John Pollack: The Foundations of Philosophical Semantics
p.3
The principle tool of philosophical semanitcs is the concept of a possible world. Carnap began it all by
talking about state descriptions.
p.4
"Today talk of possible worlds is taken literally [not just heuristically], and it is maintained tha tposs worlds
are real entities that actually exist and to which we can appeal in philophical analysis.
Distinguishes between Realistic and formal semantics. The former makes explicit appeal to possible
worlds, the latter doesn't. "Possible worlds loom faintly in the background, guiding the hand of the formal
logician, but formal semanitcal theories are not framed directly in terms of possible worlds. Foraml
semantics is carried on entirely within set theory. The basic tool of formal semanitcs is the concept of a
model, which is a kind of set-theoretic structure."
p.5
"Despite its popularity and apparent fruitfulness, the foundations of philosophical semantics are in doubt.
Although philosophers of widely differeing persusions make free use of possible worlds, there is less than
complete agreement about the nature of possible worlds and how they are reated to, for instance, necessary
truth, propositions, and meaning. Theses question bear most directly on realistic semantics, and they will
occupy our attention throught most of this book. In the final chapter we will take up the foundations of
formal semantics, which will turn out to be a morass. There are established techniques used in formal
semantics, but it is not clear what to make of the results obtained by these techniques. It is remarkable that
few practitioners of formal semantics have even raised the question of its significance. I will attempt to
answer this question by drawing precise connectons between formal semantics and realistic semantics.
p.7
Sketch of a Theory of Language
Propositions are fine-grained objects of belief. "…propositions are possible objects of belief or
disbelief…in order for  to be a proposition, it must be oss for there to be a person who either believes  or
disbelieves ."
p.8
This is a necessary, but not sufficient condition for being a proposition.
Two types of propositions: transient (truth values vary with times) and nontransient.
p.9
Distinguishes between directly referential propositions and indirectly referential. The former involve the
same object, but the object may be thought of in different ways—just as long as the same property of the
object is believed to be exemplified.
Maximally fine-grained objects of belief involve objects being thought of in the same way, at the same
time, and sometimes by the same person. Pollack reserve the term proposition for this sort of object of
belief.
p.10
We might identify coarse-grained objects of belief with sets of propositions. Note: Pollack speaks of
objects of belief vis-à-vis content of belief vis-à-vis intentional objects. "For example, the directly
referential proposition that Herbert has a mustache would be identified with the set of all propositions
ascribing having a mustache to Herbert under different modes of representation." Is a mode of
representation the same as a mode of presentation?
The sense of an expression is a ‘mode of presentation’ of its referent. The sense
associated with ‘the City of Light’ presents Paris in a particular way: as the City of Light.
To understand an expression is to grasp its sense (see Sense and reference). Frege
called the sense of a sentence a ‘thought’. To have a propositional attitude is to stand in
a certain relation to a thought. For example, to believe that Paris is beautiful is to stand
in the believing relation to the thought that Paris is beautiful. Frege claimed that senses
and thoughts are not psychological, mind-dependent things, but rather are Platonic,
abstract objects. If two people believe that Paris is beautiful, then it is the very same
thought that they both have. And this thought would exist even if there had been no
minds (see Frege, G. §4).
Looks similar. Would 'Paris' be a mode of presentation for the City of Light? It would seem so.
It is commonly (but not universally) supposed that propositions have structures and constituents; this
doctrine is problematic. "I assume that it is to be spelled out in terms of the corresponding notion of the
structure of our thoughts…I will simply assume that propositions an be individuated in terms of their sturc
and constituents and attribute to propositions however much structure that requires."
1.2 Concepts
The most familiar sort of propsitional constituent. "Objects "fall under" or exemplifyconcepts. What, if
anything, is the difference between a property and a concept for Pollack? He will take the finest-grained
criteria to individuate concepts (concepts can be general or specific in being believed of objects).
Concepts are what we believe or disbelieve of objects.
1.3 Propositional designators
A variety of logical operators will also serve as constituents of propositions. It seems like a clear notion of
intensional constitution needs to be specifed in clarifying an anti-existentialist position of intensional
entities likes propositions and states of affairs. This what I've been meaning by 'involve'. A fine-grained
proposition may entail many other propositions, but it must be individuated by its content. Propositional
designators are those constituents of propositions that pick out the objects (or objects presumably) that the
proposition is about. But must all propositions be about an object? What is he including as objects?
