The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of
imperatives
http://www.ffst.hr/~logika/implog
Berislav Žarnić
University of Split, Croatia
Warszawa 2011
http://www.ffst.hr/~logika/implog
The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Two prominent features of philosophical view on
imperatives
Use of modal logic, from static beginnings to recent dynamic trends.
Investigation of connections between imperatives and actions.
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
Agentives and imperatives
Possible worlds semantics has been established as one of the main
tools in philosophical analysis in the last third of 20th century. Using
the possible worlds semantics it became possible to explicate the
meaning of some words, which have permeated the philosophical
discussion over the centuries (e.g. necessary, possible, obligation,
permission, action, knowledge, etc.).
In this way the “logical terminology” ceased to be limited to the
small collection of just a few words (truth-functional connectives,
simple quantifiers, and identity predicate), but included an open
collection of words, and even the sentence moods (imperative logic,
erotetic logic). In this way, after a short, post-Fregean period of
limitation to the language of mathematics and natural sciences, logic
has turned back to its full scope of investigation. Given the fact that
modal logic deals with the logic of language being used in
philosophy, as well as in social sciences and humanities, modal logic
is sometimes colloquially called ‘philosophical logic.’
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
A too short introduction to modal logic
Possible worlds semantics
The idea of possible worlds was introduced in Gottfried Wilhelm
Leibniz (1646–1716), but remained theoretically inert until Rudolf
Carnap introduced explication for possible worlds in terms of
formally consistent and complete set of sentences (“state
descriptions”), and Stig Kanger and Saul Kripke introduced the
notion of accessibility relation that shows which possible worlds are
to be taken into consideration.
Use calculator!
Notions to introduce: possible world, accessibility relation, frame,
valuation.
Heuristic idea to capture: the meaning of some logical words is a
‘semantic space’ which has certain structure.
Example S5
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
A too short introduction to modal logic
Truth
Definition
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
ϕ is true at w in M iff
M, w |= p iff w ∈ V (p ), if p is propositional atom
M, w |= ¬p iff not M, w |= p
M, w |= p ∧ q iff M, w |= p and M, w |= q
M, w |= p ∨ q iff M, w |= p or M, w |= q
M, w |= p → q iff not M, w |= p or M, w |= q
M, w |= p iff M, v |= p
for all v such that Rwv
M, w |= ♦p iff exists v such that Rwv and M, v |= p
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
A too short introduction to modal logic
Exercise
Build a model M = hhW , R i , V i:
W = {w1 , w2 , w3 }: press buttons +1, +2, +3;
R = {hw1 , w2 i, hw1 , w3 i, hw2 , w1 i, hw3 , w3 i}: press buttons > 2,
> 3, > 1, and > 3 in appropriate columns;
V (p ) = {w2 , w3 }, V (q ) = {w1 }: press buttons p and q in
appropriate columns (the color of the letter in the world changes to
green if the letter is true there).
Enter the formula p → p and see in which worlds it is true (green
worlds) and in which false (red worlds). E.g. the formula is true in w2 ,
and we write M, w2 |= p → p. Explain why!
Is R transitive?
Build an arbitrary model which is transitive and inspect the formula by
choosing the valuations at random! What do you notice?
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
A too short introduction to modal logic
Logical truth in modal logic
Logical truth is a general truth, “holding always and everywhere.” There
are number of ways to define logical truth in possible worlds semantics:
(V1) validity in the model: being true at each world;
(V2) validity in the frame: being true at each world in any
model built over the frame (This kind of validity can be
checked using calculator by changing valuations.);
(V3) validity in the class of frames: being (V2) valid on each
frame from the class;
(V4) validity in all models.
In modal logic (V3) notion is used. It shows, so to speak, that the
semantics of modal operators (e.g. words like ’necessary,’ ’obligatory,’
’known’ etc.) is captured by diverse structures (their ”conceptual space”
is given by a particular structure).
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
A too short introduction to modal logic
Characterization of frames
Definition
Formula p characterizes class S of frames iff
for each frame F , for each valuation V and each world w
hF , V i , w |= p iff F ∈ S
In other words p characterizes S iff p is valid (V3) in S.
Some binary relations can be characterized both by modal logic
formula and first-order formula, some only by one of the languages.
