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Axiomatizations of Temporal Logic . 10723029 Xu Zhaoqing. I. Content. Introduction Basic temporal logic Branching time logic Conclusions. II. Introduction. Temporal Logic Broadly : all approaches to the representation of temporal information within a logical framework;

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i content
I. Content
  • Introduction
  • Basic temporal logic
  • Branching time logic
  • Conclusions
ii introduction
II. Introduction
  • Temporal Logic
    • Broadly:all approachesto the representation of temporal information within a logical framework;
    • Narrowly:themodal-style of temporal logic;
iii basic temporal logic
III. Basic Temporal Logic
  • 1. Syntax and semantics
  • 2. The Minimal logic Kt
  • 3. The IRR rule
  • 4. The logic of linear time
1 syntax and semantics
1. Syntax and Semantics
  • Language:¬,∧,G,H

Fϕ =df ¬G¬ϕ

Pϕ =df ¬H¬ϕ

Atemporal frame (or flow of time) F=(T,<),where T is non-empty,<is a binary relation which is irreflexive and transitive;

A valuation V:Ф→P(T); A model M=(F,V);



    • M, t ||- p iff t∈V(p), where p∈Ф,
    • M, t ||- ¬ϕiff not M, t╟ϕ,
    • M, t ||- ϕ∧ψ iff M, t╟ ϕ and M, t╟ ψ,
    • M, t ||- Gϕiff for all s∈T, if t<s then M, s ||- ϕ,
    • M, t ||- Hϕ iff for all s∈T, if s<t thenM, s ||- ϕ.
  • The definitions of validities are as usual.
2 the minimal logic kt
2. The Minimal Logic Kt
  • Axioms:
    • (1) All classical propositional tautologies;
    • (2) G(p→q)→(Gp→Gq); and mirror-image;
    • (3) p→GPp; and mirror-image;
    • (4) Gp→GGp.
  • Rules:
    • US: ϕ/ϕ[ψ/p]; MP: ϕ,ϕ→ψ/ψ; TG: ϕ/Gϕ;and ϕ/Hϕ.
  • The deduction is defined as usual.

Theorem 3.2.1

  • Kt is sound and complete for the class of all temporal frames.

Lemma 3.3.1

  • IRR rule is valid on the class of all temporal frames.


  • Theorem 3.3.2
    • Kt’ is sound and complete for the class of all temporal frames.
4 the logic of linear time
4. The Logic of Linear Time
  • Linearity: ∀x∀y(x<y∨x=y∨y<x)
  • Formulas: a. Fp∧Fq→F(p∧Fq)∨F(p∧q)∨F(Fp∧q);

b. Pp∧Pq→P(p∧Pq)∨P(p∧q)∨P(Pp∧q);

  • Or c. PFp→(Pp∨p∨Fp); d.FPp→(Fp∨p∨Pp);

LTL=Kt+a+b(or +c+d).

  • Theorem 3.4.1
  • LTL is sound and complete for the class of all linear temporal frames.
iv branching time logic
IV. Branching Time Logic
  • 1. Branching time
  • 2. Definitions of the F
  • 3. The basic branching time logic
  • 4. The logic of Peircean branching time
  • 5. The logic of Ockhamist branching time
1 branching time
1. Branching Time
  • Why consider branching time?
    • The argument for determinism:
    • 1. p→ □p (ANP)
    • 2. Fp→ □Fp
    • 3. F¬p→ □F¬p
    • 4. Fp∨F¬p (EMP)
    • 5. Fp∨F¬p→ □Fp∨□F¬p
    • 6. □Fp∨□F¬p

Definition 4.1.1

  • A treelike frame F=(T,<) is a temporal frame, where < satisfying the tree property:∀x∀y∀z(y<x∧z<x→(y<z∨y=z∨z<y)).






Definition 4.1.2

  • Where (T, <) is a treelike frame and t∈T, a branch (or history) b is a maximal linearly ordered subset of T.





2 definitions of f
2. Definitions of F
  • Why consider other definitions?
    • The Linear future :
      • M, t ||- Fϕiff there exists s∈T, such that t<s and M, s ||- ϕ;
    • then
      • Fp∨F¬p is valid; Fnp∧Fn¬p is satisfiable; {¬Pp, ¬p,¬Fp,PFp} is satisfiable.

