conclusions
THE LOGICAL
LANDSCAPE
Yoshihiro Maruyama
co-taught with Pascal Bercher
on the legacy of John Slaney
SUMMARY – WHAT DID YOU LEARN?
➤ 1. Natural deduction (propositional; first-order; equality; restricted quantifiers)
➤ 2. Semantic tableaux (propositional; first-order)
➤ 3. Semantics (propositional; first-order; and three-valued models)
➤ 4. Sequent calculus (multiple-conclusion; and single-conclusion)
➤ 5. Relevant logic and the paradoxes of implications
➤ 6. Fuzzy logic and the sorites paradox
➤ 7. Intuitionistic logic and the constructive view of truth
THE LOGICAL LANDSCAPE
Classical LogicIntuitionistic Logic
Fuzzy Logic
Relevant Logic(Int. Rel. Log.)
(Int. Fuz. Log.)
Full Lambek Calculus
(single-conclusion SC
without contr./weak.)
REFLECTION – YOUR FAVOURITE LEADS YOU TO A DEEPER WORLD
➤ What did you like?
➤ Proof system or semantics?
➤ Classical or non-classical (relevant, fuzzy, intuitionistic)?
➤ Logical pluralism or monism?
➤ Paradox of implication or Sorites paradox (or Liar paradox)?
➤ Multiple- or single-conclusion? Realism or antirealism?
➤ Semantic denotationalism or inferentialism?
PRIOR’S TONK PARADOX
➤ You can do whatever you want to do in logic; there is no predetermined law in logic.
➤ Arthur Prior, born in New Zealand, defined a logical connective “tonk” in ND.
➤ We can then prove anything implies anything, which basically makes logic inconsistent.
➤ What’s wrong with Prior’s tonk? It has both intro. and elim. rules like any other
connectives. Why can’t we define tonk in natural deduction? Or can we actually?
➤ Prior’s tonk paradox is still debated in philosophy of logic.
tonkE
B
AtonkB
tonkI
A
AtonkB
https://plato.stanford.edu/entries/prior/
THE LOGICAL LANDSCAPE IN ARITHMETIC AND SET THEORY
Zermelo
Set Theory
Peano
Arithmetic
Primitive
Recursive
Arithmetic
Robinson
Arithmetic
Second-Order
Arithmetic
Finite-Order
Arithmetic
Higher-Order
Set Theory
ZF
Set Theory
Inconsistent
Infinitary
Set Theory
Inconsistent
Naive
Set Theory
All this is systems over classical logic.
If we change the underlying logic into FL calculus,
naive set theory is consistent (no Russell paradox).
Be careful of large infinities,
which sometimes solve your problems
but often lead you to inconsistencies.
THE 0=1 LANDSCAPE IN SET THEORY
https://ncatlab.org/nlab/show/large+cardinal
https://mathoverflow.net/questions/194486/is-there-a-compendium-of-the-consistency-strength-between-the-most-important-for https://mathoverflow.net/questions/194486/is-there-a-compendium-of-the-consistency-strength-between-the-most-important-for
PEANO ARITHMETIC
➤ How to prove 1+1=2? Note: 1=s0 and 2=ss0.
➤ Then: 1+1=s0+s0=s(s0+0)=ss0=2. Also: 1*1=s0*s0=s0*0+s0=s0=1.
https://www.chegg.com/homework-help/questions-and-answers/relation-u-greater-equal-v-natural-numbers-n-defined-formula-ez-u-z-v–give-formal-proof-u-q64888301
=: binary relation symbol
0: constant symbol
s: “+1” function symbol
+: “add” function symb.
・:“multiply” fun. sym.
PA7: induction axiom
ZF SET THEORY WITH THE AXIOM OF CHOICE
https://cute766.info/the-axioms-of-zermelo-fraenkel-set-theory-with-choice-zfc/
GÖDEL’S FIRST INCOMPLETENESS THEOREM
➤ Gödel’s first incompleteness theorem: if a logical system is finitary and rich enough to
express basic arithmetic, there is always an undecidable proposition A within the
system (i.e., A is not provable, and its negation is not provable, either).
➤ It can happen that the human mind still recognises the truth of that formally
undecidable proposition A, and thus Lucas and Penrose argued AI is impossible.
➤ Lucas-Penrose have been criticised so much, but Gödel himself argued: “the human
mind (even within the realm of pure mathematics) infinitely surpasses the power of
any finite machine” or “there exist absolutely unsolvable diophantine problems”
➤ Gödel was a Platonist about math, and did not believe in the second disjunct, so he
believed the human mind “infinitely surpasses the power of any finite machine.”
GÖDEL’S SECOND INCOMPLETENESS THEOREM
➤ Gödel’s second incompleteness theorem: under the same assumptions, it is
impossible to prove the consistency of the system within the same system.
➤ You cannot prove your own consistency, which can only be proven with a stronger
system. This means there are different strengths of consistency of logical systems.
➤ If a logical system S1 can prove the consistency of another system S2, then S1 is
stronger in terms of consistency strength than S2.
➤ The second incompleteness is called intensional whereas the first incompleteness
is extensional, due to certain subtleties in the second incompleteness (with a
subtle trick, we can actually prove the consistency of a system with the system).
THE PHILOSOPHICAL LOGICAL LANDSCAPE
…
Quine
Putnam
Frege
Russell
Pierce
Carnap
Vienna Circle
Wittgenstein
Dummett
Davidson
Kripke
D. Lewis
T. Williamson
and many others
His famous books (further reading):
Vagueness (1994)
Knowledge and its Limits (2000)
The Philosophy of Philosophy (2007)
Modal Logic as Metaphysics (2013)
Martin-Löf
Hintikka
and many others
HOW TO USE LOGIC IN ORDINARY LIFE
➤ When someone says “It is logically correct”, you can ask “Which logic do you mean?”
➤ When someone says “It is obvious like 1+1=2”, you can ask “Can you prove it logically?”
➤ When someone says “Everything is unambiguously defined in mathematics”, you can ask
“How do you define conjunction without using any conjunction at the meta-level?”
➤ When someone says “There are always answers to mathematical questions”, you can ask
“What do you think about the first incompleteness theorem?”
➤ When someone says “Machines replace humans”, you can ask the same question.
➤ When someone says “Mathematics is always consistent and coherent”, you can ask “What
do you think about the second incompleteness theorem?”
IT’S FINALLY DONE
Thank you!
Hope you enjoyed logic and continue to do so.
Try to use logic in your life so that you don’t forget.
Chat with your friends about logic like guys above.
Good luck on exam, but grade is grade, logic is logic.
https://www.mentalfloss.com/article/501995/12-fascinating-facts-about-rick-and-mortyhttps://www.brainpickings.org/2015/05/07/rebecca-goldstein-incompleteness-godel-einstein-time/