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
A too short introduction to modal logic
Axiom schemata and frames
Axiom schema
T
4
5 or E
B
D
1 ∀x ∀y ∀z
characterizes
p → p
p → p
♦p → ♦p
♦p → p
p → ♦p
class of frames
reflexive
transitive
Euclidian1
symmetric
serial
((R (x, y ) ∧ R (x, z )) → R (y , z ))
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
A too short introduction to modal logic
S5
Logics of logics
For modality ’. . . logically true’ and for some other modalities too,
S5 = KT 4B = KT 4E logics seems appropriate.
Axiom schemata of S5 characterizes the class of reflexive, transitive
and symmetric frames, i.e. class of frames where accessibility is an
equivalence relation.
Thanks to equivalence of R, when using S5 we can disregard
accessibility relation.
In S5 , ♦ and ¬ create classical “square of opposition.” Segerberg
interpreted logic of action of St. Anselm (1033–1109) in that way.
See calculator!
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
Chelass’ modal logic of imperatives
Different modalities
Philosophical analysis typically require multiple modalities as its very
name ([desire to] [know]) suggest.
In the seminal paper in modal imperative logic Brian Chellas uses
two binary accessibility relations, St for world lines that overlap up
to the time point t, and Rt for relation of “imperative alternative.”
Modalities ! and are standardly defined as universal (holding in all
Rt alternatives) and existential (holding in some Rt alternatives).
!
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
Chelass’ modal logic of imperatives
Text analysis
Now the three conditions on the relation of imperative alternativeness
may be stated precisely: for each w , w ′ , w ′′ ∈ W , and t ∈ T ,
(I) there is a w ′ ∈ W such that Rt (w , w ′ );
(II) if Rt (w , w ′ ), then St (w , w ′ );
(III) if St (w , w ′ ), then Rt (w , w ′′ ) iff Rt (w ′ , w ′′ ).
Brian Chellas. 1971. Imperatives.
Theoria 59:114–128.
Given the fact that Chellas directly (i.e. in the model) characterizes
the relations, is axiomatic presentation of imperative logic to be
expected in his text?
Look at the condition (I)! Name the relational property it describes!
State the axiom that characterizes it! Is Rt transitive?
Chellas reads !p as ‘Let it be the case that p’ and interprets it as an
expression of imperative obligation. What reading and interpretation
seems appropriate for ?
Explain why !p ⇔ ¬ ¬p holds!
Use Chellas notation for “historical necessity” (true in all St
relata) and represent condition (II)!
!
!
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
Chelass’ modal logic of imperatives
Labeled natural deduction for Chellas’ logic
Basin, Matthews and Vigano have developed a system of deduction for normal logics
with one modality lying within “Geach’s hierarchy” (i.e. those whose relational theory
is representable in first order language). The rules are general to any universal and
existential modality (e.g. rules are applicable to Chellas’ ! and ); the differences
between logics are introduced using relational theory which describes frame properties.
At each w we use classical rules.
!
Rules for operators ( = !, , ♦ = , ♦)
Intro
Elim
♦Intro
♦Elim
Γ, Rwv ⊢ v : p ⇒ Γ ⊢ w : p
Γ ⊢ Rwv , w : p ⇒ Γ ⊢ v : p
Γ ⊢ Rwv , v : p ⇒ Γ ⊢ w : ♦p
Γ, Rwv , v : p ⊢ ϕ ⇒ Γ, w : ♦p ⊢ ϕ
v does not occur in Γ
v does not occur in Γ ∪ { ϕ}
Relational theory
(I) ⊢ Rt (w , f (w ))
(II) Rt (w , v ) ⊢ St (w , v )
(III) St (w , v ), Rt (w , u ) ⊢ Rt (v , u ) and
St (w , v ), Rt (v , u ) ⊢ Rt (w , u ).
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
Chelass’ modal logic of imperatives
Stoic theorem: desiring the unavoidable
1
2
w : p
3
v
assumption
Rt ( w , v )
assumption
4
St (w , v )
3/ (I)
5
v :p
2, 4/ Elim
6
w :!p
7
w : p → !p
2–5/ !Intro
2–6/ →Intro
Problem
Is ‘Let it be the case whatever is necessary the case’ a theorem of logic or
a thesis of particular normative system?
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
Chelass’ modal logic of imperatives
Ross’ paradox
1
2
w : !p
3
v
assumption
Rt ( w , v )
assumption
4
v :p
2, 3/ !Elim
5
v : p∨q
4/ ∨Intro
6
7
w :!(p ∨ q )
w :!p →!(p ∨ q )
3–5/ !Intro
2–6/ →Intro
Problem
Connective ∨ is problematic on both introduction and elimination side in
imperative context.