Anything?
p.11 Propositional designators include definite descriptions. De re designators are nondescriptive
designators. There are other nondescriptive designators as well. Personal designators are nondescriptive
designators of ourselves. The present time: temporal designators.
p.13
Pollack takes necessity and possibility to be properties of propositions. Necessity and possibility can also
be employed as modal operators.
"If we are to identify  with some proposition to the effect that  is necessary, we must seize upon a
particular propsitional designator δ that designates  and then, letting N be the concept of being necessarily
true, identify  with (N:δ). But what designator might δ be? Clearly, δ cannot be the same designator as
in either (1) [the proposition that John's favorite proposition is necessary] or (2) [the proposition that the
first proposition entertained by Bertarand Russell on the morning of Apirl 7, 1921, is necessary)] above.
(1) is about John, and (2) is aobut Bertrand Russell, but  is about neither. Similar
p. 14
difficultlties arise for any designator that designate  in terms of contingent properties it happens to have.
If this general proposal for the analysis of  is to work , it must proceed in terms of some kind of
prepositional desinator that designatres  necessarily. Is there such a designator, and does it make the
analysis plausible?
The answer to this question turns upon our having a special way of thinking about propositions.
There is a difference btrwn entertaining a proposition and thinking about it. If I think to myself that 2+2=4,
I am entertaining the proposition [this depends on how we take 'that'; this is ambiguous between that p and
that p is true]. But I can also think to myself that the propositions that 2+2=4 is true. This is to entertain a
proposition about the proposition that 2+2=4 is true. There is a distinction here, I think. Whaen I do this, I
don not normally think of the proposition that 2+2=4 in terms of some contingent description of it. I think
of it in terms of its content. This is a special way of thinking of a prp. If we c ould not think of
propositions in this way, we would have no way of judging that a proposition is true or necessary. Suppose
I think about  in this special way, perhaps believing that  true. I tereby entertain a proposition about , so
the proposition I entertain contains a propsitioanl designator designating . That designator corresponds to
my special way of thinkg about . Let us call such designators logical designator. I will write the logical
designaotr for  as [<>]. There must be analogous logical designators for concepts. My proposal is now
that we can identify modal propositions with propositions employing logical designators:
(1.4)  = (N: <>)
(1.5) ◊ =~~
Pollack here is not using '=' consistently, since, presumably he is asserting an identity here. He says, "I
earlier distinguished between  and the proposition that  is necessary by saying that  is about whatever
 about, while the propositions that  necessary is about  itself. (1.4) seems to fly in the face of that
distinction. But it does so only partially. To think of in terms of <> is to think of  in a very special
way—in terms of its content. Thus there is a sense
p.15
in which, although (N: <>) is about , it is about  in terms of whatever  is about, and so can also be
viewed as being about whatever is about. My suggestions in then that it is the existence of logical
designators that makes modal propositions and modal operators possible."
It seems to me that all Pollack needs to say here is that there is no distinction between necessity as a
property of propositions or as an operator as long as the proposition in question is logically designated.
Thus, I don't think that Pollack has here made clear the distinction between modality as an operator and
modality as a property of propositions. Thus it would seem that 1.4 is in fact a true identity statement,
taking  as a property of  and  as <>.
[From Language and Thought:  is a proposition which is about , because it contains a propositioanl
designator designating , but it is about  in a very special way. To think of in terms of the designator
<> is to think of  in terms of its content. thus there is a resasonable sense in which  is also about
whatever is about. It is important to recognize that  an operator, not a concept, because  is appended to
the propositions itself to form the proposition .  is not appended to adesignator designating . It is
only possible to construct such an operator, however, because there are the privileged designators <>."
—p.177.
Pollack seems to take necessity as a primitive property.
p.19
Pollack thinkgs it most likely that de dicto necessity can be reduced to de re necessity. A de dicto
necessary proposition is really just a de re necessary about the object (in this case the proposition) which
necessilary exemplified the property of truth.
p.22
Statements  propositions since the senten and received propositions of statements can vary because of the
mode of representation that we deploy for the objects referred to. E.g., Jessi and me would think of Janet
Garlow differently in Janet Garlow is resting right now than I might. Directly referential
propositions(which, may—I'm not sure—be identical to directly referential propositions or perhaps include
these as one sort of statement there is) are coarse-grained objects of belief; but, propositions are maximally
fine-grained objects of belief. Hence, the former is not what Pollack means by propositions.
p.25
Statements are constructed oout of various statemental constituents, the simplest of which is an attribute.
