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ESSLLI 2021 Course on Temporal Logics
Lecture 5
Introduction to first-order temporal logics
Valentin Goranko
Department of Philosophy, Stockholm University
ESSLLI 2021 July 30, 2021
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First-order temporal logics: introduction
Objects exist in time, and they change their properties over time. Propositional temporal logics are not expressive enough to capture adequately the dynamic aspects of the world.
Instead, a full-fledged temporal model of the world is needed, tracing the temporal history and evolution of objects in it and their relationships. Accordingly, the logical language should contain names for objects, variables and quantifiers ranging over objects, as well as predicates for denoting properties and relations – in addition to the usual temporal operators for reasoning about changes over time.
This is what first-order temporal logics provide.
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Outline
• Introduction: temporality, existence and quantification.
• Technical preliminaries: first-order relational languages and models.
• First-order temporal logics (FOTL): languages and semantics.
• Eternalist and presentist semantics of first-order temporal logics.
• Eternalist semantics and constant domain models.
• Presentist semantics and varying domain models.
• Axiomatic systems for FOTL. Temporal Barcan formulae and their converses.
• Concluding remarks.
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Existence and quantification over time
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Existence and quantification over time
Existence in time is a major topic in the philosophy of time.
Usually, objects come into being at one point in time and go out of being at some later time.
But, what does it mean for an object to exist at a given time instant? Do only present objects exist (presentism), or is existence understood in a broader sense, comprising past and future objects as well (eternalism)? E.g. is “Socrates exists” true?
Yes, if we read “exists” in the eternatist sense. Not, if we read “exists” in the presentist sense. But, it was true in that sense in 400 B.C.
Related debate on persistence: the question how objects exist through time. See details and references on these debates in the SEP article on
Temporal logic.
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Temporality, existence and quantification
Interpretation problems arise from the interaction of time and existence. Consider the sentence “A philosopher will be a king”.
It can be interpreted in several different ways, such as:
∃x(Philosopher (x) ∧ FKing (x)): Someone who is now a philosopher will be a king at some future time.
∃xF(Philosopher (x) ∧ King (x)): There exists now someone who will be both a philosopher and a king at some future time. F∃x (Philosopher (x ) ∧ FKing (x )): There will exist a philosopher at
some future time, who will later be a king. F∃x (Philosopher (x ) ∧ King (x )): There will exist someone who is
at the same time both a philosopher and a king.
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Existence over time: four natural views
1. Objects come into being at one point in time and go out of being at a later time, so they actually exist only over a certain period of time. Formally captured by models with varying domains, where the local domain at a given time instant comprises those, and only those, objects that presently exist at that instant.
In this sense, the sentences “Every human is born after 1888” and
“Every human will die before 2220” are (in all likelihood) true. 2. Objects actually exist over a period of time, but they remain in the
temporal history of the world once they have ceased to actually exist. Formally captured by models with increasing domains, containing not only those objects that presently exist but all objects existing in the past, as well.
In this sense, the sentence ”Socrates exists and will always exist” is true, while “Every human is born after 1888” is false.
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Existence over time: four natural options, cont’d
3. All objects that will ever exist are initially part of the temporal history of the world but drop out once they have ceased to actually exist. Technically, this amounts to the idea that the local domain at a time instant comprises all present and future objects.
Captured by models with decreasing domains.
In this sense, ”Socrates existed, but will never exist again” is true, while “Every human will die before 2220” is (hopefully) false. 4. Past, present, and future objects always exist.
This is the notion of existence in an eternalist sense.
It requires that the local domain associated with a time instant contains all objects that are part of the temporal history of the world. Formally captured by models with constant domains.
Here, ”Socrates exists, and so does his mother, as well as every descendant of the current king of Sweden born after 2050” is true. Question: what about existence of fictitious entities, such as Pegasus,
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The scope of quantification over time
What do we quantify over in a temporal setting?
• Only over those objects that exist at the current time instant?
• Over all present, past, and future objects in the temporal history of the world?
The two options coincide when ‘existence over time’ is interpreted in eternalist sense, but not in general.
What about “Every human being has lived, is alive, or will ever live”? (Think again of Santa Claus, Sherlock Holmes, Pippi Longstocking, etc.)
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Modality and quantification: further issues
Many more challenging issues and problems arise.
• Individuals and individual concepts. Definite descriptions and rigid designators.
