Download What*s the Right Logic

What’s the Right Logic
What Is Logic?
Joe Lau
The laws of biology might be true
only of living creatures, and the
laws of economics are only
applicable to collections of agents
that engage in financial
transactions. But the principles of
logic are universal principles
which are more general than
biology and economics.
Alfred Tarski
“[Logic is] ... the name of a
discipline which analyzes the
meaning of the concepts common
to all the sciences, and establishes
the general laws governing the
concepts.”
Frege
“To discover truths is the task of
all sciences; it falls to logic to
discern the laws of truth. ... I
assign to logic the task of
discovering the laws of truth, not
of assertion or thought.”
Different Logics
Extensions of First-Order Logic
FOL with identity
FOL with infinitary conjunction and disjunction
FOL with ϵ and axioms of set theory
FOL with modal operators □, ◊
FOL with tense operators
Many-sorted FOL with <
Second-order logic
Two Ways of Doing the Logic of Time
Tense operators:
• P(φ): ‘at some time in the past, φ’
• F(φ): ‘at some time in the future, φ’
• H(φ): ‘it has always been true that φ’
• G(φ): ‘it is always going to be true that φ’
Operators are Duals
P(φ) ↔ ~H(~φ)
H(φ) ↔ ~P(~φ)
F(φ) ↔ ~G(~φ)
G(φ) ↔ ~F(~φ)
Definitions
A(φ) = H(φ) & φ & G(φ)
‘it is always true that φ’
S(φ) = P(φ) v φ v F(φ)
‘it is sometimes true that φ’
A(φ) ↔ ~S(~φ)
S(φ) ↔ ~A(~φ)
Now
We can also introduce a ‘now’ operator:
P ⊢ N(P)
N(P) ⊢ P
(Note that you can’t substitute N(P) for P everywhere!)
Interpretation
Standard FOL interpretation:
We introduce the domain D of individuals.
Models M = <I, s> are composed of an interpretation function I, and an
assignment to the values of variables s.
I is a function from individual constants to objects in D, and from
monadic predicate letters to sets of objects in D.
s is a function from variables to objects in D.
M ⊨ Fa iff I(a) ϵ I(F)
New Interpretation
Now we introduce two domains De and Dt, the domain of entities and
of times.
Statements are no longer true or false relative to a model, they are true
or false relative to a model at a time
M, t ⊨ Fa iff I(a) ϵ I(F, t)
The idea is that different predicates are satisfied by different objects at
different times.
M, t ⊨ P(φ) iff for some t* < t, M, t* ⊨ φ.
Different Way
In addition to our old variables x, y, z… we introduce t, t1, t2, …
In addition to our old predicates and relation symbols F, G, H… we
introduce a new relation symbol <
For each of the old predicates we add a new argument ‘Fb’ becomes
‘Fbt’
We add a new name n.
Past and Future
Let φ(t) be some formula with free variable t.
∃t t < n & φ(t)
‘there is a time before now at which φ’
∃t n < t & φ(t)
‘there is a time after now at which φ’
Interpretation
This time we still have two domains, De and Dt, the domains of entities
and times.
t-variables range over Dt.
The interpretation function I assigns a relation between times to ‘<’,
and we specify various properties that the relation satisfies.
Everything else is standard.
Which Is Right Logic of Time?
Right Logic?
• Semantic considerations
• Metaphysical considerations
• Physical considerations
Semantics: Expressive Power
Once, everyone who is now happy was going to be miserable.
Good translation: ∃t (t < n & ∀x(HAPPYxn → ∃t* (t < t* & Mxt*)))
Bad translation: P(∀x(N(HAPPYx) → F(Mx)))
Semantics: Reference to a Time
“I left the stove on!”
“John came in. Then he sat down.”
It was true that I was happy.
Metaphysics: Two Theories of Time
The A-Theory
The important metaphysical
notions regarding time are: the
present, the past, and the future.
The B-Theory
The important metaphysical
notions regarding time are: before
and after.
A-Theory is strongly allied with
Presentism, the claim that only
the present exists.
B-Theory is strongly allied with
Eternalism, the claim that all times
exist. ‘Now’ is like ‘here’: it’s not
special, it’s just where we are.
Metaphysics: Two Theories of Time
The A-Theory
A-Theorists like an operator logic
of time, because it doesn’t
commit us to the existence of
times!
“It’s true that in the past
dinosaurs existed, but there does
not exist a time at which they
did.”
The B-Theory
B-Theorists like a quantifier logic
of time, because it’s not
committed to a privileged present.
The constant “n” receives
different interpretations in
different models. It can be any
time you like in Dt.
The Rietdijk-Putnam Argument
According to the theory of relativity, different observers that are
moving relative to each other, will judge the timing of events
differently.
A might experience X and Y as happening at the same time, B might
experience X before Y, and C might experience Y before X.
The Rietdijk-Putnam Argument
Let the 3D universe as A experiences it now from her frame of
reference be the A-universe, and the same for the B- and C-universes.
If A, B, and C are all experiencing truly, then there must be multiple 3D
universes, that is, we must be living in a 4D universe that contains
multiple 3D universes as parts.
So there exists more than one time, and there is no special present: the
B-Theory is correct.
Joe Lau
The laws of biology might be true
only of living creatures, and the
laws of economics are only
applicable to collections of agents
that engage in financial
transactions. But the principles of
logic are universal principles
which are more general than
biology and economics.
Alfred Tarski
“[Logic is] ... the name of a
discipline which analyzes the
meaning of the concepts common
to all the sciences, and establishes
the general laws governing the
concepts.”
Frege
“To discover truths is the task of
all sciences; it falls to logic to
discern the laws of truth. ... I
assign to logic the task of
discovering the laws of truth, not
of assertion or thought.”
• Is the logic of time topic neutral?
• Is it logic?
• Is it about concepts and meanings?
• Is it universal?
• Is it necessary?
• Are we doing it right?
• Is ANY logic topic neutral?
• Is it about concepts and meanings?
• Is it universal?
• Is it necessary?
• How do we do it?
Quantum Logic
The Distribution Law
P & (Q v R) ↔ (P & Q) v (P & R)
Proof.
1
1. P & (Q v R)
1
2. P
1
3. Q v R
4
4. Q
1,4 5. P & Q
1
6. Q → (P & Q)
A
1 &E
1 &E
A for →I
2, 4 &I
4,5 →I
The Distribution Law
P & (Q v R) ↔ (P & Q) v (P & R)
Proof.
1
6. Q → (P & Q)
7
7. R
1,7 8. P & R
1
9. R → (P & R)
1
10. (P & Q) v (P & R)
Other direction cont’d on whiteboard.
4,5 →I
A for →I
2, 7 &I
7, 8 →I
1, 6, 9 PC
Video Time!
https://www.youtube.com/watch?v=a8FTr2qMutA
Putnam on Logic
Logic is as empirical as geometry.
We live in a world with a nonclassical logic. Certain statements
– just the ones we encounter in
daily life – do obey classical logic,
but this is so because the
corresponding subspaces… [form]
a so-called ‘Boolean lattice’.
Putnam on Logic
Quantum Mechanics explains the
apparent validity of classical logic
‘in the large’, just as nonEuclidean geometry explains the
approximate validity of Euclidean
geometry ‘in the small’.