Attributes are more general than concepts, which are constituents of propositions. Attributes are like
coarse-grained concepts. "Just as we can talk about sentence and received porps for statements, we can talk
about sentence and received concepts of rattributes. In the statemnt 'John is a brother of Robert' the
concept 'brother of Robert', but since Robert can have different prepositional designators, the attribute
covers all of these..
p.26
Sent and received concepts comprise the diagram of the attribute of being aluminum. E.g., I might think of
aluminum differently than a metallurgist.
p.37
The meaning of sentences (most of which are identical for at least time, if not because they contain
indexical nouns) are all of statements that can be made by the sentence.
"It is useful to distincguish between (1) the meaning of a senp. 38
tence and (2) its senseon a particular occasion. The sense of a sentence on a particular occasion is the
statement made by using it on that occasion. The meaning of a sentence (i.e., its S-intension) is the
function that determines its sense on each possible occasion of its use."
p.43
Possible Worlds
p.52 Possible worlds as maximal states of affairs has the advantage over possible worlds as world-books in
that it can say how an object can be identified across worlds though it has varying properties in various
worlds; the object becomes part of the specification of the world. On the world-book account, one must
appeal to haecceities, which Pollock denies for some objects like billiard balls. He may be right here;
haeceities (qua properties like being Socrates), if they are individuators are essential to the their objects.
Plantinga seems to solve this with properties like being red at alpha, though. This issue isn't critical to my
research.
"Plantinga has often talked about another kind of essence. For example, he talks about the property of
being Socrates. This is supposed to be what is expressed by the predicate [x = Socrates]. His remarks
suggest that he takes this property to be directly referential. It involves Socrates "directly" rather than via
some representation. I do not wan to deny that there is such a property, but I do insist that it is neither a
concept nor an attribute. Once we have a workable notion of a possible world, we will be able to define a
broad notion of a property that includes properties like that of being Socrates. But such properties cannot
be used in constructing world books, because they are not constituents of either propositions or statements.
I will grant that these properties are constituents of proposition-like objects—what we might call 'truths'
(and what Ploantinga calls 'propositions'). But these are neither fine-grained objects of belief nor products
of assertion." Why this move to truths? What can't these properties be constituents of propositions?
"It has become customary to "explain" necessary truth as truth at all possible worlds." Possible worlds are
employed in a variety of ways, but it has never been entirely clear just what possible worlds are supposed
to be. Lewis says possible worlds are ways things could have been. Kripke: conuterfactual situations.
"The two most popular moves have been to identify possible worlds with maximal consistent sets of porps,
and to identify them with maximal possible staes of affairs.
p.52
Possible Worlds as Maximal States of affairs
"We normally express states of affairs in English by employing gerund clasues. "
Pollack thinks that my not existing is a state of affairs. He admits the existence/obtain distinction.
This would have to be an impossible state of affairs; if there's any state of affairs I'm inclined to
disallow, it's these.
Pollock symbolizes x's being  [x|]. If  is an assertion, then [Ø|] is 's being true.
"In many way, states of affairs resemble propositions. In
p.53
particular, they are truth bearers of a sort. States of affairs are not literally true or false, but
obtaining and not obtaining are truth-like properties. There is at least one respect, however, in
which states of affairs differ importantly from propositions. States of affairs are directly
referential in a strong sense in which propositions are not. For example, if Mary is the girl in the
red hat, then (on at least one construal of the definite description) the girl in the red hat's looking
lost is the same state of affairs as Mary's looking lost. States of affairs may involve objects
directly rather than under a description. This seems like constituent talk; not necessarily; this
could just be de re talk. Because states of affairs are directly referentioal, they are suitable for the
construction of possible worlds satisfying the desideratum of explaining de re necessity.
State of affairs are very much like directly referential propositions [which Pollack doesn't think
exist]. they are "about" objects but not in terms of some mode of representation. States of
affairs, in some sense, contain objects as direct constituents. Whether we distinguish between
directly referential propositions and states of affairs seems to me mainly a matter of
conveniences.
It is generally (though not universally) acknowledged that distinct propositions can be logically
equivalent. The question arises whether this is also true of states of affairs; i.e., can two distinct
states of affairs be necessarily such that one obtains iff the other does? I do not see any basis in
intuiation for
p.54 deciding this matter, so I am going to follow the simpler course and suppose that logically
equivalent states of affairs are identiacal. In other owords, I shall assume:
If S and S* are states of affairs, S = S* iff and S and S* are necessarily sucht that one
obtains iff the other does.