• Necessity and equality. Identity of individuals over time. The Ship of Theseus problem.
• Existence and necessity de re vs de dicto.
• Quantification and necessitation de re vs de dicto. Quine’s interpretation problem(Quine, 1953):
quantifying into modal contexts is incoherent.
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First-order temporal logics:
a technical introduction
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Preliminaries: first-order relational structures
A first-order relational structure consists of:
• A non-empty set, called adomain (of discourse) D;
• Designated predicates (relations) in D;
• Designated individuals (constants)in D;
(I will not consider structures with designated functions here.) Examples:
– numerical structures, e.g. N, with domain the set of natural numbersN, predicates=and<, and the constant0.
– the structureHon the set of all humans, e.g. with unary predicatesM (‘man’), W(‘woman’), binary predicatesP(’parent of ’),C(’child of ’),L (‘loves’), and constants (names), e.g. ‘Adam’, ’Eve’, ‘Joe’, ‘Mia’ etc.
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Preliminaries: first-order relational languages
Vocabulary of a relational FO language L:
• Non-logical symbols: constant symbols and predicate (relational) symbols of specified arities.
• Individual variables: x , y , z, possibly with indices.
• Logical symbols, including:
– propositional connectives: (a sufficient subset of) ¬, ∧, ∨, →, ↔ – equality=; quantifiers: ∀, ∃;auxiliary symbols: ( , )
Terms in L are just constant symbols and individual variables.
Atomic formulae in L: p(t1, ..., tn) where t1, ..., tn are terms and p is an n-ary predicate symbol in L. In particular, t1 = t2 is an atomic formula. Formulae in L:
φ ::= α | ¬φ | (φ ∧ φ) | ∀x φ
where α is an atomic formula. Now, ∨, →, ↔, ∃ are defined as usual. Also, scope of quantifiers, free and bound variables in formulae,
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Language of FOTL
FOTL extends FOL with temporal operators, as expected. E.g., the language of the Priorean FOTL:
ϕ := R(τ1, . . . , τn) | τ1 = τ2 | ⊥ | ¬ϕ | (ϕ ∧ ϕ) | ∀xϕ | Hϕ | Gϕ, where R(τ1, . . . , τn) and τ1 = τ2 are atomic formulae.
The connectives ∨, →, and ↔, as well as the temporal operators P and F, and the quantifier ∃ are defined as usual.
Nexttime, Since and Until can be added, as in propositional temporal logic.
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First-order temporal logic: models
The models of FOTL are based on temporal frames where each time instant is associated with a first-order relational structure.
Formally, a first-order temporal modelis a tuple: M = (T , ≺, D, Dom, I) where:
• T = hT , ≺i is a temporal frame;
• D is theglobal domain (universe) of the model;
• Dom : T → P(D) is a domain function, assigning to each time instant t ∈ T a local domainDomt ⊆ D, such that
D =St∈TDomt.
• I is aninterpretation function, assigning for each t ∈ T :
• an object It(c) ∈ D to each constant symbol c;
• an n-ary relation It(R) ⊆ Dn to each n-ary predicate symbol R. The tuple F = (T , ≺, D, Dom) is called the augmented temporal frame
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First-order temporal models: some remarks
• The assumption that the global domain is the union of all local domains intuitively excludes the possibility of fictitious individuals that do not actually exist at any time instant (such as Santa Claus, Sherlock Holmes, Superman, etc.).
Several possible solutions. Is there a best one?
• The interpretations of the constant and predicate symbols are defined locally, i.e. with respect to a given time instant, while their respective extensions range over the global domain.
This allows for reference to objects that do not currently exist and enables a proper treatment of cross-temporal relations, e.g. like the sentence “C.S. Pierce had a longer beard than Aristotle”.
• Since a FOTL model is supposed to represent the temporal evolution of the world, the local domains at the different time instants in the underlying augmented temporal frame must be suitably connected. How? Recall the four notions of existence over time.
Which one of them is the best to handle this? Or, something else?
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Formal semantics for first-order temporal logic:
variable assignments and term valuations
Given a fixed FOTL language L, we denote the set of individual variables in L by VAR and the set of terms in L by TERM.
Given a model M = (T , ≺, D, Dom, I) for L, a variable assignment in Mis a mapping v : VAR → D that assigns to each variable x an element v(x ) of the global domain D of the model.