Permissions distribute over disjunctions (‘You may take an apple or pear’
is a “free-coice permission”). In Chellas’ system (p ∨ q ) 6⇒ p. Create a
counterexample using calculator!
!
!
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
Chelass’ modal logic of imperatives
No obligation w.r.t. p implies permission w.r.t. ¬p
w : ¬!p
assumption
!
1
2
w : ¬ ¬p
3
v
assumption
Rt ( w , v )
assumption
4
v : ¬p
5
w : ¬p
3, 4/ Intro
6
w :⊥
2, 5/ ⊥Intro
!
w :p
!
7
assumption
4–6/ ¬Intro
8
w :!p
3–7/ !Intro
9
w :⊥
1, 8/ ⊥Intro
!
2–9/ ¬Intro
10
w : ¬p
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Modal logic and theory of imperatives
Chelass’ modal logic of imperatives
Deconstructing Chellas’ semantics
Thesis
The semantics of imperatives is multi-layered.
Proof.
The abundance of different systems, each convincible in its own right, modeling different aspects,
e.g. wishes of the Speaker (Chellas), will of the Imperator (Segerberg), actions commanded
(Belnap et al.), preferences of the Hearer (Van Benthem and Liu), obligations of the Hearer
(Yamada), etc.
Also, our discussion yesterday.
Semantic dimensions in Chellas’ system
Semantic dimensions Speaker Hearer
bouletic
Yes
No
doxastic
No
No
Yes
No
deontic
agentive
No
Do Chellas’ dimensions mix well? It is the theorem that ¬!p ⇔ ¬p: is it
really so that ‘negated wish’ equals ‘permission’ ?
!
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Imperatives and action
Let us recall!
Imperative content thesis.
Regardless of its force on an occasion of use, the content of every
imperative is agentive.a
Nuel D. Belnap. 2001. Facing the Future: Agents and Choices in
Our Indeterminist World, p. 10.
Oxford University Press
a ‘Agentive’
denotes ‘agency ascribing sentence.’
The problem of modeling: to respect ‘imperative content thesis,’ to
preserve multi-layered semantics (to a certain extent), to get “logical
geography” right.
Which logic of action to take into consideration for incorporation
into logic of imperatives?
There is number of logics of action to choose from. In particular,
Krister Segerberg’s, and Nuel Belnap’s theories stand out.
But we will turn to Georg Henrik von Wright. Why?
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
The father of logic of action
Von Wright has been titled “the father of logic of action” by
prominent authors in the field (v. Segerberg 1992; Hilpinen 1997)
and rightly so.
Von Wright’s semantics of action is simple, reduced to basic
elements, and yet strong enough to explicate important distinction.
Von Wright’s semantics can be easily adapted to dynamic semantics.
We will discuss his logic of action in its final phase:
Georg Henrik von Wright. 1966. The logic of action : a sketch.
In The Logic of Decision and Action, ed. Nicholas Rescher
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Act and forbearance
To act is intentionally (“at will”) to bring about or prevent a change in
the world (in nature). On this definition, to forbear (omit) action is
either to leave something unchanged or to let something happen.
Georg Henrik von Wright. 1966. The logic of action : a sketch.
In The Logic of Decision and Action, ed. Nicholas Rescher
Remark
There are two types of action: productive action, preventive action; and
two types of forbearance (omission): letting something happen, leaving
something unchanged.
‘To act’ refers both to “productive or preventive interference with the
world” and to forbearance.
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
A sequence of definitions
G. H. von Wright gives a sequence of definitions, ending with ‘generic
state of affairs’ as a primitive term.
Action Action is bringing about or preventing a change in the
world (in nature).
Change Change is transformation of states (of affairs).
Non-changes Changes occur when a state of affairs
ceases to be or comes to be or continues to
be. “Non-changes” are immediate
progressions in time with the same initial
and end-state, and non-changes are
included among changes.
State of affairs ‘State of affairs’ is introduced by way of an example:
“the sun is shining” is an example of a generic state of
affairs,2 which can be “instantiated on a certain occasion
in space and time,” and “instantiated state of affairs” is
individual state of affairs.