This is a safe assumption even if there is another sense of 'state of affairs' in which equivalent
states of affairs need not be identical, because we can always regard states of affairs in the present
sense as being equivalence classes of the more finely individuated kind of states of affairs. Well,
then we should designate a sort of state of affairsfg which is, like a proposition, maximally finegrained, it seems to me and an intensional entity.
Essentially, it looks like states of affairs are not fine-grained sorts of entites on Pollack's view.
p.56
"Obviously, it follows from these definitions that states of affairs satisfy the asxioms of rthe
propsitional calculus, or what comes to the same thing, states of affairs forma Boolean algerbra
under the operations &, ~, and v.
p.57
Pollock's notion of inclusion: S is a subset of (symbol—backwards C) S* iff S and S* are
necessarily such that if S* obtains then S obtains.
Equivalently:
S is a subset of S* iff (S*  S) is necessarily such that it obtains.
p.58
How do we know that there are maximal states of affairs and not just bigger and bigger ones.
This is Jubien's objection, I think. "Ass far as I can see, the only way to defend the existence of
possible worlds is by acknowledging the existence of infinite conjuinctions of states of affairs.
Pollock conceives of an infinite set in which every member is a state of affairs that obtains; this in
turn is put in states of affairs terms.
p.62
A state of affairs S is necessary iff S is necessarily such that ti obtains. I think he's going to take
necessity as primitive.
"Because logically equivalent states of affairs are identical, there is just one necessary state of
affairs."
A proposition  is necessary iff [Ø|] is necessary;  is possible iff [Ø|] is possible.
Entailment between propositions:
If  and θ are propositions,  entails θ iff, necessarily, if  is true then θ is true.
Entailment between states of affairs:
IF S and S* are states of affairs, S entitals S* iff, necessarily, if S obtains the S* obtains.
p.63
For any state of affairs S, S is possible iff S obtains at some possible world.
p.64
For any states of affairs S, S is necessary iff S obtains at every possible world.
Lewis takes possible worlds as primitive, since they're needed to define necessity.
p.65
Ways things could have been, says Pollock, at not possible worlds as Lewis says, but states of
affairs. I think this is exactly right.
Even if we take states of affairs as primitive, which Pollock does, we cannot give a noncircular
account of necessity in terms of possible worlds since we must still say what states of affairs are
possible worlds, viz., nontransient maximal and possible.
The most promising account of necessity is not in terms of possible worlds, but is epistemological
says Pollock.
Herbrand’s theorem
According to Herbrand’s theorem, each formula F of quantification theory can be
associated with a sequence F1, F2, F3,… of quantifier-free formulas such that F is
provable just in case Fn is truth-functionally valid for some n. This theorem was
the centrepiece of Herbrand’s dissertation, written in 1929 as a contribution to
Hilbert’s programme. It provides a finitistically meaningful interpretation of
quantification over an infinite domain. Furthermore, it can be applied to yield
various consistency and decidability results for formal systems. Herbrand was
the first to exploit it in this way, and his work has influenced subsequent research
in these areas. While Herbrand’s approach to proof theory has perhaps been
overshadowed by the tradition which derives from Gentzen, recent work on
automated reasoning continues to draw on his ideas.
Herbrand’s theorem says that each formula F of quantification theory can be associated
with a sequence
of quantifier-free formulas such that F is provable just in
case is truth-functionally valid for some n. If F is prenex – that is, consists of a
quantifier-free matrix G preceded by a string of quantifiers – then each will be a
disjunction of instances of G obtained by substituting terms for free variables. To
simplify the exposition we consider only prenex formulas F. With each such F we can
associate an existential formula
as follows: (1) substitute new constants for the free
variables of F; (2) remove each universal quantifier and replace the resulting free
occurrences of x by a term
, where f is a new function symbol and
are all the variables bound by existential quantifiers to the left of in F. For
example, applied to
(with R quantifier-free) this procedure yields
Although Herbrand would not have described their relationship in these terms,
can
be said to express the meaning of F in the sense that F is true under an interpretation
just in case
is true under any expansion of to the new constants and function
symbols in
. To see this, consider our example. Let , range over the domain of ,
over functions on this domain and let ‘ ’ indicate that the value is assigned to the
variable x. Then the following are equivalent:
is true under .
There is no such that for all
.
There is no and no function such that for all
.
For every and every function there is some such that
.
No matter which individual is chosen to interpret c and which function to interpret f,
is true under .
