoxes were produced

parados emotion were produced by Zenon

from Greek philosopher 5th or 6th

century BCE and his paradox emotion had

been discussed extensively by Western

philosophers it's only one of those that

I want to focus on today and that's

usually called the paradox of the arrow

so you must imagine that there is an

arrow shot from the bone and it's on its

way to the target now take the arrow at

any instant of its motion at the instant

it makes no progress at all in its

journey advance equals zero but the

whole journey is constituted by the sum

total of these instants so if the

progress made at any one instant is zero

then the progress made in the subtotal

of instance is zero

if you add not to note as many times as

you like

even infinitely many times you still get

nought okay so this was Zeno's paradox

of the arrow the arrow cannot move it

cannot make any progress in this motion

now one way you can resolve the paradox

is essentially as Hegel did so Hegel

said well look if you take the arrow at

an instant of its motion it's both here

and not here at this same instant and

it's not here precisely because due to

the fact it's in motion it's already got

a bit further and it's already lagging a

bit behind so the very fact of motion

tells you that the arrow being here and

not here it's already gone a bit further

so the arrow actually does make progress

in the instant and so Zeno's paradox is

broken we talked about hey gold before

and this is essentially a gradient

solution to the paradox of motion let me

give you a second kind of paradox of a

very different kind this is sometimes

called well there's a family of paradox

called the paradoxes of self-reference

and the easier these to get the easiest

of these to get your head around is

something called the liar paradox it was

invented or discovered by another

ancient Greek philosopher abilities at

the same time as you know as per his

little bit later and it goes as follows

suppose that I tell you this this very

statement that I'm now telling you is

false I ask you is that true or is it false

well if it's true well it says it's

false so it must be false and if it's

false well hey that's what it says so

it's true so it seems to be true and

false and that's exactly what the

dial-in theists about matters takes it

to be that statement the statement then

what I'm now telling you is false is

both true and false now you might think

that the liar paradox is a bit of a

party game

although logicians have taken it

seriously seriously auditions have taken

it seriously for two thousand and a half

thousand years now but in the 20th

century it became impossible to treat it

as not serious and this is because it

turned out to be the paradox turned out

to be one of a whole family of paradox

of self reference which turned up in the

foundations of earth magic orbs so what

the issue here is the very nature and

the ground of mathematics and let me

give you one of the other paradox in

this family which is relatively simple

compared with some of them but let me

try to explain it so this is a paradox

that was discovered by Bertrand Russell

and so it's called a Russell's paradox

and it's about sets collections of

things which are mathematical objects of

a certain kind now some sets are not

members themselves so for example the

set of countries in the world is not a

country so the set of countries in the

world is not a country it can't be a

member of a set of countries but some

sets are members themselves so the set

of all abstract ideas for example is an

abstract idea and so it's a member of

the set of all abstract ideas so it

would seem some sets are members

themselves and some are not so far so good

good

now what about the set of all those sets

which aren't members themselves okay so

this is a set which contains just those

ask yourself whether this is or is not a

member itself

well if it is a member itself it's one

of those things which is not a member

itself so if it is it isn't and if it's

not a member itself well then it's one

of those things offender set so it is a

member itself so it was sent to you both

in the set and not in the set so this is

Russell's paradox

now the paradox of self reference have

occasioned a enormous literature in the

twentieth century and this is not the

place to go into it

of course people have suggested

non-diabetic solutions but often these

seem to run into problems let me just

illustrate this with the liar paradox so

recall the liar paradox was a claim that

this very sentence I'm now telling you

is false what we saw is that if it's

true it's false and if it's false is

true okay that's fine

one natural response at this point is to

say well yes if it's false is true if

it's true it's false maybe it's neither

so this is not a dialectic view it's not

saying it's both true and false it's

saying kind of the dual of that it's

neither true nor false and some people

have endorsed this kind of solution to

the liar paradox now the problem with

this kind of solution is that if middie

seems to shift the problem let me

explain why this tweak the liar sentence

a little bit let's suppose that indeed

it's neither a nor false but now let's

consider this sentence the sentence

which I'm now telling you is either

false or neither true an awful

what we'd say about that well if it's

true it's either false or neither true

nor false that's a contradiction right

if it's false well then if it's false

it's either false or neither true nor

false but that's what it says

so it's true and we're back with the

contradiction and now suppose we try to

