3. Semantics (study of formal theories of meaning)

In recent years semantics has become to mean the study of formal theories of meaning rather than simply the study of the semantic properties of some expression. Paradigmatically, a formal semantics for a fragment of a natural language consists first in assignments of semantic value to various subsentential portions of the language, such as objects to names and extensions to predicates, and truth functions to various operators. Second, the semantic theory provides interpretations for complex sentences relative to a time, possible worlds and index. The notion of an index is crucial to the interpretation of sentences including indexical terms, whose reference is not fixed independently of a context.

Semantics, according linguists and philosophers such as Jeffrey C. King,^[1] is the discipline that studies linguistic meaning generally, and the qualification ‘formal’ indicates something about the sorts of techniques used in investigating linguistic meaning. More specifically, formal semantics is the discipline that employs techniques from symbolic logic, mathematics, and mathematical logic to produce precisely characterized theories of meaning for natural languages (that is, naturally occurring languages such as English, Urdu, and so on) or artificial languages (that is, first-order predicate logic, computer programming languages and so on).

1. Formal Semantics

Formal semantics which first arose in the twentieth century was made possible by certain developments in logic during that period. What follows will chronicle those developments and how they led to the development of formal semantics. This will provide the reader with a preliminary understanding of the discipline. As presented by King (2006), though the works of Gottlob Frege, Bertrand Russell, and Ludwig Wittgenstein in the late 1800s and early 1900s were important precursors, the development of formal semantics really begins with the work of the Polish logician Alfred Tarski on the notion of truth. During and immediately prior to the late 1920s and early 1930s, the time when Tarski produced his seminal work on truth, a movement called logical positivism was beginning to dominate scientifically minded philosophy. Scientifically minded philosophers and logicians of the time thought that the use of techniques from mathematics and symbolic logic in philosophy was the way to move the discipline forward. The logical positivists generally very much shared this vision of philosophy. However, they also thought that much traditional philosophy was nonsense, and they had formulated criteria of meaningfulness according to which much traditional work in philosophy failed to satisfy these criteria and so was meaningless (Carnap, 1932). As a result of this outlook, the positivists were extremely suspicious of the use of any terms in philosophy that appeared to attempt to make reference to things that in some sense were beyond human experience. Thus, talk of ‘things in themselves’, ‘the absolute, and so on were dismissed as gibberish. Talk of truth made the positivists nervous as well, perhaps because it seemed to them to presuppose some mind independent reality that in principle could extend beyond human experience and that served as that which makes true things true.

Tarski’s work on truth needs to be viewed in this context. In (1935) Tarski takes as its goal the definition of ‘true sentence’ for a range of formal languages. It should be mentioned that Tarski thought that it was not possible to coherently define the notion of ‘true sentence’ for natural languages such as German or Polish. Because such languages contain, or allow for the introduction of, names for its own sentences, and contain the expression ‘true sentence’ as well as other semantic vocabulary (‘names’, ‘denotes’ etc.), these languages allow the formulation of the liar paradox (and others as well). Thus, Tarski regarded such languages as logically inconsistent and thought that as a result there could be no correct definition of ‘true sentence’ for such languages. Tarski also produced a definition of ‘true sentence’ for what he called the languages of the calculus of classes.

This language is a first-order language containing two sentential connectives (the negation sign, disjunction), a universal quantifier, and a two-place predicate (‘I’) whose meaning is ‘is included in’. Considered as the language under study, the language for which ‘true sentence’ will be defined, we call this the object language. The sentences of this language are the well-formed formulas lacking free occurrences of variables. The metalanguage, the language in which we construct the definition of truth, Tarski did not attempt to formalize, though he did clearly describe its important features. The crucial point is that for every sentence Samyutta Nikāya of the object language, the metalanguage contained a name (or structural description -a linguistic description of the sentence of the object language in the vocabulary of the metalanguage) of Samyutta Nikāya and a translation of Samyutta Nikāya Tarski’s famous convention T stated that an adequate definition of ‘true sentence’ for a language must have as consequences all sentences obtained from ‘x is true iff p’ by substituting for ‘x’ a structural description of a sentence of the object language and for ‘p’ the translation of this sentence in the metalanguage. Tarski notes that for languages with infinite numbers of sentences, the idea suggests itself of defining ‘true sentence’ by recursion, (that is, one defines ‘true sentence’ for the simplest sentences, and then shows how whether a complex sentence is true depends on whether the simpler sentences it is made up of are true). One would consider all the ways complex sentences can be built out of simpler ones, and then specify how the truth or falsity of a complex depends on the truth or falsity of its component sentences. The problem is that some sentences are not built up out of sentences, but rather formulae with free to which the notion of truth simpliciter is not applicable. As a result, Tarski suggests defining another semantic notion that is applicable to all wellformed formulae and that can be used to define truth directly. This is the notion of satisfaction. To define it, Tarski considered infinite sequences f of objects, in the present case classes. As indicated, in Tarski’s object language, ‘I’ is the two-place predicate meaning ‘is included in’ and individual variables was subscripted as follows: ‘x₁’, ‘x₂’, and so on. Tarski stipulated that element of the sequence f be associated with the variable ‘x_n’. Let f_n be the nth element of the sequence f.