Every such assignment v is uniquely extended to a term valuation (again, denoted by v), defined as a mapping v : W × TERM → D as follows:
vw(x ) := v(x ), vw(c) := Iw(c).
(When the language contains function symbols, the valuation is extended world-wise over all terms, like in FOL.)
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Formal semantics for first-order temporal logic:
truth valuations of formulae
As in propositional temporal logic, formulae of FOTL are evaluated in FOTL models locally, in possible worlds, inductively as follows:
• M, w |=v p(t1, ..., tn) iff (vw(t1), ..., vw(tn)) ∈ Iw(p), for any n-ary predicate symbol p and terms t1, ..., tn in L.
Intuitively: p(t1, ..., tn) is true for v in w if the tuple of individuals (Iw(t1), . . . , Iw(tn)) designated by (assigned as values by v to) the terms t1, ..., tnin w has the extensional property Iw(p) designated by p at w , i.e., belongs to the extension of the predicate p in w in M.
• M, w |=v t1 = t2 iff vw(t1) = vw(t2), for any terms t1, t2 in L.
• M, w |=v ¬ϕ iff M, w 6|=vϕ
• M, w |=v ϕ1∧ ϕ2 iff M, w |=vϕ1 and M, w |=vϕ2
• M, w |=v Gϕ iff M, u |=vϕ for every u ∈ W such that w ≺ u.
• M, w |=v Hϕ iff M, u |=v ϕ for every u ∈ W such that u ≺ w .
• to be continued...
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Two approaches to quantification in FO modal logic:
possibilist and actualist
– possibilistapproach: quantification over the global domain that contains all the possible objects, not just those that happen to exist in the current/given world.
In the context of FOTL, this can be called eternalist approach.
– actualist approach: quantification over the local domain of the current possible world, i.e. only over the objects that actually exist in that world. In this view, the quantifier ∃x reflects commitment to what is actual in the given world, rather than to what is merely possible there.
In the context of FOTL, this can be called presentist approach. Respectively, two different semantics of FOTL emerge:
eternalist and presentist semantics.
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Formal semantics for first-order temporal logic, completed
• M, w |=v ∀xφ iff ...
– eternalist semantics: ... M, w |=v[a/x ]φ for every a ∈ D, where v[a/x ] is the variant of the assignment v such that v(x ) = a.
– presentist semantics: ... M, w |=v[a/x ]φ for every a ∈ Domw, where v[a/x ] is the variant of the assignment v such that v(x ) = a.
Thus, ∀x φ is true for v in w iff
φ is true in w for every re-assignment of value to x :
• in the global domain D, in the eternalist semantics,
• respectively, only in the local domain Domw, in the presentist semantics.
NB: like in FOL, the truth of a formula for a given valuation (resp., in any possible world) only depends on the assignment of values of the constant symbols and free variables in that formula (resp., in that possible world). In particular, the truth of a sentence only depends on the possible world.
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Quantification in first-order temporal logic:
derived truth definitions
The derived truth definitions for F, P, and ∃ are:
M, w |=v Fϕ iff M, u |=v ϕ for some u ∈ W such that w ≺ u. M, w |=v Pϕ iff M, u |=vϕ for some u ∈ W such that u ≺ w . M, w |=v ∃xφ iff M, w |=v[a/x ]φ ...
• ... for some a ∈ D, in the eternalist semantics,
• ... for some a ∈ Domw, in the presentist semantics.
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Formal semantics for first-order temporal logic:
validity and satisfiability
For each of the two versions of the semantics we define the following notions of validity likewise.
A FOTL formula φ is:
• valid in a model M, if M, w |=vφ for every assignment v in M and every possible world w ∈ M.
• valid in a class of models, if it is valid in every model in that class.
• valid, if it is valid in every model.
• satisfiable, if its negation is not valid.
NB. Adopting one or the other semantics of FOTL essentially affects the validities, even for non-modal principles, e.g. the Universal Instantiation:
∀xφ(x) → φ(x)
This is valid in the eternalist semantics, but not in the presentist sense.