This means, in particular, that F is true under every interpretation iff
called the validity functional form of F.
is; the latter is
We can use the functional terms appearing in
to generate a series of finite domains
. These will consist of constants which can themselves be substituted into the
quantifier-free matrix
of
. For convenience, let
consist of a single constant d,
of a distinct constant (its ‘denotation’) for each term obtained from one occurring in
by substituting d for its free variables; in particular, constants occurring in
are
assigned denotations in
. In general,
consists of distinct denotations for each
term obtained from one occurring in
by substituting members of
for its free
variables. , the Herbrand expansion of F of order n, is a disjunction whose disjuncts
represent all possible ways of substituting members of
for the variables of
and
replacing functional terms, where possible, by their values in
. For example, if we
allow each closed term to denote itself, then the first three domains generated by our
earlier formula will be
,
and
. The Herbrand
expansions of orders 1, 2 and 3 are:
Notice that is not an interpretation of the quantifiers of
in
in the usual sense
because some of the terms occurring in only acquire a denotation in
.
Now, for provable F, Herbrand showed how to compute a number n such that would
be a tautology. He argued by induction on the proof of F. The only axioms in his system
are the quantifier-free tautologies, so his method is to analyse each rule of inference
and show how the order of a tautologous expansion of its conclusion can be calculated
from the order(s) of such expansions of its premises. Conversely, he showed how to
construct a proof of F from any tautologous . Because the only rules required here
are forms of generalization and simplification (in particular, modus ponens is not used),
this yields a normal form for proofs in quantification theory analogous to Gentzen’s cutelimination theorem and lends itself to many of the same applications. It allows a bound
to be placed on the complexity of formulas appearing in a normal proof depending on
the complexity of its conclusion (and assumptions, if any).
Herbrand’s proof contains ideas of great interest but, in contrast to Gentzen’s, it is not
easily adapted to non-classical systems. Furthermore, it is difficult to follow and turns
out to contain a defect. Although this has been repaired by Dreben and his
collaborators, the proof has not been reproduced and nowadays the theorem (for
prenex formulas only) is often established as a corollary to a refinement of the cutelimination theorem. (Since a cut-free proof of a prenex formula can be separated into a
truth-functional and a quantificational part, by working back from the conclusion we can
discover a truth-functionally valid formula from which it follows.)
While he accepted set-theoretic arguments in mathematics, Herbrand was committed to
using finitary methods in metamathematical investigations. His work was influenced by
Löwenheim (1915), but he rejected the naïve notion of satisfiability in an infinite domain
the latter assumed. In fact, Herbrand claimed that his result was just a more rigorous
version of Löwenheim’s famous theorem to the effect that a sentence is satisfiable if
and only if it is satisfiable in a denumerable domain. It would be more accurate,
however, to describe it as a finitist variant thereof. To understand why this is so, notice
that we can reformulate Herbrand’s theorem using a construction dual to the above. We
remove existential quantifiers from F, replacing the newly freed variables by functional
terms to obtain a universal formula (the satisfiability functional form of F), which can
then be expanded as a series of finite conjunctions over the
. The theorem then
states that F is irrefutable iff is truth-functionally satisfiable for every n. (This is the
original theorem expressed as a criterion for the non-provability of
.) Constrained by
his finitist scruples, Herbrand explicated validity in terms of provability and, a fortiori,
satisfiability in terms of irrefutability. He also suggested that F’s having truthfunctionally satisfiable expansions for every n could serve as a constructive account
of what it means for F to be satisfiable in a (denumerable) domain. Replacing the
informal set-theoretic notions in the statement of Löwenheim’s theorem by their finitist
analogues transforms it into a statement of Herbrand’s theorem – even though the
theorems themselves remain distinct.
Herbrand’s ideas have yielded numerous metamathematical applications. The normal
form for proofs mentioned above supplies a tool for obtaining consistency proofs.
Furthermore, the elimination of quantifiers in favour of functional terms suggests how to
exhibit the constructive content of the theorems of a formal theory – as in the ‘no
counterexample’ interpretation of arithmetic, for example (Kreisel 1951). Herbrand’s
theorem also allows us to prove a theory consistent by approximating a model for it, that
is, by establishing that, for every (finite) conjunction F of the theory’s axioms, is truthfunctionally satisfiable for every n. Herbrand himself gave a consistency proof for a
fragment of arithmetic along these lines, and his methods have been extended to yield
proofs for the full system (see Dreben and Denton 1970, Scanlon 1972). These proofs,
while no less complex than Gentzen’s, yield directly the sort of constructive
interpretations alluded to above.