Other choices:

    • The Peircean future :
      • M, t||- Fϕ iff for any branch b through t, there exists s∈b, such that t<s, and M, s ||- ϕ;
    • Then
      • Fp∨F¬p is invalid; p||-/PFp;

The Ockhamist future:

    • M, t, b ||- p iff t∈V(p), where p∈Ф,
    • M, t, b ||- Fϕ iff there exists s∈b, such that t<s and M, s,b ||- ϕ.
  • Then
    • Fnp∧Fn¬p is invalid; {¬Pp, ¬p,¬Fp,PFp}is not satisfiable; Fp∨F¬p is valid.


    • M, t||- ϕ iff for any branch b through t, we have M, t, b||- ϕ.
  • Then
    • Fnp∧Fn¬p is invalid; {¬Pp, ¬p,¬Fp,PFp}is not satisfiable; Fp∨F¬p is valid.
  • The Linear future:
    • “it possibly will be case”, too weak;
  • The Peircean future:
    • “it necessarily will be the case ”, too strong;
  • The Ockhamist future:
    • “it will be the case in the actual future”, the most promising.
3 the basic btl
3. The Basic BTL
  • BTL=Kt+b (or d)+IRR
  • Theorem 4.3.1
  • BTL is sound and complete for the class of all treelike frames.
4 the logic of pbt
4. The logic of PBT
  • Language:
    • G, H, F□;
    • The dual of F□ is defined as:

G◇ϕ=df.¬ F□¬ϕ.



    • Peircean frame is treelike frame.
    • For satisfaction, we only add:
      • M, t||- F□ϕ iff for any branches b through t, there exists t∈b, such that t<s and M, s ||- ϕ.

PBTL=BTL+the following axioms:

    • a. G (p→q)→(F□p→F□q)
    • b. Hp→Pp ; Gp→F□p
    • c. Gp→G◇p
    • d. F□F□p→F□p
    • e. Hp→ (p→ (G◇p→G◇Hp))
    • f. F□Gp→GF□p

Theorem 4.4.1

  • PBTL is sound and complete for the class of all endless Peircean frames.

Definition 4.4.2

    • A bundle B on a treelike frame is F=(T,<) is a collection of branches through T containing at least one branch through each t ∈T.

Definition 4.4.3

  • We define weak satisfaction with respect to a bundle B much as ordinary satisfaction was defined above, changing only the last clause of the definition:
    • M, t||- F□ ϕ w.r.t. B iff for any branches b∈B through t ,there exists s∈b with t<s, such that M, s ||- ϕ w.r.t. B.

Definition 4.4.4

  • ϕ is weakly satisfiable if M, t||- ϕ w.r.t. B for some M, t and B; ϕ is strongly valid if ¬ϕ is not weakly satisfiable.
5 the logic of obt
5. The logic of OBT
  • The language:
    • G,H,□;
    • The dual of □ is defined as:

◇ϕ=df.¬ □¬ϕ.

F≤ϕ =df ϕ∨Fϕ,G≤ϕ =df ϕ∧Gϕ,P≤ϕ =df ϕ∨Pϕ,H≤ϕ =df ϕ∧Hϕ.



    • Ockhamist frame is a treelike frame.
    • We define satisfaction inductively:
      • M, t, b ||- p iff t∈V(p), where p∈Ф,
      • M, t, b ||- ¬ϕiff not M, t, b╟ϕ,
      • M, t, b ||- ϕ∧ψ iff M, t, b ╟ ϕ and M, t, b||-ψ,
      • M, t, b ||- Gϕ iff for all s∈T ,if s∈b and t<s then M, s,b ||- ϕ,
      • M, t, b ||- Hϕ iff for all s∈T ,if s<t thenM, s,b||- ϕ.
      • M, t, b ||- □ϕ iff for all branches b’⊆T,if t∈b’ thenM, t,b’ ||- ϕ.

