Elements of Logic 18
Start studying Traditional (Formal) logic: chapter 2 Comprehension and Extension. whats the difference between comprehension and extension?. The historical account usually given of comprehension and extension is this, "that the difference between distinctness and clearness is uniformly overlooked. Was a logical extension of history as it was in ? The book There is a difference between hearing or reading and trulyunderstanding the material.
Extension of a term, on the other hand,refers to the totality of individuals or classes to which the intension is applicable. Many philosophers are now using the words connotation and denotation to signify intension and extension respectively. The intension and extension of terms are inversely related to each other. The greater the intension of a term, the lesser its extension, and vice-versa. A young dog for instance, stands as an intension of the term puppy.
If we add Dalmatian to the intension, then the extension is reduced for it does not involve non-Dalmatian puppies. If we further increase the intension by adding newly born, the extension then decreases all the more for it now excludes those young Dalmatian dogsthat are not newly born. Logically therefore, if we lessen the intension, the result would be the increase in the extension.
Kinds of terms In these various classifications of terms, a term might receive a place in each of the classes since categorizations are based on different aspects: Singular, Particular, Universal, and Collective Terms 1.
A term is singular if it refers to only one individual or thing. The indicators of singularity are: A term is particular if it stands for an indefinite part of a whole. The following indicates particularity: Subject terms in propositions that are true to only some of the denotations of the term, e. Belgians are religious; Men have sense of chivalry.
A term is general or universal if it refers to all individuals signified by the term. The following indicates universality: A man is a rational being. A term is collective if itrefers to a group of individuals considered as a single unit. Collective nouns such as audience, committee, crowd, flock, government, jury, gang, and orchestra are collective terms.
A collective term may be universal, e. Collective term is not applicable to the objects taken singly and individually, unless used figuratively. The term family, as an example, is collective, since it is predicable of the family members taken collectively, and not individually. A term is affirmative or positive if it expresses what is real, true, or essential of a thing, e.
A term is also affirmative when it affirms the presence of desirable traits, e. There are two kinds of affirmative term: A term is negative if it indicates the non-appearance of some trait, e.
There are two kinds of negative term: Immediate and Mediate 1. Immediate terms are formed through direct perception of things. Mediate terms are formed indirectly, that is, through the mediation of other ideas.
God, soul, spirit, universe D. According to the nature of referents: Concrete, Abstract, Logical, and Null 1. A term is concrete if its referent is tangible or can be perceived by the senses. Concrete term also refers to that which indicates a quality or characteristic as inherent in a subject.
A term is abstract if its referent is intangible or can be understood only by the mind and cannot be perceived by the senses. Abstract term also refers to the quality or characteristic considered independently from the subject in which it inheres.
A term is logical if it was formulated to serve as linguistic device to aid learning. A term is null or empty if it has no actual or real referents but is only imaginary.
According to definiteness of meaning: Univocal, Equivocal, and Analogous 1. A term is univocal if it exhibits exactly identical sense and meaning in different incidents. This has been further developed by subsequent researchers, of course with modern possible world semantics added to the mix. The Carnap approach is not the only one around, but it does take us quite a bit of the way into the intensional thicket. Even though it does not get us all the way through, it will be the primary version considered here, since it is concrete, intuitive, and natural when it works.
Axiomatically presented propositional modal logics were well-established, so it was important to see how or if they could be extended to include quantifiers and equality.
At issue were decisions about what sorts of things quantifiers range over, and substitutivity of equals for equals. Ruth Barcan Marcus began a line of development in Marcus by formally extending the propositional system S2 of C. Lewis to include quantification, and developing it axiomatically in the style of Principia Mathematica. It was clear that other standard modal logics besides S2 could have been used, and S4 was explicitly discussed.
Especially significant for the present article, her system was further extended in Marcus to allow for abstraction and identity.
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Two versions of identity were considered, depending on whether things had the same properties abstracts or necessarily had them. In the S2 system the two versions were shown to be equivalent, and in the S4 system, necessarily equivalent. In a later paper Marcus the fundamental role of the deduction theorem was fully explored as well. Marcus proved that in her system identity was necessary if true, and the same for distinctness.