say well it's neither true nor false

well if it's neither true nor false then

it's either false or neither true nor

false and so since that's what it says

it's true so we're back with this

contradiction so there isn't an easier

way out by supposing that the liar

sentence is neither true nor false so so

far I've given you two examples of

paradoxes and what a diabeetus might say

about them one was you know the paradox

of the arrow the other was paradox of

self-reference such as the liar paradox

let me give you a third this is a

paradox sometimes called the server I

tease it was discovered by your Beulah

DS the same as the liar paradox and

essentially it goes as follows suppose

you are completely sober and I give you

one cent a liter of alcohol well then

you're still sober it has no effect on

you in fact whatever your state if

you're sober and I'll give you one cent

a liter of alcohol you're still sober

one centimeter just doesn't have a

difference so let's suppose you're sober

and I give you one centiliter alcohol

then you're still sober I'll give you

another you're still sober like you know

you're still sober well so eventually of

course I give you several liters of

alcohol and you're blind drunk but this

is the sorites paradox that adding one

centimeter of alcohol to your

bloodstream really cannot change you for

being so good being drunk yet

eventually by

adding single Center leads of alcohol to

your bloodstream you become drunk let me

give you another example this is to do

with colors sometimes you see a color

spectrum so I want you to imagine this

you've got a color spectrum between say

red and blue so at this end of the

spectrum it's very deep red and then

it's what goes red changes through sort

of a ready blue into blue and at this

end it's completely blue now I want you

to imagine that we cut this spectrum up

into little slices and that the changes

are so slow that you can't tell the

difference between the color of one

strip and the next you can easily

construct sorites sequences such as this

I leave it as an exercise for you to do

if you'd like now the strip on this end

is red and if any strip is red then the

strip next to it is indistinguishable in

color so that must be red - so we have a

psoriasis situation the first trip is

red the second strip is

indistinguishable so that strip is red

but the next trip is indistinguishable

so that strip is red - so it's red all

the way down but of course at this st.

it's not red it's blue now this is the

sorites paradox and it's a paradox of

vagueness the thing about predicates

like is drunk is red is that they're

vague they don't seem to have any

precise cutoff points this is what

causes the writers paradox now what is

one to say about solutions the sorites

paradox well a very standard thought is

this if you've got one of these sorites

sequences there's a kind of middle area

so take the color sorites for example at

this end the strips are definitely read

this and the strips are definitely not

read they're blue and in the middle

there's a kind of an area where well

it's it's sort of symmetric so

that the status of these things in the

middle they seem to be as much red as

blue as much rate as not ready so it

doesn't make much sense to say the

things in the middle I read and only

wrote it doesn't make much sense to say

the things in the middle of blue and

only blue because the situation is

symmetric okay there are two symmetric

possibilities one is that these things

in the middle are neither red nor blue

neither red nor not read the other

symmetric possibility is they're both

red and not red both red and blue so

that possibility is the diabetics

solution but there is the other

possibility namely that the things in

the middle are neither red nor not red

so you might wonder is there any reason

suppose the diabetic possibility is

better than the other possibility that

the things in the middle are neither red

nor not red and matters are contentious

but here's at least one reason if you

think about the things in the middle

well you wouldn't want to say they're

green you wouldn't want to say they're

yellow I mean they're either red or blue

there's no third possibility right but

if they're blue they're not red so if

they're either red or blue they're

either red or not red and if the claim

that is red was neither true nor false

well then the claim that it was not red

will be neither true nor false so the

claim that it's either red or not red

wouldn't be true so if that's right you

must have the other possibility okay so

this is the third example of a Dietetics

Ellucian to a well-known paradox I've

given you three I've given you so you

knows paradox of the arrow I've given

you the liar paradox and I've given you

the sorority's paradox now I must stress

that all these paradoxes are contentious

there is an enormous literature on these things

things and

and

additions argue at great length about

the possible solutions these paradoxes

and particularly about weather and a

daily thing paradox can be accepted

however you can at least now see why you

might be inclined to endorse dilute ISM

well this is the final nature on logic I

I hope now that you have some

understanding of the nature of logic and

some of the reasons why it's important

some of the things that engages with

some of the issues in semantics and

metaphysics and wile auditions consider

logic to be a really important subject

and if you've enjoyed thinking about

some of the things we've been talking

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YouTube Transcript:
Graham Priest - 6. Paradoxes