Meta-Logic: Theorising about Logic

Question 1

What is the correspondence theory of truth?

Answer 1

A sentence is true iff it corresponds to the facts. It answers the question: what is truth?

Question 2

what are sentences in logic?

Answer 2

strings of symbols from a vocabulary built according to the rules of grammar. It is a language-relative notion.

Question 3

what is the difference between an interpreted and an uninterpreted sentence?

Answer 3

interpreted: makes sense - corresponds in some sense to natural language
uninterpreted: doesn’t necessarily mean anything

Question 4

truth-apt

Answer 4

capable of truth or falsity

Question 5

What is Russel’s view on facts?

Answer 5

Facts consist of worldly entities e.g. FM won best actress at the oscars -> FM has the property of won best actress at oscars.

Question 6

What is Frege’s view on facts?

Answer 6

Fact doesn’t consist of the person and the property but their meanings or senses.

Question 7

what is the T-Schema?

Answer 7

Truth Schema:

• has infinitely many instances: one for every sentence.
• the correspondence theory implies all instances of the T-Schema ‘s’(mention) is true iff s(use)
• The T-Schema is presented in the metalanguage.

Question 8

what are the potential problems for the correspondence theory?

Answer 8

1. The facts to which sentences correspond may be metaphysically suspect
2. the concept of ‘corresponds to the fact’ is no more clear than the concept of ‘truth’ - is there a circularity here?
3. The Liar Paradox:
   - L: ‘L’ is not true
   - is L true? Yes, then it is not. No, then it is.

Question 9

What is the Liar Paradox and how could you potentially resolve it?

Answer 9

The Liar Paradox:

• L: ‘L’ is not true
• is L true? Yes, then it is not. No, then it is.

Assumptions used:

• classical logic
• unrestricted T-Schema (L is an instance of the T-Schema)
• self-reference as L talks about itself
• truth predicate - does L need ‘is true’

you must amend or give up one of these to resolve the paradox:

• you don’t want to give up classical logic or the T-Schema applying unrestrictedly
• if we give up self-reference: i.e. you disallow sentences from speaking about themselves - but you still have a problem because you could have: [A: ‘B’ is true][B: ‘A’ is not true]→paradox
• so you must tackle the truth predicate: Tarksi does this by tackling the object/ meta-language distinction.

Tarski:
truth must be relativised to a language. So “L: ‘L’ is not true” really means “L’ ‘L’ is not true in Language L”. When we talk about truth, that is a semantic notion, so we semantically ascend to the metalanguage. ‘L ‘L’ is not true in language L’ is therefore not a sentence of Language L, it is a sentence of the metalanguage. L’’: ‘L’ is not true in language L0-> L” is a sentence of L1. But there is no problem since it is not a sentence of L0. There is an infinite hierarchy of languages so there is no contradiction, L0 has L1 as a meta-language, which has L2 as a meta-language ad infinitum.

Question 10

What did Tarski do?

Answer 10

Tarski: attempted to resolve the Liar paradox
truth must be relativised to a language. So “L: ‘L’ is not true” really means “L’ ‘L’ is not true in Language L”. When we talk about truth, that is a semantic notion, so we semantically ascend to the metalanguage. ‘L ‘L’ is not true in language L’ is therefore not a sentence of Language L, it is a sentence of the metalanguage. L’’: ‘L’ is not true in language L0-> L” is a sentence of L1. But there is no problem since it is not a sentence of L0. There is an infinite hierarchy of languages so there is no contradiction, L0 has L1 as a meta-language, which has L2 as a meta-language ad infinitum.

Question 11

What criticisms have been levelled against Tarksi?

Answer 11

1. a ban on self-reference is too drastic
2. self-referential sentences work in natural language, they should work in logic too e.g. if you read this, you’re too close.

Question 12

What is the use/mention distinction? mention object language and meta-language

Answer 12

The distinction between use and mention can be illustrated for the word cheese:
Use: Cheese is derived from milk.
Mention: ‘Cheese’ is derived from the Old English word ċēse.

English is the metalanguage in which we study the object language which is in this case logic. Things in the object language are mentioned e.g. ⌝ etc.

Question 13

what are sets?

Answer 13

sets are characterised by their members. one object related to many objects in a special way, membership is denoted by ∈

Question 14

what is a subset of a set ?

Answer 14

A is a subset of set B, all of set A is contained within B: A⊆B iff ∀x(x∈A→x∈B)

Question 15

what is a proper subset?

Answer 15

A⊂B: A is a proper subset of B if, A is contained with B but not the same as B: A≠B

Question 16

what is the axiom of extensionality?

Answer 16

sets are identical iff they have exactly the same members e.g. for set x and y: x=y↔∀z(z∈x↔z∈y)

Question 17

what is the axiom of comprehension?

Answer 17

for any predicate Ø, the set {x:Øx} exists, given the axiom of extensionality there is exactly one empty set Ø

Question 18

What is Russel’s Paradox?

Answer 18

It is a paradox that demonstrates the inconsistency of set theory:
Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell’s paradox. Symbolically:

Let R={x | x∉x}, then R∈R ↔ R∉R

Question 19

Cantor’s Paradox

Answer 19

both the set of even numbers and the set of natural numbers have an infinite cardinality (there are infinite elements), this means the members of the set of natural numbers can be paired off with the members of the set of even numbers. The set of real numbers |R = {x | -∞ < x

Question 20

cardinality

Answer 20

the number of elements in the set

Question 21

The Naive Principle of Comprehension (sometimes of Abstraction)

Answer 21

Every property determines a set – or, more formally, for any predicate Px, there is a set y, such that x(x ε y Px). (Here x ε y means, as before, x is an element (or member) of the set y.)

Question 22

bijections

Answer 22

one-to-one correspondence: without necessarily counting the number of elements of a set, you can establish a parallel cardinality by matching elements one to one

Question 23

equinumerous

Answer 23

to have the same cardinality

Question 24

Galileo’s Paradox

Answer 24

there are as many even natural numbers as there are natural numbers.This is because f(x) = 2x is a one-one correspondence between the whole set of natural numbers N and the set of even natural numbers E. This is only a ‘paradox’ in the sense that it is rather odd (“paradox” means “outside or
beyond orthodoxy”); but no formal inconsistency is involved

Question 25

powerset

Answer 25

the set of all subsets of a set. If a set has n members, then its powerset has 2^n members. Denoted by a crazy p
Note: the empty set is a subset of every set.

Question 26

the iterative conception

Answer 26

a response to Russel’s and Cantor’s paradoxes:

• we begin with objects and then build all possible sets of things formed at earlier stages.
• no set can be a member of itself since it is formed at earlier stages - therefore Russell’s set does not exist
• there is no set of all sets therefore no cantor’s paradoc