Finally, Herbrand’s theorem supplies an approach to the decision problem for classes of
quantificational formulas, that is, to discovering an algorithm which, for any formula in
the class, computes whether or not it is satisfiable. The idea here is to investigate the
relationship between syntactic properties of quantificational formulas and the structure
of their expansions, and to discover properties of the latter which ensure the existence
of a decision procedure (see Dreben and Goldfarb 1979). His theorem and the methods
introduced in its proof are relevant not only to theoretical decidability results, but also to
the design of implementable automated reasoning procedures. These employ
unification algorithms to improve the efficiency of testing quantifier-free expansions for
satisfiability, and Herbrand is credited with being the first to devise such an algorithm
(Snyder 1991).
A.M. UNGAR
Copyright © 1998-2002 Routledge, an imprint of the Taylor & Francis group. All rights reserved.
p.70
Modal logic
Modal logic, narrowly conceived, is the study of principles of reasoning involving
necessity and possibility. More broadly, it encompasses a number of structurally
similar inferential systems. In this sense, deontic logic (which concerns
obligation, permission and related notions) and epistemic logic (which concerns
knowledge and related notions) are branches of modal logic. Still more broadly,
modal logic is the study of the class of all possible formal systems of this nature.
It is customary to take the language of modal logic to be that obtained by adding
one-place operators ‘ ’ for necessity and ‘ ’ for possibility to the language of
classical propositional or predicate logic. Necessity and possibility are
interdefinable in the presence of negation:
hold. A modal logic is a set of formulas of this language that contains these
biconditionals and meets three additional conditions: it contains all instances of
theorems of classical logic; it is closed under modus ponens (that is, if it
contains A and
it also contains B); and it is closed under substitution (that
is, if it contains A then it contains any substitution instance of A; any result of
uniformly substituting formulas for sentence letters in A). To obtain a logic that
adequately characterizes metaphysical necessity and possibility requires certain
additional axiom and rule schemas:
By adding these and one of the – biconditionals to a standard axiomatization
of classical propositional logic one obtains an axiomatization of the most
important modal logic, S5, so named because it is the logic generated by the fifth
of the systems in Lewis and Langford’s Symbolic Logic (1932). S5 can be
characterized more directly by possible-worlds models. Each such model
specifies a set of possible worlds and assigns truth-values to atomic sentences
relative to these worlds. Truth-values of classical compounds at a world w
depend in the usual way on truth-values of their components.
is true at w if A
is true at all worlds of the model;
, if A is true at some world of the model. S5
comprises the formulas true at all worlds in all such models. Many modal logics
weaker than S5 can be characterized by models which specify, besides a set of
possible worlds, a relation of ‘accessibility’ or relative possibility on this set.
is
true at a world w if A is true at all worlds accessible from w, that is, at all worlds
that would be possible if w were actual. Of the schemas listed above, only K is
true in all these models, but each of the others is true when accessibility meets
an appropriate constraint.
The addition of modal operators to predicate logic poses additional conceptual
and mathematical difficulties. On one conception a model for quantified modal
logic specifies, besides a set of worlds, the set
of individuals that exist in w,
for each world w. For example,
is true at w if there is some element of
that
satisfies A in every possible world. If A is satisfied only by existent individuals in
any given world
thus implies that there are necessary individuals;
individuals that exist in every accessible possible world. If A is satisfied by nonexistents there can be models and assignments that satisfy A, but not
.
Consequently, on this conception modal predicate logic is not an extension of its
classical counterpart.
The modern development of modal logic has been criticized on several grounds,
and some philosophers have expressed scepticism about the intelligibility of the
notion of necessity that it is supposed to describe.
1 History
2 Propositional S5
3 Philosophical questions about S5
4 Quantified S5
5 Weaker systems
6 General results
1 Indicative and material conditionals
In general an ‘indicative conditional’ has the form ‘If A then C’, where A is called the
antecedent and C the consequent. A central issue is the relationship between the truthvalue of a conditional and the truth-values of its antecedent and consequent. This
much is immediately plausible: if A is true and C is false, then the conditional is false. If
I say ‘If it rained, the match was cancelled’, and what happened was that it rained but
the match went ahead, then what I say is clearly false. There are three other
possibilities: A and C are both true, A is false and C is true, and A and C are both false.
There are a number of arguments designed to show that in each of these cases the
conditional is true. Here is one. ‘If A then C’ is intuitively equivalent to the disjunction
‘Either not-A, or A and C’. (Instead of saying that if it rained the match was cancelled, I
could have said ‘Either it did not rain, or it did and the match was cancelled’.) But the
latter is true in all three cases: in each, either the first disjunct (‘not-A’) or the second
disjunct (‘A and C’) is true. A conditional that is false when its antecedent is true and its
consequent false, and true in all other cases, is called a ‘material conditional’, and is
symbolized ‘
’ (read ‘A hook C’). ‘
’ is definitionally equivalent to ‘Not-A or C’.