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Varying, expanding, shrinking, and constant domains
Recall that an augmented temporal frame is a tuple F = (T , ≺, D, Dom). Four types of augmented temporal frames:
1. with varying domains: no restrictions apply;
2. with expanding (increasing) domains: for all t, t0 ∈ T , if t ≺ t0, then Domt ⊆ Domt0;
3. with shrinking (decreasing) domains: for all t, t0 ∈ T , if t ≺ t0, then Domt0 ⊆ Domt;
4. with constant domains: when Domt = Domt0 for all t, t0 ∈ T . The combination of increasing and decreasing domains implies
locally constant domains: for all t, t0 ∈ T , if t ≺ t0 then, Domt = Domt0. Semantically, this is equivalent to assuming constant domains
(in languages without global modality).
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Relating the types of semantics and types of domains
The eternalist semantics essentially amounts to the assumption that the model has one fixed domain for all possible worlds,
i.e. to constant domain models.
That is why, the FOTL with eternalist semantics is also called semantics on constant domains, or CD-semantics.
Respectively, the actualist semantics amounts to the assumption that each possible world has its own domain, i.e. to varying domain models. So, the FOTL with actualist semantics is also called semantics on varying domains, or VD-semantics.
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First-order temporal logics:
Barcan formulae and axiomatic systems
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Barcan formulae in FOTL
Named after Ruth Marcus Barcan, who studied them in the context of FO modal logic in her 1946 doctoral thesis. The FOTL versions:
• TheFuture Barcan Formula scheme
BFG ∀xGϕ(x) → G∀xϕ(x) Equivalent version: F∃x ϕ(x ) → ∃x Fϕ(x ).
• TheConverse Future Barcan Formula scheme: CBFG G∀x ϕ(x ) → ∀x Gϕ(x ) Equivalent version: ∃x Fϕ(x ) → F∃x ϕ(x ).
Exercise: define the respective Past Barcan formulae. Exercises: show that:
• BFG is valid in every temporal frame with decreasing domains; in particular, in every temporal frame with constant domains.
• CBFG is valid in every temporal frame with increasing domains;
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Relating the temporal Barcan formulae
with conditions on the domains
We denote validity in the varying domain semantics by |=VD. Theorem
For any augmented FOTL frame F , the following are equivalent: 1. F has expanding
domains.
2. F |=VDBFH. 3. F |=VDCBFG.
Theorem
For any FOTL frame F , the following are equivalent: 1. F has shrinking
domains.
2. F |=VDBFG. 3. F |=VDCBFH.
Consequently, an augmented temporal frame F has locally constant domains iff either of the following formulae is VD-valid in F : BFG∧ BFH, CBFG∧ CBFH, BFG∧ CBFG, or BFH∧ CBFH.
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Axiomatic system FOTL(CD) for the minimal FOTL
with semantics on models with constant domains
I. Axiom schemes:
1. All axioms of the minimal propositional temporal logic Kt
2. Universal Instantiation (∀-Elimination):
∀xϕ(x) → ϕ(τ ), for any term τ free for substitution for x in ϕ 3. Reflexivity of equality: ∀x (x = x )
4. Extensionality: ∀x ∀y (x = y → (ϕ[x /z] → ϕ[y /z]))
5. Future necessity of non-equality: ∀x ∀y (x 6= y → G(x 6= y )) II. Inference rules (where `CD denotes derivability in FOTL(CD)):
1. Modus Ponens: If `CDϕ → ψ and `CD ϕ, then `CDψ 2. G-Necessitation: If `CD ϕ, then `CDGϕ
3. H-Necessitation: If `CDϕ, then `CDHϕ 4. Universal Generalization (∀-Introduction):
If `CDψ → ϕ(x ), then `CDψ → ∀x ϕ, where x does not occur free in ψ
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Some important theorems of FOTL(CD)
1. Symmetry of equality: `CD∀x∀y (x = y → y = x) (Derived from Extensionality, applied to ϕ := (z = x ).)
2. Future necessity of equality: `CD∀x∀y (x = y → G(x = y )) (Derived from Extensionality, applied to ϕ := G(x = z).)
3. Past necessity of equality: `CD∀x∀y (x = y → H(x = y )). (Derived likewise.)
4. Past necessity of non-equality: ∀x ∀y (x 6= y → H(x 6= y )). (Derived from the future necessity of equality.)
5. Transitivity of equality: `CD∀x∀y ∀z(x = y ∧ y = z → x = z) 6. The Converse Future Barcan formula CBFG:
`CDG∀x ϕ(x ) → ∀x Gϕ(x )(Derived just like in FOML.) 7. The Converse Past Barcan formula CBFH:
`CD∀xHϕ(x) → H∀xϕ(x)
(Derived by replacing G by H in the derivation of CBFG.) 8. `CDP∀x ϕ(x ) → ∀x Pϕ(x ).