Following King (2006), let us now consider a formula containing free variables, such as:

(1) Ix₁ x₂

Traski’s definition of satisfaction has it that f satisfies (1) iff (the class) f₁ is included in f₂. For a disjunctive formula, a sequence satisfies it iff it satisfies one of the disjuncts, (satisfaction of negations of formulae works in the obvious analogous way). Finally, a sequence f satisfies a universally quantified formula whose quantifier’s variable is ‘x_n’ iff for every sequence f’ like f except that f’_n might not be the same as f_n, f’ satisfies formula resulting from stripping off the universal quantifier being treated whose variable is ‘x_n’.

Further, for sentences, Tarski’s definition of satisfaction has the consequence that either every sequence satisfies it or none does. Thus, one can immediately define a ‘true sentence’ (of the language in question) as one that is satisfied by every sequence. Tarski goes on to note that his definition satisfies convention T and that ‘truth sentence’ on his definition has a variety of properties that one would expect it to have. Finally, he discusses defining ‘true sentence’ for a variety of other languages.

It should be noted that Tarski (1935) did not define the now more familiar ‘true relative to a model Majjhima Nikāya’. He held the meaning of the non-logical symbols (I) fixed (it means ‘is included in’) and so simply defined ‘true sentence’ (for that language) . Because assigning the conditions under which sentences of a language are true and false to those sentences has come to be viewed as the central task of formal semantics, it is hard to overstate the significance of the fact that Tarski showed for the first time how to do this for a range of formal languages.

Carnap was very much aware of Tarski’s work, and very much influenced by it. Carnap’s 1947 Meaning and Necessity was the first work that, using Tarski’s techniques, provided formal semantics for languages that go beyond first-order logic in including devices akin to some of the more problematic devices present in natural languages, including expressions for expressing modality (Necessarily) and verbs of propositional attitude (‘believes). Carnap (1942, 1947) explicitly understood the task of providing what he called a ‘semantical analysis’ of meaning. Meaning and Necessity was an important work in the history of formal semantics, in that it offered formal semantic analyses of recalcitrant expressions like ‘It is necessary that’ and ‘believe’. Though the details of Carnap’s proposals have been abandoned, his proposals were influential and elements of his accounts have been preserved in the accounts of others.

2. Two Dimensional Semantics

Two-dimensional approaches to semantics, broadly understood, recognize two ‘dimensions’ of the meaning or content of linguistic items. On these approaches, expressions and their utterances are associated with two different sorts of semantic values, which play different explanatory roles. Typically, one semantic value is associated with reference and ordinary truth-conditions, while the other is associated with the way that reference and truth-conditions depend on the external world. The second sort of semantic value is often held to play a distinctive role in analyzing matters of cognitive significance and/or context-dependence.

In this broad sense, even Frege’s theory of sense and reference might qualify as a sort of two-dimensional approach. More commonly, twodimensional approaches are understood more narrowly to be a species of possible-worlds semantics, on which each dimension is understood in terms of possible worlds and related modal notions.

In possible-world semantics, linguistic expressions and/or their utterances are first associated with an extension. The extension of a sentence is its truth-value: for instant, the extension of ‘Plato was a philosopher’ is true. The extension of a singular term is its referent: for example, the extension of ‘Don Bradman’ is Bradman. The extension of a general term is the class of individuals that fall under the term: for example, the extension of ‘cat’ is the class of cats. Other expressions work similarly.

One can then associate expressions with an intension, which is a function from possible worlds to extensions. The intension of a sentence is a function that is true at a possible world if and only if the sentence is true there: the intension of ‘Plato was a philosopher’ is true at all worlds where Plato was a philosopher. The intension of a singular term maps a possible world to the referent of a term in that possible world: the intension of ‘Don Bradrnan’ picks out whoever is Bradman in a world. The intension of a general term maps a possible world to the class of individuals that fall under the term in that world: the intension of ‘cat’ maps a possible world to the class of cats in that world.

