Meta-Logic: Theorising about Logic Flashcards
What is the correspondence theory of truth?
A sentence is true iff it corresponds to the facts. It answers the question: what is truth?
what are sentences in logic?
strings of symbols from a vocabulary built according to the rules of grammar. It is a language-relative notion.
what is the difference between an interpreted and an uninterpreted sentence?
interpreted: makes sense - corresponds in some sense to natural language
uninterpreted: doesn’t necessarily mean anything
truth-apt
capable of truth or falsity
What is Russel’s view on facts?
Facts consist of worldly entities e.g. FM won best actress at the oscars -> FM has the property of won best actress at oscars.
What is Frege’s view on facts?
Fact doesn’t consist of the person and the property but their meanings or senses.
what is the T-Schema?
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.
what are the potential problems for the correspondence theory?
- The facts to which sentences correspond may be metaphysically suspect
- the concept of ‘corresponds to the fact’ is no more clear than the concept of ‘truth’ - is there a circularity here?
- The Liar Paradox:
- L: ‘L’ is not true
- is L true? Yes, then it is not. No, then it is.
What is the Liar Paradox and how could you potentially resolve it?
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.
What did Tarski do?
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.
What criticisms have been levelled against Tarksi?
- a ban on self-reference is too drastic
- self-referential sentences work in natural language, they should work in logic too e.g. if you read this, you’re too close.
What is the use/mention distinction? mention object language and meta-language
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.
what are sets?
sets are characterised by their members. one object related to many objects in a special way, membership is denoted by ∈
what is a subset of a set ?
A is a subset of set B, all of set A is contained within B: A⊆B iff ∀x(x∈A→x∈B)
what is a proper subset?
A⊂B: A is a proper subset of B if, A is contained with B but not the same as B: A≠B
what is the axiom of extensionality?
sets are identical iff they have exactly the same members e.g. for set x and y: x=y↔∀z(z∈x↔z∈y)
what is the axiom of comprehension?
for any predicate Ø, the set {x:Øx} exists, given the axiom of extensionality there is exactly one empty set Ø
What is Russel’s Paradox?
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
Cantor’s Paradox
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
cardinality
the number of elements in the set
The Naive Principle of Comprehension (sometimes of Abstraction)
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.)
bijections
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
equinumerous
to have the same cardinality
Galileo’s Paradox
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
powerset
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.
the iterative conception
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