(Derived like CBFH by replacing H by P and applying contraposition.)
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Derivation of The Future Barcan formula in FOTL(CD)
The Future Barcan formula BFG: ∀x Gϕ(x ) → G∀x ϕ(x ) is derivable, too. Proof sketch:
1. `CDPGϕ(x ) → ϕ(x ) (Instance of a TL validity) 2. `CD∀xPGϕ(x) → PGϕ(x) (Universal Instantiation) 3. `CD∀xPGϕ(x) → ϕ(x) (From (a) and (b) by propositional logic) 4. `CDP∀x Gϕ(x ) → ∀x PGϕ(x ) (Instance of Thm 8, previous slide) 5. `CDP∀x Gϕ(x ) → ϕ(x ) (From (c) and (d) by propositional logic) 6. `CDP∀x Gϕ(x ) → ∀x ϕ(x ) (From (e) by Universal Instantiation) 7. `CDGP∀x Gϕ(x ) → G∀x ϕ(x ) (From (f) by NECG, KG, and MP) 8. `CD∀xGϕ(x) → GP∀xGϕ(x) (Instance of a TL validity) 9. `CD∀xGϕ(x) → G∀xϕ(x) (From (g) and (h) by propositional logic) The Past Barcan formula BFH: ∀x Hϕ(x ) → H∀x ϕ(x )
is derivable likewise (by replacing P and G by F and H in the proof above).
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On the presentist quantification
and the semantics over models with varying domains
In the varying domain semantics, variables range over the global domain, whereas quantifiers are given a presentist reading, i.e., they quantify over the local domains only.
Consequently, some important formulae that are valid in the semantics with constant domains are not valid in the varying domain semantics. In particular, the varying domain semantics invalidates Universal
Instantiation ∀x ϕ(x ) → ϕ(τ ) as well as both the Future and Past Barcan Formula schemata and their converses.
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Axiomatic system FOTL(VD) for the minimal FOTL
with semantics over models with varying domains
I. Axiom schemes:
1. All axioms of the minimal propositional temporal logic Kt
2. Restricted Universal Instantiation (∀-Elimination):
∀y (∀xϕ(x) → ϕ[y /x]), for any y free for substitution for x in ϕ 3. Vacuous Generalization: ∀x ϕ ↔ ϕ, if x does not occur free in ϕ 4. ∀-Distributivity: ∀x(ϕ → ψ) → (∀xϕ → ∀xψ)
5. ∀-Permutation: ∀x∀y ϕ → ∀y ∀xϕ 6. Reflexivity of equality: ∀x (x = x )
7. Extensionality: ∀x ∀y (x = y → (ϕ[x /z] → ϕ[y /z])) 8. Necessity of non-equality: ∀x ∀y (x 6= y → A(x 6= y ))
(Recall that Aϕ = Hϕ ∧ ϕ ∧ Gϕ.)
II. Inference rules (where `VD denotes derivability in FOTL(VD)): 1. Modus Ponens: If `VD ϕ → ψ and `VDϕ, then `VDψ
2. G-Necessitation: If `VD ϕ, then `VD Gϕ 3. H-Necessitation: If ` ϕ, then ` Hϕ
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Inter-reducibility of the two semantics
While presentism and eternalism are alternative theories in the philosophy of time, their respective formal semantics are interreducible in some sense.
• The varying domain semantics can be simulated in the constant domain semantics by adding to the language of FOTL an existence predicatefor ‘existence at the current time instant’, which can be defined in the varying domain semantics by E (τ ) := ∃x (x = τ ).
• The constant domain semantics can be obtained from the varying domain semantics with presentist quantification by imposing constraint on the models that the Past and Future Barcan Formula schemata and their converses be valid. Using the existence predicate that constraint can be imposed in the language.
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Using the existence predicate
The predicate letter E has interpretation of E (x ) as “x actually exists”. Thus, “Some man exists who signed the Declaration of Independence” can be formalised as
∃x(Man(x) ∧ Signed (x))
which is true at the present (in the eternalist semantics), or as
∃x(E (x) ∧ Man(x) ∧ Signed (x)) which is false at the present.