Hence the argument’s conclusion is that indicative conditionals are equivalent to
material conditionals. This conclusion has the virtue of validating the two most
obviously valid inferences governing conditionals – modus ponens: ‘A, if A then C,
therefore, C’, and modus tollens: ‘Not-C, if A then C, therefore, not-A’.
Despite the appeal of the argument, there are serious problems for the equivalence
thesis that indicative conditionals are material conditionals. It entails that any
conditional with a false antecedent is true regardless of its consequent, that is, the
account validates ‘Not-A, therefore, if A then C’. This is implausible. ‘If I live in London
then I live in Scotland’ strikes us as false (it is rather ‘If I live in London then I do not live
in Scotland’ which is true), but because I do not live in London, it is true on the
equivalence thesis. Also, it entails that any conditional with a true consequent is true:
the account validates ‘C, therefore, if A then C’. But ‘If I live in London, then I live in
Australia’ strikes us as false even after we learn that I do in fact live in Australia. These
two results are known as the paradoxes of material implication (‘material implication’
being the name of the relation between A and C when ‘
’ is true).
Copyright © 1998-2002 Routledge, an imprint of the Taylor & Francis group. All rights reserved.
Is it because of the paradoxes of material implication that iff is not necessarily to read modally such that in a philsophical
definition of the form x is a y iff…. , this form may not be equivalent to x is a y iff necessarily….?
Conceptual analysis
A distinction must be made between the philosophical theory of conceptual
analysis and the historical philosophical movement of Conceptual Analysis.
The theory of conceptual analysis holds that concepts – general meanings of
linguistic predicates – are the fundamental objects of philosophical inquiry, and
that insights into conceptual contents are expressed in necessary ’conceptual
truths’ (analytic propositions). There are two methods for obtaining these truths:
(1) direct a priori definition of concepts;
(2) indirect ’transcendental’ argumentation.
The movement of Conceptual Analysis arose at Cambridge during the first half of
the twentieth century, and flourished at Oxford and many American departments
of philosophy in the 1950s and early 1960s. In the USA its doctrines came under
heavy criticism, and its proponents were not able to respond effectively; by the
end of the 1970s the movement was widely regarded as defunct. This reversal of
fortunes can be traced primarily to the conjunction of several powerful
objections: the attack on intensions and on the analytic/synthetic distinction; the
paradox of analysis; the ‘scientific essentialist’ theory of propositions; and the
critique of transcendental arguments. Nevertheless a closer examination
indicates that each of these objections presupposes a covert appeal to concepts
and conceptual truths. In the light of this dissonance between the conventional
wisdom of the critics on the one hand, and the implicit commitments of their
arguments on the other, there is a manifest need for a careful re-examination of
conceptual analysis.
1 Origins and career of Conceptual Analysis
2 The theory and methods of conceptual analysis
3 Five fundamental objections
4 The inescapability of conceptual analysis
Copyright © 1998-2002 Routledge, an imprint of the Taylor & Francis group. All rights reserved.
P.73 Pollock takes properties to be functions that assign to each object x the state of affairs
consisting of x's having the property P. For example, the property of being red is identified with
the function othat assigns x's being red to each object x. More generally, an n-place property is
taken to be a function from n-tuples of objects to states of affairs.
Here Pollock will take states of affairs as more basic than properties, which seems wrong. I.e., he
has to define properties in terms of states of affairs. Better: either they are both equally
primitive, or states of affairs are to be defined in terms of properties.
Pollock distinguishes between functions in extension and functions in intension. The former are
sets of ordered pairs; the latter are intensional entities akin to concepts; there "functional"
concepts:
f is a function in intension iff f is a two-place concept that is necessarily such that for any x,y,z, if
<x,y> and <x,z> both exemplify f then y=z.
If <x,y> exemplifies f, we write [f(x) = y].
p. 74
Properties must be identified with functions-in-intension.
p.76 Pollock defines an essence of an object to be a property that the object has at every possible
world and that no other object has at any possible world.
p.80
Plantinga and Pollock disagree on whether not existing is a property. I'm inclined to agree with
Plantinga. It's like the property of not being a property. This doesn't necessarily exclude all
negative properties, but at least those of this sort.
p.81 "The upshot of this is that serious actualism is either false or uninteresting, depending upon
just how we choose to use the term 'prpoerty'.
p.84
Actualism
p.91
Pollock develops a possibilistic quantifier and an actuality operator to produce the conclusion that
"…I think it must be concluded that actualism, taken as the claim that no sense can be made of
quanitifaction voer possible objects, is false."
