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 = 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 D e and D t, 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, t 1, t 2, … 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, D e and D t, the domains of entities and times. t-variables range over D t. 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. A-Theory is strongly allied with Presentism, the claim that only the present exists. The B-Theory The important metaphysical notions regarding time are: before and after. 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 D t.

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. 11. P & (Q v R)A 12. P1 &E 13. Q v R1 &E 44. QA for →I 1,45. P & Q2, 4 &I 16. Q → (P & Q)4,5 →I

The Distribution Law P & (Q v R) ↔ (P & Q) v (P & R) Proof. 16. Q → (P & Q)4,5 →I 77. RA for →I 1,78. P & R2, 7 &I 19. R → (P & R)7, 8 →I 110. (P & Q) v (P & R)1, 6, 9 PC Other direction cont’d on whiteboard.

Video Time!

Putnam on Logic Logic is as empirical as geometry. We live in a world with a non- classical 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 non- Euclidean geometry explains the approximate validity of Euclidean geometry ‘in the small’.