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Axioms for the minimal FOTL with E :
the Free Logic version
Free Logic allows for terms that do not refer to any existing entity. That can be captured in the language by using the existence predicate E . Some valid principles of FOL are not valid here and need to be modified.
• The axiom scheme of Universal Instantiation (∀-Elimination) is restricted here by using the predicate E :
` ∀xϕ(x) → (E (y ) → ϕ[y /x]), where y does not occur free in ϕ.
• The rule of Universal Generalization (∀-Introduction) is modified accordingly:
If ` ψ → (E (x ) → ϕ(x )), then ` ψ → ∀x ϕ(x ), where x does not occur free in ψ.
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Using the existence predicate generalized
For every formula ϕ of FOTL, its E -relativizationin the extended with E language can be obtained by replacing every occurrence of ∀x in ϕ with
“∀x (E (x ) → ...)” and every occurrence of ∃x with “∃x (E (x ) ∧ ...)”. Theorem
For any FOTL sentence ϕ not containing E :
ϕ is valid in the varying domain semantics if and only if its E -relativization is valid in the constant domain semantics.
The question that remains, from a philosophical point of view, is whether existence is a legitimate predicate. Many philosophers reject that.
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Further issues on first-order temporal logics
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Rigid designators and definite descriptions
The interpretation of rigid designators, in particular proper names, should not vary over time.
However, the interpretation of definite descriptions may vary over time (e.g. ‘the King of Sweden’)
In the formal semantics of FOTL, the interpretation of constant symbols is specified locally, relative to each time instant, even though it denotes globally. Constant symbols can thus be used to identify an object in time. For instance, using a as a name for Aristotle, the sentence
“Aristotle existed but no longer exists”
can be formalized in the varying domain semantics as P∃x (x = a) ∧ ¬∃x (x = a).
But, what about e.g., ”The king of France existed, but no longer exists.”? or, ”Le roi est mort, vive le roi!”?
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Rigid designators and necessity of identity
If terms (variables or constants) τ1, τ2 are interpreted as rigid designators, the temporal version of the principle of Necessity of Identity is valid:
τ1 = τ2 → H(τ1 = τ2) ∧ G(τ1 = τ2)
That, however, is generally invalid, even for definite descriptions (e.g. “the King of Sweden” and “Carl XVI Gustaf”)
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Definite descriptions, revisited
In a temporal setting, the question arises of how to deal with definite descriptions that refer to objects that no longer exist, or do not yet exist, such as “the last person who died in Sweden in the 20th century” or
“the first child to be born in Sweden in 2050”.
The problems get even more intricate in a branching time setting, where time and modality (as historical necessity) are combined.
For instance, some definite descriptions that are temporally rigid may be modally non-rigid, and vice versa.
A way to deal with definite descriptions is to switch from an extensional to an intensional account of individual terms.
That is, rather than assigning to each term at each time instant an extension, i.e. an object from the domain, one may assign to each term an intension, i.e. a function from time instants to objects.
One general framework in which individual terms are assigned both extensions and intensions is the Case Intensional First-Order Logic (CIFOL) by Belnap and M¨uller (see paper linked to the course website).
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Some technical results on first-order temporal logics
First-order temporal logics are very expressive, and this often comes with a high computational price. These logics can be deductively very complex and are typically highly undecidable.
For example, the FOTL with constant domain semantics over the natural numbers, with only two variables and unary relation symbols is not only undecidable but not even recursively axiomatizable.
Few axiomatizable, and even fewer decidable, natural fragments of first-order temporal logics have been identified and investigated so far. One such case is the monodic fragment of FOTL, only allowing formulae with at most one free variable in the scope of a temporal operator. For more, see references in the SEP article.
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FO temporal logics and models: some research questions
Both the presentist and the eternalist semantics can apply to models linear and with branching time temporal flows.
Some research questions:
• Which is the right, or the better one, if any, for each of the notions of existence over time?
• What are the right axioms and conditions on FO temporal models for each of these and the various temporal semantics:
Priorean, Ockhamist, Peircean?
Many open technical problems on axiomatizations, completeness and decidability.
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Concluding remarks for the course
There is a rich and diverse variety of temporal logics, used in philosophy, computer science, artificial intelligence and linguistics.
Many deductive systems and algorithmic decision procedures have been developed for them.
The field is now mature, but still actively developing, with many new conceptual and technical problems and new applications arising. The purpose of this course was to offer a flavour of these.
THE END
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