Names were understood as tags. They might have their designation specified through an initial use of a definite description, or by some other means, but otherwise names had no meaning, only a designation. Thus they did not behave like definite descriptions, which were more than mere tags. The essential point had been made. One could develop formal modal systems with quantifiers and equality. The ideas had coherence.
Still missing was a semantics which would help with the understanding of the formalism, but this was around the corner.
All treated intensions functionally. In part, Bressan wanted to provide a logical foundation for physics. The connection with physics is this. When we say something has such-and-such a mass, for instance, we mean that if we had conducted certain experiments, we would have gotten certain results. This does not assume we did conduct those experiments, and thus alternate states or cases, as Bressan calls them arise.
Hence there is a need for a rich modal language, with an ontology that includes numbers as well as physical objects. In Bressanan elaborate modal system was developed, with a full type hierarchy including numbers as in Principia Mathematica.
The treatment is semantic, but in Gallin an axiom system is presented. The logic Gallin axiomatized is a full type-theoretic system, with intensional objects of each type. Completeness is proved relative to an analog of Henkin models, familiar for higher type classical logics. Unfortunately his work did not become widely known. For one thing, intensions depend not only on worlds, but also on times.
The idea is that expressions determine intensions and extensions, and this itself is a formal process in which compound expressions act using the simpler expressions that go into their making; compositionality at the level of constructions, in other words.
Somehow, based on its sense intension, meaning a designating phrase may designate different things under different conditions—in different states.
In neither state were people wrong about the concept of planet, but about the state of affairs constituting the universe. If we suppress all issues of how meanings are determined, how meanings in turn pick out references, and all issues of what counts as a possible state of affairs, that is, if we abstract all this away, the common feature of every designating term is that designation may change from state to state—thus it can be formalized by a function from states to objects.
This bare-bones approach is quite enough to deal with many otherwise intractable problems. In order to keep things simple, we do not consider a full type hierarchy—first-order is enough to get the basics across.
The first-order fragment of the logic of Gallin would be sufficient, for instance. The particular formulation presented here comes from Fittingextending Fitting and Mendelsohn Predicate letters are intensional, as they are in every version of Kripke-style semantics, with interpretations that depend on possible worlds.
The only other intensional item considered here is that of individual concept—formally represented by constants and variables that can designate different objects in different possible worlds. The same ideas can be extended to higher types, but what the ideas contribute can already be seen at this relatively simple level. Intensional logics often have nothing but intensions—extensions are inferred but are not explicit. However, an approach that is too minimal can make life hard, so consequently here we explicitly allow both objects and individual concepts which range over objects.
There are two kinds of quantification, over each of these sorts. Both extensional and intensional objects are first-class citizens. Basic ideas are presented semantically rather than proof-theoretically, though both axiom systems and tableau systems exist.
Even so, technical details can become baroque, so as far as possible, we will separate informal presentation, which is enough to get the general idea, from its formal counterpart, which is of more specialized interest. A general acquaintance with modal logic is assumed though there is a very brief discussion to establish notation, which varies some from author to author.
It should be noted that modal semantics is used here, and generally, in two different ways.
Often one has a particular Kripke model in mind, though it may be specified informally. For instance, we might consider a Kripke model in which the states are the present instant and all past ones, with later states accessible from earlier ones.
But besides this use of informally specified concrete models, there is formal Kripke semantics which is a mathematically precise thing. Informal models pervade our discussions—their fundamental properties come from the formal semantics. These operators can be thought of as alethic, deontic, temporal, epistemic—it will matter which eventually, but it does not at the moment. Kripke semantics for propositional modal logic is, by now, a very familiar thing.
A more detailed presentation can be found in the article on modal logic in this encyclopedia. States could be states of the real world at different times, or states of knowledge, or of belief, or of the real world as it might have been had circumstances been different. We have a mathematical abstraction here.
Then more complex formulas are evaluated as true or false, relative to a state. At each state the propositional connectives have their customary classical behavior. For the modal operators. These are, by now, very familiar ideas. This is a way of thinking that goes back to Frege, who concluded that the denotation of a sentence should be a truth value, but the sense should be a proposition. He was a little vague about what constituted a proposition—the formalization just presented provides a natural mathematical entity to serve the purpose, and was explicitly proposed for this purpose by Carnap.