It can easily happen that two expressions, as argued by Chalmers,^[2] have the same extension but different intensions. For example, Quine’s terms ‘cordate’ (creature with a heart) and ‘renate’ (capture with a kidney) pick out the same class of individuals in the actual world, so the same extension. But there are many possible worlds where they pick out different classes (any possible world in which there are creatures with hearts but no kidney, for example), so they have different intensions. When two expressions have the same extension and a different intension in this way, the difference in intension corresponds to an intuitive difference in meaning. So it is natural to suggest expression’s intension is at least an aspect of its meaning.

Carnap (1947) suggested that an intension behaves in many respects like a Fregean sense, the aspect of an expression’s meaning that corresponds to its cognitive significance. For example, it is cognitively significant that all renates are cordates and vice-verse (this was a non-trivial empirical discovery about the world), so that ‘renate’ and ‘cordate’ should have different Fregean senses. One might naturally suggest that this difference in sense is captured more concretely by a difference in intension, and that this pattern generalizes. For example, one might suppose that when two singular terms are cognitively equivalent (so that ‘a = a’ is trivial or at least knowable a priori, for example), then their extension will coincide in all possible worlds, so that they will have the same intension. And one might suppose that when two such terms are cognitively distinct (so that ‘a = b’ is knowable only empirically, for example), then their extensions will differ in some possible world, so that they will have different intensions. If this were the case, the distinction between intension and extension could be seen as a sort of vindication of a Fregean distinction between sense and reference.

However, the work of Kripke (1980) is widely taken to show that no such vindication is possible. According to Kripke, there are many statements that are knowable only empirically, but which are true in all possible worlds. For example, it is an empirical discovery that Hesperus is Phosphorus, but there is no possible world in which Hesperus is not Phosphorus (or viceversa), as both Hesperus and Phosphorus are identical to the planet Venus in all possible worlds. If so, then ‘Hesperus’ and ‘Phosphorus’ have the same intension (one that picks out the planet Venus in all possible worlds), even though the two terms are cognitively distinct. The same goes for pairs of terms such as ‘water’ and ‘H₂O’: it is an empirical discovery that water is H₂O but but according to Kripke, both ‘water’ and ‘H₂O’ have the same intension (picking out H₂O in all possible worlds). Something similar even applies to terms such as ‘I’ and ‘Ho K. Phong’, at least as used by me on a specific occasion: ‘I’ and ‘Ho K. Phong’ expresses non-trivial empirical knowledge, but Kripke’s analysis entails that I am Ho K. Phong in all worlds, so that my utterances of these expressions have the same intension. If this is correct, then intensions are strongly dissociated from cognitive significance.

Still, there is a strong intuition that the members of these pairs (‘Hesperus’ and ‘Phosphorus’, ‘water’ and ‘H₂O’, ‘I’ and Ho K. Phong’) differ in some aspect of meaning. Further, there remains a strong intuition that there is some way the world could turn out so that these terms would refer to different things. For example, it seems to be at least epistemically possible (in some broad sense) that these terms might fail to co-refer. On the face of it, cognitive differences between the terms is connected in some fashion to the existence of these possibilities. So it is natural to continue to use an analysis in terms of possibility and necessity to capture aspects of these cognitive differences. This is perhaps the guiding idea behind twodimensional semantics.

Two-dimensional approaches to semantics start from the observation that the extension and even the intension of many of our expressions depend in some fashion on the external world. As things have turned out, my terms ‘water’ and ‘H₂O’ have the same extension, and have the same (Kripkean) intension. But there are ways things could have turned out so that the two terms could have had a different extension, and a different intension. So there is a sense in which for a term like ‘water’, the term’s extension and its Kripkean intension depend on the character of our world. Given that this world is actual, it turns out that ‘water’ refers to H₂O, and its Kripkean intension picks out H₂O in all possible worlds. But if another world had been actual (for example, Putnam’s Twin Earth world in which XYZ is the clear liquid in the oceans), ‘water’ might have referred to something quite different (for example, XYZ), and it might have had an entirely different Kripkean intension (for example, one that picks out XYZ in all worlds).