2 2.01
A statehttp://www.ffst.hr/
of affairs (a state
of things) The
is amainstream
combination
of objects
in philosophical semantics of imperatives
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The mainstream in philosophical semantics of imperatives
Logic of action
Modeling worlds
Von Wright identifies the notion of ‘total state of of the world on a
given occasion’ with a description that indicates “for every one of a
finite number of n states p1 , . . . , pn whether it obtains or does not
obtain on that occasion.”
In the formal sense we will reduce full state descriptions to sets of
non-negated propositional letters.
Definition
Let A = {p1 , . . . , pn } be a finite set of propositional letters. Any subset
w ⊆ A is a “Wittgenstein world,” total state, “state description.”
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Lindenbaum’s sets
The reduced description can be easily expanded to the full state
description either in semantic or syntactic terms. In semantic terms,
a truth assignment can be defined as binary function determined by
w
h(p, w ) = t iff p ∈ w
In syntactic terms, set w can be expanded to set of literals
lt(w ) = w ∪ {¬p | p ∈ A − w }. The conjunction of all the literals
will be called ‘state description’
^
lt(w )
where literals are assumed to be listed in the conjunction according
to their alphabetic order.
It is well known fact that a valuation of propositional letters
determines the valuation of all sentences in the language of
propositional language, as well as the fact that the set of all literals
is syntactically complete. Therefore, subsets w of propositional
letters provides a minimal representative of formally complete and
consistent sets.
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
A digression: time, action, imperative
There is a strong ontological presupposition in the notion of action,
and consequently in use of imperative sentences.
Acts include counterfactual element: if it not were for the agent’s
interference with the nature, the proper change would not have
occurred (in the case of productive act), or the proper change would
have occurred (in the case of preventive act).
So, the notion of time that lies at the bottom of the concept of
action, the notion of time that makes our “imperative language
practice” possible, is the notion of time with the “open future.”
Let W denote a set of total states, and R ⊆ W × W an ordering: a
“history,” and I set of indexes:
Deterministic time: |{R | R ⊆ W × W }| = 1;
Relativistic time: |{R | R ⊆ W × W }| > 1; for each i, j ∈ I
mem1 (Ri ) ∪multiset mem2 (Ri ) = mem1 (Rj ) ∪multiset mem2 (Rj );
Common-sense time (open future, “time of imperatives”) belongs to
the family: |{R | R ⊆ W × W }| > 1; for some i, j ∈ I
mem1 (Ri ) ∪multiset mem2 (Ri ) 6= mem1 (Rj ) ∪multiset mem2 (Rj ).
This is just the sketch; more rigorous modeling is needed.
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Change expression
V
V
lt(wi )T lt(wj ) is a “change expression” describing either a
proper change or a continuation (non-change) of a total state. It is
the shortest possible history.
T is to be read ‘and next.’ The T expressions can be concatenated
T( T(...T )...) and if the empty places are filled with state
descriptions, then a “history” (i.e. sequence of total states) will be
depicted.
There is a need to index T connective in order to compare different
histories since change expressions only show that a “generic state” is
immediately followed by another in the row of “discrete time points.”
Hence, the time enters the Von Wright’s modelling in two ways:
1
2
there is an ordering of time points (time as ordering),
according to which a sequence of state-descriptions can be composed
(time as history).
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
History
Example
0
1
2
V
V
lt(wi ) T
lt(wj ) T
lt(wk ) T
a full description of a history of length n
V
3
lt(wl )
T
...
...
3
¬p ∨ ¬q
T
...
...
V
n
V
Example
0
1
2
p ∧ q T ¬p ∧ q T p ∨ q
a partial description of a history
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T
The mainstream in philosophical semantics of imperatives
lt(wm )
The mainstream in philosophical semantics of imperatives
Logic of action
The descriptions of different histories cannot be combined.
Example
(pTp ) ∧ (¬pT¬p ) is inconsistent expression if understood as a
description of a single history. On the other hand, if (pTp ) ∧ (¬pT¬p ) is
| {z } | {z }
h1
h2
taken to be a description of two histories, h1 and h2 , the conjunction is
consistent one. Moreover, if T is understood as the border between same
1/2
1/2
h1
h2
time points, e.g. (p T p ) ∧ (¬p T ¬p ) , than the conjunction describes
| {z } | {z }
two concurrent histories.