Translation (ϕ)o from Peircean formulas to Ockhamist ones:

    • The only non-trivial clause of this map concerns the future operators:

(fϕ)o = □Fϕo and (Gϕ)o = □Gϕo

    • It is straightforward to prove that for all tree models M, all points t in M and all branches b with t∈b, we have that:

M, t||- ϕ iff M,t, b||- ϕo


Definition 4.5.1

  • Weakly satisfaction:
    • M, t, b||- □ϕ w.r.t. Biff for any branches b’ ∈B, if t∈b’ thenM, t,b’ ||- ϕ w.r.t. B.
  • Strong validity is defined similarly.

The Logic of strong Ockhamist validities(SOBTL):

    • Axioms:
      • A0. All substitution instance of propositional tautologies;
      • L1: G(α→β)→(Gα→Gβ) and mirror image;
      • L2: Gα→GGα;
      • L3: α→GPα and mirror image;
      • L4: (Fα∧Fβ)(F(α∧Fβ)∨F(α∧β)∨F(Fα∧β)) and mirror image;
      • BK: □(α→β)→(□α→□β);
      • BT: □α→α;
      • BE: ◇α→□◇α;
      • HN: Pα→□P◇α;
      • MB: G⊥→□G⊥;
    • Rules: MP, GT, GN, IRR, and ANF: p→□p, for each atomic proposition p.

Theorem 4.5.2

  • SOBTL is sound and complete for the class of all strong validities.

We’ve known that every strongly valid Ockhamist formula is valid, but the converse is not right.

  • Counterexamples:
    • □G◇F□p→◇GFp (Burgess, 1978);
    • GH□FP(H¬p∧¬p∧Gp)→FP◇FP(¬p∧□Gp) (Nishimura,1979);
    • (p∧□GH(p→Fp))→GFp (Thomason,1984);
    • □G(p→◇Fp)→◇G(p→Fp) (Reynolds,2002).
  • All formulas above are valid but not strongly valid, soSOBTLis incomplete for the class of all Ockhamist frames.

The logic of OBT:

  • Theorem? 4.5.3
  • OBTL is sound and complete for the class of all Ockhamist frames.
v conclusion
V. Conclusion
  • The most promising suggestion was given by Reynolds, and if the completeness can be proved, the long standing open problem gets closed eventually.

Open problems:

    • Ockhamist logic with until and since connectives;
    • Ockhamist logics over trees in which all histories have particular properties such as denseness or being the real numbers.
vi references
VI. References
  • [01] J. Burgess, The Unreal Future, Theoria ,44, 157-179,1978.
  • [02] J.Burgess, Decidability for Branching Time. StudiaLogica, 39, 203–218, 1980.
  • [03] D.Gabbay, I. Hodkinson, and M. Reynolds, Temporal Logic: Mathematical Foundations and Computational Aspects, Volume 1. Oxford University Press, 1994.
  • [04] R. Goldblatt. Logics of Time and Computation. CSLI Lecture Notes. Center for the Study of Language and Information, Stanford University, second edition, 1987.

[05] Y. Gurevich and S. Shelah. The decision problem for branching time logic. In The Journal of Symbolic Logic, 50, 668-681,1985.

  • [06] H. Nishimura. Is the semantics of branching structures adequate for chronological modal logics? Journal of Philosophical Logic, 8, 469–475, 1979.
  • [07]A. Prior, Past, Present and Future, Oxford University Press, 1967.
  • [08] M. Reynolds. Axioms for branching time. Logic and Computation, Vol. 12 No. 4, pp. 679–697 2002.

[09] M. Reynolds,An Axiomatization of Prior’s Ockhmist Logic of Historical Necessity,to appear.

  • [10] R. Thomason, Indeterminist Time and Truth-value Gaps. Theoria, 36, 264–281, 1970.
  • [11] R. Thomason. Combinations of tense and modality. In Handbook of Philosophical Logic, Vol II: Extensions of Classical Logic, D. Gabbay and F. Guenthner, eds, pp. 135–165. Reidel, Dordrecht, 1984.

[12] Y. Venema, Temporal Logic, in The Blackwell Guide to Philosophical Logic, Blackwell publishers, 2001.

  • [13] A. Zanardo. A finite axiomatization of the set of strongly valid Ockamist formulas. Journal of Philosophical Logic, 14, 447–468, 1985.
  • [14] A. Zanardo. Axiomatization of ‘Peircean’ branching-time logic. StudiaLogica, 49, 183–195, 1990.