This suggests a natural formalization. If an expression’s (Kripkean) intension itself depends on the character of the world, then we can represent this dependence by a function from worlds to intensions. As intensions are themselves functions from worlds to extensions, this naturally suggests a two-dimensional structure.

1. Two-Dimensional Approaches^[3] : Kaplan’s Character and Content

Perhaps the best-known broadly two-dimensional approach is Kaplan’s analysis of the character and content of indexicals (Kaplan, 1979, 1989). According to Kaplan, his work is partly grounded in work in tense logic byKamp (1971) and Vlach (1973), which gives a sort of two-dimensional analysis of the behavior of ‘now’. Kaplan applies his analysis to indexicals such as ‘I’, ‘here’, and ‘now’, as well as to demonstratives such as ‘this’ and ‘that’. Kaplan’s analysis is well-known, so this subsection will describe it only briefly here.

For Kaplan, the ‘worlds’ involved in the first dimension are contexts of utterance: these can be seen as at least involving the specification of a speaker and a time and place of utterance, within a world. The ‘worlds’ involved on the second dimension are circumstances of evaluation: these are ordinary possible worlds at which the truth of an utterance is to be evaluated. Let us consider an expression as (2).

(2) I am hungry now

According to Kaplan’s analysis, when this expression is uttered by John at time t₁, it expresses a proposition that is true if and only if John is hungry at t₁. We can call this proposition expressed the content of the utterance. This content can naturally be represented as an intension that is true at all and only those worlds (those circumstances of evaluation) in which John is hungry. (Kaplan regards propositions as structured entities rather than intensions, but the difference does not matter much here.) In a different context -say, a context with Diana speaking at t₂ - an utterance of the same expression will have a different content. This content will be a proposition that is true at a world if and only if Diana is hungry at t₂ in that world.

The character of an expression is a function from contexts to contents, mapping a context of utterance to the content of that expression in that context. (If content is seen as an intension, then character is a sort of twodimensional intension.) So the character of ‘I am hungry’ maps the first context above to the proposition that John is hungry at t₁, and the second context above to the proposition that Diana is hungry at t₂. Extending this idea to subsentential indexical terms, we can say that the character of ‘I’ maps the first context to John and the second context to Diana; more generally, it maps any context into the speaker in that context. Similarly, the character of ‘now’ maps any context into the time specified in that context.

The above definition of character is still somewhat imprecise, and many tricky issues come up in giving a precise definition. But to a rough first approximation, one can say that the character of an expression maps a context to the content that the expression would have if uttered in that context. There is more to say than this (especially as Kaplan intends his analysis to apply even to contexts in which there is no token of the relevant utterance), but this is enough for now. In general, character is associated with an expression type rather than with an expression token, although this matter is complicated somewhat by the case of demonstratives such as ‘this’ and ‘that’, whose character may vary between different utterances.

On Kaplan’s analysis, the character of indexicals such as ‘I’, ‘now’, and ‘here’, as well as the character of demonstratives such as ‘this’ and ‘that’, reflects their cognitive significance. For example, ‘I am here now’ has a propositional content that is true in only some worlds, but its character yields a proposition that is true in all contexts of utterance. (Kaplan does not “diagonalize” character into an intension, but it would be easy enough to do so. If one did so, then ‘I am here now’ would be associated with a diagonal intension that is necessarily true.) So the character rather than the content seems to reflect the fact that the sentence can be known a priori (or near enough). Likewise, when a true utterance of ‘this is that’ is cognitively significant, the utterances of ‘this’ and ‘that’ will refer to the same object, but their characters will differ. So at least in these domains, character behaves a little like a Fregean sense.

However, this behavior does not extend to other expressions. For example, Kaplan holds that names refer to the same individual in any context of utterance. On this view, co-extensive names such as ‘Mark Twain’ and ‘Samuel Clemens’ will have exactly the same character, and an identity such as ‘Mark Twain is Samuel Clemens’ will have a character that yields a true proposition in every context, even though the identity appears to be a posteriori and cognitively significant. Something similar applies to natural kind terms such as ‘water’. So on Kaplan’s analysis, names and natural kind terms have a “constant character” that is dissociated from cognitive roles.

One can diagnose the situation by noting that character is most closely tied to the patterns of context-dependence associated with an expression, rather than to the expression’s cognitive significance. In the case of indexicals, the patterns of context-dependence of an expression are themselves closely associated with the expression’s cognitive significance. But for many other expressions, such as names and natural kind terms, cognitive significance is strongly dissociated from patterns of context-dependence. As a result, in the general case, Kaplan’s framework is better suited to the analysis of the contextdependence of expressions than to an analysis of their cognitive significance.