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Example
Let hAg be a history in which the agent is present as agent (i.e. as one
having the ability to interfere with nature), and Nt a history in which the
agent is not present as an agent, but only as “natural object.” The
comparison shows that state of affairs p at the moment after is due to
agency, the agent is responsible for p at the moment after .
hAg
hNt
before
V
lt(w )
V
lt(w )
http://www.ffst.hr/~logika/implog
T
after
p
¬p
The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Actions, intentions
621. Let us not forget this: when ’I raise my arm’, my arm goes up. And
the problem arises: what is left over if I subtract the fact that my arm
goes up from the fact that I raise my arm?
Wittgenstein, PI: 161
. . . an event is an action if and only if it can be described in a way that
makes it intentional.
Davidson, 2001: 229
In the famous Davidson’s definition of ‘action,’ actions are a subset of
events. It seems that, according to Davidson, an action α is an event e
together with intention i:
Action(e ) iff Event (e ) ∧ Intentional (e )
If the Davidson’s ontology of events is abandoned, then events must be
modeled. It seems quite natural to identify theoretically events with
changes, and these with transformations of states of affairs.
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Actions, intentions
Example
Let C stand for ‘the window is closed’. The event of opening the window
is described by change expression C T¬C . In order to describe the action
of opening the window, a notion of ‘intentionality’ is needed, which
presumably will turn out to be a very complex, involving not only an
appropriate mental state of the agent, but also a notion of causation.
An act is not a change in the world
It would not be right, I think, to call acts a kind or species of events. An
act is not a change in the world. But many acts may quite appropriately
be described as the bringing about or effecting (’at will’) of a change. To
act is, in a sense, to interfere with ’the course of nature’.
Von Wright, 1966: 36
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Actions, intentions
Modeling causation
Problem
“An act is not a change in the world,” writes von Wright. To each act
there corresponds a change in the world, but the act is not identical to it.
The formal representation of act requires taking into account concurrent
histories (i.e. sequences of total states). The pre-theoretical idea of
causation, bringing it about, seeing it to that a certain generic state of
affairs obtains is captured by parallel histories: if p occurs in all after
instant of agency histories, and in no after instant of nature histories,
then agency is necessary and sufficient condition of p.
p obtains in
p obtains in
p obtains in
agency
all
some
all
nature
none
none
some but not all
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suff.
suff.
nec.
nec.
G. H. von Wright
Belnap et al.
The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Actions, intentions
An act is not a change, but . . . a cluster of changes
So at least two histories must be taken into account to represent action:
1
agency history: it is the (proper or vacuous) change for which the
agent is responsible,
2
nature history: it is the counterfactual element in the concept of
action, the change that would have occurred if the agent had not
interfered with the world, the history in which the agent has been
removed as agent (i.e. has no intentions), but still present as
physical object.
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Actions, intentions
The minimal cluster of act and omission
E
C
Agency
Nature
I
E
C
Agency
Nature
I
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Actions, intentions
Thus we have in the simplest form the action semantics consist of three
different points in a branching or convergent configuration:
1
2
3
initial point is common for all acts (either productive or porventive),
end-point, results from agent’s action,
‘counter point’, which would have happend had the agent remained
passive.
There are two modes of action: act and forbearance. While act is
characterized by the fact that end-point and counter-point are different,
in forbearance they coincide. One might call productive and preservative
acts — proper acts, and forbearances — letting things happen.
See calculator!
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The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Actions, intentions
Von Wright in Norm and Action uses d ( T ) and f ( T ) notation for acts
and forbearances; while in The Logic of Action: A Sketch he uses
connective I, to be read ‘instead’. Let us use full state descriptions of the
V
form lt(w ).
act
forbearance
V
V
V
lt(wi )T(V lt(wj )I V lt(wk )) where j 6= k
V
lt(wi )T( lt(wj )I
http://www.ffst.hr/~logika/implog
lt(wj ))
The mainstream in philosophical semantics of imperatives
The mainstream in philosophical semantics of imperatives
Logic of action
Actions, intentions
Interesting configurations
There is a number of useful distinctions that one can make on the basis
of three point semantics, i.e. two concurrent shortest histories.
Let |A| = n. Then the number of shortest histories of length m = 2 is
2n × 2n .
determinism in nature
indeterminism in nature
impotence
omnipotence
V
lt(wi )T(⊤Is )
if only one total description s satisfies the formula
V
lt(wi )T(⊤Is )
if more then one total description s satisfies the form
V
lt(wi )T(sI⊤)
if only one total description s satisfies the formula
V
lt(wi )T(sI⊤)
if any total description s satisfies the formula
http://www.ffst.hr/~logika/implog
The mainstream in philosophical semantics of imperatives