Sets like these are commonly referred to as propositions in the modal logic community.
Intension and extension
Intensions will be introduced formally in Section 3. The material discussed here can be found more fully developed in Fitting and MendelsohnHughes and Cresswellamong other places. At the propositional level truth values play the role of things, but at the first order level something more is needed. In classical logic each model has a domain, the things of that model, and quantifiers are understood as ranging over the members of that domain.
It is, of course, left open what constitutes a thing—any collection of any sort can serve as a domain. That way, if someone has special restrictions in mind because of philosophical or mathematical considerations, they can be accommodated.
It follows that the validities of classical logic are, by design, as general as possible—they are true no matter what we might choose as our domain, no matter what our things are. A similar approach was introduced for modal logics in Kripke Domains are present, but it is left open what they might consist of. But there is a complication that has no classical counterpart: Should there be a single domain for the entire model, or separate domains for each state?
Both have natural intuitions. Consider a version of Kripke models in which a separate domain is associated with each state of the model. At each state, quantifiers are thought of as ranging over the domain associated with that state. This has come to be known as an actualist semantics. Think of the domain associated with a state as the things that actually exist at that state.
Thus, for example, in the so-called real world the Great Pyramid of Khufu is in the domain, but the Lighthouse of Alexandria is not. If we were considering the world of, say,both would be in the domain. In an actualist approach, we need to come to some decision on what to do with formulas containing references to things that exist in other states but not in the state we are considering.
Intension and extension | logic and semantics | index-art.info
Several approaches are plausible; we could take such formulas to be false, or we could take them to be meaningless, for instance, but this seems to be unnecessarily restrictive. So, the formal version that seems most useful takes quantifiers as ranging over domains state by state, but otherwise allows terms to reference members of any domain.
The resulting semantics is often called varying domain as well as actualist. Suppose we use the actualist semantics, so each state has an associated domain of actually existing things, but suppose we allow quantifiers to range over the members of any domain, without distinction, which means quantifiers are ranging over the same set, at every state.
What are the members of that set? They are the things that exist at some state, and so at every state they are the possible existents—things that might exist. Lumping these separate domains into a single domain of quantification, in effect, means we are quantifying over possibilia.
Thus, a semantics in which there is a single domain over which quantifiers range, the same for every state, is often called possibilist semantics or, of course, constant domain semantics. Possibilist semantics is simpler to deal with than the actualist version—we have one domain instead of many for quantifiers to range over.
And it turns out that if we adopt a possibilist approach, the actualist semantics can be simulated. This gives an embedding of the actualist semantics into the possibilist one, a result that can be formally stated and proved. Free and bound occurrences of variables have the standard characterization.
Note that first order valuations are not state-dependent in the way that interpretations are. We also have mixed cases such as the Barcan and converse Barcan formulas: Much has been made about the identity of the number of the planets and 9 Quineor the identity of the morning star and the evening star Fregeand how these identities might behave in modal contexts.
But that is not really a relevant issue here. Intensional issues will be dealt with shortly. Quantification is possibilist—domains are constant. But, as was discussed in Section 3. With possibilist quantifiers, this is valid and reasonable. Now a second kind of quantification is added, over intensions. As has been noted several times, an intensional object, or individual concept, will be modeled by a function from states to objects, but now we get into the question of what functions should be allowed.
Intensions are supposed to be related to meanings. If we consider meaning to be a human construct, what constitutes an intension should probably be restricted. There should not, for instance, be more intensional objects than there are sentences that can specify meanings, and this limits intensions to a countable set. This is an issue that probably cannot be settled once and for all. Instead, the semantics about to be presented allows for different choices in different models—it is not required that all functions from states to objects be present.
It should be noted that, while this semantical gambit does have philosophical justification, it also makes an axiomatization possible. The fundamental point is the same as in the move from first to second order logic.
If we insist that second order quantifiers range over all sets and relations, an axiomatization is not possible. If we use Henkin models, in which the range of second order quantifiers has more freedom, an axiomatization becomes available. Formulas are constructed more-or-less in the obvious way, with two kinds of quantified variables instead of one: But there is one really important addition to the syntactic machinery, and it requires some discussion.
Both versions are useful and correspond to things we say every day. We will allow for both, but the second version requires some cleaning up. This is the de re reading, in which a possible property is ascribed to a thing.