But is this the claim? It's not necessarily a claim about meaning, but rather whether possibilism is
true. Of course, if the proposition used to state possibilism is meaningless, it would seem that it's
false. But producing concepts to make possibilism meaningful does not then convince one that it
is true. No doubt, one can create a possibilistic quantifier, but the question is, does it exist—is it
something that is real in the world. To assume so begs the question. When we speak of fictitious
objects, our intuition is that the object is nothing more than a combination of concepts that do not
apply to anything—nothing exemplies them. The claim of the actualist seems to me to be the
claim that "There are objects that do not exist" not to be meaningless on all readings, but to come
out false on all readings. Certainly false statements are still meaningful statements.
p.91
Possibilistic Set Theory
"What is of even more interest is that possibilistic quantifiers allow us to talk about setsof
nonexistence but possible objects." It would seem that the predicate 'exists' and the predicate 'is actual' are
the same predicate for the possibilist. I want to deny this.
p. 92
Pollock speaks of possible sets given the axiom of extensionality.
p.98
Existentialism
Pollock is a possibilist, but not an existentialist.
"States of affairs "involve" object in them. For example Socrates' being snubnosed involves
Socrates in a special way. Advocates of existentialism feel that Socrates is a "constituent" of this
state of affairs in much the same way that members of a set are constituents of the set, and
accordingly they feel that the state of affairs cannot exist without Socrates existing. More
generally, and state of affairs contiaining Socrates' being snubnoed will fail to exist if Socrates
does not exist. Ultimately, the defense of existentialism comes down to this intuiation.
Plantinga's response is to object that the notion of a constituent is too vague and unclear to be of
much use here. It must be admitted that there is a certain amount of justice to this charge. But it
must also be admitted that the
p.99
existentialist intuition is a fairly compelling one.
Pollack thinks he has an argument showing existentialism to be incoherent. "The argument turns
upon the observation that is existentialism is correct then there is a distinction between a possible
world's obtaining and its being actual." This argument, however, involves using the property of
non existing, which Plantinga doesn't take to be a property—and I'm inclined to agree with him
p.102
"I am convinced on the basis of epistemological considerations of the preceding sort that all talk
of abstract entities must be analyzable in terms of (possibly modal) talk of noabstract entities..
p. 175
Formal Semantics
"The upshot of this is that we seem unable, at this point, to give any simple answer to the question of how
this formal semantics represents the notion of truth at all possible worlds, and the philosophical
significance of the semantics is left in doubt. Perhaps all the semantics amounts to is a mathematical
characterization of the theorems of the modal logic, devoid of phlosophcial significance. But if that is true,
lgicians have been pulling the wool over the eyes of their colleagues for years by convicnign them they
were doing something philosophically important. As I am still doing logic, the reader will surmoise that I
believe there is more to formal semantics than the skeptical view I have just suggested, but the true
significance of formal semantics involves a long story."
I was recently surporised to discover that there are logicicans who do not believe that formal semantics
stands in need of any justification or foundations. They regard formal semantics as a "philosophical
aoccomplishement in its own right". Such a view strikes me as absurd. Me too!
p.181 State of
affairs logics
"As states of affairs are not (except indirectly) objects of belief, apriority is inapplicable to states of
affairs."
p.228
"We are now in a postion to answer the rather general question with which we began this chapter. Formal
semantics does have a limited philosophical significane, although it does not have the all-pervasive
significance sometimes attributed to it by logicians. To begin with, formal semantics attempts to
characterize with mathematical precisions the sets of formulas that are valid of rthe different concepts of
validity we have
p.229
been able to make precise. The extent to which a particular semantics accomplishes this can be objectively
evaluated, and we have done that for a number of the most popular formal semantical theories.
It is often alledge that formal semantics provide us with anaysies of logical concepts. That is not entirely
accurate. A formal semantics by itself acannot provide us with an analysis. It can do that, however, when
it is coupled with a characterizion of the suuogate relation. Such a characterization cannot be purely
mathematical. It must proceed by relation ghte mathematical concept of a model to the philosophical
concepts of an interpretation and a possible world. The effect of this is to turn the formal semantics into a
realistic semantics.