3. Semantics, Assertibility Conditions

It is Dummett who first elaborated an account of meaning in terms of the assertibility conditions associated with statements or sentences. Roughly speaking, the meaning of a statement is what is known by the person who understands it. Dummett (1996) contends that what that person would have knowledge of is the conditions that warrant asserting that statement; in other words, its assertibility conditions.

4. Semantics, Conceptual Role

Conceptual role semantics is a theory that explains the contents of mental states in terms of their conceptual connections to other mental states. More specifically, the contents thus attributed to mental states are a matter of the conceptual roles played by those states in the whole economy of mental states. Conceptual roles are often explained inferentially in terms of the roles played by the states in reasoning, their connections to perceptual inputs and behavioural outputs. The content thus attributed typically is understood as a narrow content.

5. Semantics, Inferentialist

Inferentialist semantics is a theory that explains the meaning of a sentence or utterance in terms of its inferential connections. Thus, the meaning of ‘Leo is a mammal’ is understood in terms of its entailing ‘Leo is an animal’, being incompatible with ‘Leo is a plant’, and being entailed by ‘Leo is a lion’. This is a version of meaning holism because the meaning of each sentence is determined by its connections to the meanings of other sentences.

6. Semantics, Possible-world

The possible-world semantics was first elaborated by Kripke. It provides a way of assigning truth conditions to, and understanding the logical relations between, sentence expressing counterfactuals or involving various modalities. Thus, for instance, necessary truths are interpreted as being true in all possible worlds.

7. Semantics, truth-conditional

The truth-conditional semantics which includes all formal theories of the meanings of linguistic sentences or utterances in terms of their truth conditions is very important in the study of semantics. According to Tanesini, the basic idea is that if a person knows that the Italian sentence ‘la neve ē bianca’ is true if and only if snow is white, one knows what that sentence means. Davidson (1991) developed this idea in detail. He argued that any adequate formal theory of meaning for a natural language such as English or Italian should generate T-sentences for each sentence of the target language as theorems. Thus, an adequate theory of meaning for Italian formulated in English should have as theorems, T-sentences such ‘“la neve ē Bianca” is true if and only if snow is white’, ‘“l’erba e verde” is true if and only if grass is green’, and so forth for each sentence in the language. It has been objected that Davidson’s adequacy requirement is too lax. It would seem possible to have a theory which generates T-sentences which are all true but which, intuitively, do not seem to capture the meanings of the relevant sentences. Thus, for example, the following T-sentences are all true: ‘“la neve ē Bianca” is true if and only if snow is white’; ‘“la neve ē Bianca” is true if and only if snow is white and 2 + 2 = 4’; ‘“la neve ē bianca” is true if and only if grass is green’. Yet they cannot all be giving the meaning of the Italian sentence. These different sentences are all true because any sentence of the form ‘P if and only if Q’ is true provided P and Q are both true or both false. In this instance the biconditional T-sentences are equivalent because it is true that snow is white, that grass is green, and that snow is white and 2 + 2 = 4. Davidson has replied to this objection by arguing that in his view the T-sentences must be generated by a recursive theory. As a result it generates T-sentences in which parts of sentences such as the noun neve ‘snow’ make the same contribution to the meanings of the sentences in which it occurs. Thus the theory respects the principle of the compositionality of language, and rules out the two rogue T-sentences above. Davidson also argues that a theory of meaning as a theory of truth is an empirical theory, evidence for which must be found when engaged in the project of radical interpretation.

Footnotes and references:

[1]:

Jeffrey C. King. [2006] 2008. Formal Semantics. In The Oxford Handbook of Philosophy of Language. ed. Ernest Lepore and Barry C. Smith. New York: Oxford University Press, pp. 558-73.

[2]:

David J. Chalmer. 2006. Two-Dimensional Semantics. In The Oxford Handbook of Philosophy of Language. ed. Ernest Lepore and Barry C. Smith. New York: Oxford University Press, pp. 574-606.

[3]:

More detailed discussions of all of these two-dimensional frameworks and their interrelations can be found in two recent collections: the March 2004 special issue of Philosophical Studies on “The Two-Dimensional Framework and its Applications,” and the book Two-Dimensional Semantics (Garacia-Carpintero and Macia, 2006). See especially Chalmers, 2006; Davies, 2004; and Stalnaker, 2004, and also the discussion in Soames, 2005.