This is the de dicto reading, possibility applies to a sentence. The de re and de dicto readings are different, both need representation, and we cannot manage this with the customary syntax.
An abstraction mechanism will be used to disambiguate our syntax. It should be noted that one could simply think of abstraction as a scope-specifying device, in a tradition that goes back to Russell, who made use of such a mechanism in his treatment of definite descriptions. Abstraction in modal logic goes back to Carnapbut in a way that ignores the issues discussed above. With two kinds of variables present, the formation of atomic formulas becomes a little more complex.
Formulas are built up from atomic formulas in the usual way, using propositional connectives, modal operators, and two kinds of quantifiers: In addition to the usual formula-creating machinery, we have the following.
To distinguish the models described here from those in Section 3.06-Extension and Comprehension of Terms
They are discussed more fully in Fitting It should be noted that the examples of designating terms just given are all definite descriptions. These pick out different objects in different possible worlds quite naturally. The situation with proper names and with mathematics is different, and will be discussed later in section 3. An intension is rigid if it is constant, the same in every state.
We might think of a rigid intension as a disguised object, identifying it with its constant value. It should not be a surprise, then, that for rigid intensions, the distinction between de re and de dicto disappears. Indeed, something a bit stronger can be shown. Instead of rigidity, consider the weaker notion called local rigidity in Fitting and Mendelsohn We have deliberately left vague the question of which ones we must have.
There are some conditions we might want to require. Here are some considerations along these lines, beginning with a handy abbreviation. It simply says intensions always designate. On the other hand, there is no a priori reason to believe that every object is designated by some intension, but under special circumstances we might want to require this.
We also might want to require the existence of choice functions. Requiring the validity of the following formula seems as close as we can come to imposing such an existence condition on FOIL models. To handle such things, the representation of an intension can be generalized from being a total function from states to objects, to being a partial function.
We routinely talk about non-existent objects—we have no problem talking about the King of France in But there is nothing to be said about the present King of France—there is no such thing. This will be our guide for truth conditions in our semantics. Using that, we introduce a further abbreviation.
It is important to differentiate between existence and designation. As things have been set up here, existence is a property of objects, but designation really applies to terms of the formal language, in a context. We have to be a bit careful about non-existence though. What we do have is the following important item. In a temporal model, the first three determine partial intensions there have been instants of time with no people ; the fourth determines an intension that is not partial; the fifth determines an intension that is rigid.
So far we have been speaking informally, but there are two equivalent ways of developing definite descriptions ideas formally. The approach introduced by Bertrand Russell RussellWhitehead and Russell is widely familiar and probably needs little explication here. Suffice it to say, it extends to the intensional setting without difficulty. As an attractive alternative, one could make definite descriptions into first-class things. Then modify the definition of formula, to allow these new intension terms to appear in places we previously allowed intension variables to appear.
That leads to a complication, since intension terms involve formulas, and formulas can contain intension terms. In fact, formula and term must be defined simultaneously, but this is no real problem. Semantically we can model definite descriptions by partial intensions. Then the conditions from section 3. But with definite descriptions available as formal parts of the language, instead of just as removable abbreviations in context, one can see they determine intensions possibly partial that are specified by formulas.
This is simply because the definite description might not designate. However, if it does designate, it must have its defining property. Indeed, we have the validity of the following: This is a problem that sets a limit on what can be handled by the Carnap-style logic as presented so far.
Two well-known areas of difficulty are mathematics and proper names, especially in an epistemic setting. But if we model sense by a function from states to designations, the functions would be the same, mapping each state to 5. So again, how could one not know this, or any other mathematical truth?
Exactly what program depends on how we were taught to add, but let us standardize on: We might identify the program with the sense, and the output with the denotation. Identifying the intension of a mathematical term with its computational content is a plausible thing to do.
It does, however, clash with what came earlier in this article. For any given expression, how do we decide which way to treat it? It is possible to unify all this. Here is one somewhat simple-minded way. This can be pushed only so far. We might be convinced by some general argument that addition is a total function always defined.
But this cannot be captured using the semantics outlined thus far, assuming arithmetic terms behave correctly. It is also possible to address the problem from quite a different direction.