Theories of Explanation | Internet Encyclopedia of Philosophy
Despite the original name of “natural philosophy”, science has little to do with I hope I have answered clearly what is see is the difference between science unquestioned assumptions being overturned and new theories being written over their ruins. . "It may appear to some scientists that they are using the logical and . The relation between logic and philosophy is discussed. problems such as those usually in the main branches of philosophy discussed below). 2. Russell's Five Minute World Hypothesis: Suppose the earth were created five minutes ago. Logic in computer science covers the overlap between the field of logic and that of computer science. The topic can essentially be divided into three main areas: to assist logicians; 3 Logic applications for computers; 4 See also; 5 . Article on Logic and Artificial Intelligence at the Stanford Encyclopedia of Philosophy.
Traditional versions of object-based theories assumed that the truth-bearing items usually taken to be judgments have subject-predicate structure. An object-based definition of truth might look like this: A judgment is true if and only if its predicate corresponds to its object i.
Note that this actually involves two relations to an object: Owing to its reliance on the subject-predicate structure of truth-bearing items, the account suffers from an inherent limitation: The problem is obvious and serious; it was nevertheless simply ignored in most writings. Object-based correspondence was the norm until relatively recently. In a number of dialogues, Plato comes up against an argument, advanced by various Sophists, to the effect that false judgment is impossible—roughly: To judge falsely is to judge what is not.
But one cannot judge what is not, for it is not there to be judged. To judge something that is not is to judge nothing, hence, not to judge at all. Therefore, false judgment is impossible. Euthydemus ea; Cratylus c-e; Republic a-c; Theaetetus de. Plato has no good answer to this patent absurdity until the Sophist dbwhere he finally confronts the issue at length.
The key step in his solution is the analysis of truthbearers as structured complexes. By weaving together verbs with names the speaker does not just name a number of things, but accomplishes something: The simple sentence is true when Theaetetus, the person named by the name, is in the state of sitting, ascribed to him through the verb, and false, when Theaetetus is not in that state but in another one cf.
Only things that are show up in this account: He emphasizes that truth and falsehood have to do with combination and separation cf. Unlike Plato, Aristotle feels the need to characterize simple affirmative and negative statements predications separately—translating rather more literally than is usual: This characterization reappears early in the Prior Analytics 24a.
Fact-based correspondence theories became prominent only in the 20th century, though one can find remarks in Aristotle that fit this approach see Section 1 —somewhat surprisingly in light of his repeated emphasis on subject-predicate structure wherever truth and falsehood are concerned.
Fact-based theories do not presuppose that the truth-bearing items have subject-predicate structure; indeed, they can be stated without any explicit reference to the structure of truth-bearing items. The approach thus embodies an alternative response to the problem of falsehood, a response that may claim to extricate the theory of truth from the limitations imposed on it through the presupposition of subject-predicate structure inherited from the response to the problem of falsehood favored by Plato, Aristotle, and the medieval and modern tradition.
The now classical formulation of a fact-based correspondence theory was foreshadowed by Hume Treatise, 3. It appears in its canonical form early in the 20th century in Moorechap. The self-conscious emphasis on facts as the corresponding portions of reality—and a more serious concern with problems raised by falsehood—distinguishes this version from its foreshadowings.
Somewhat ironically, their formulations are indebted to their idealist opponents, F. Joachimthe latter was an early advocate of the competing coherence theory, who had set up a correspondence-to-fact account of truth as the main target of his attack on realism.
FieldPopper It has become customary to talk of truthbearers whenever one wants to stay neutral between these choices. Five points should be kept in mind: It is intended to refer to bearers of truth or falsehood truth-value-bearersor alternatively, to things of which it makes sense to ask whether they are true or false, thus allowing for the possibility that some of them might be neither.
One distinguishes between secondary and primary truthbearers. Secondary truthbearers are those whose truth-values truth or falsehood are derived from the truth-values of primary truthbearers, whose truth-values are not derived from any other truthbearers.
This is, however, not a brute ambiguity, since the secondary meanings are supposed to be derived, i. For example, one might hold that propositions are true or false in the primary sense, whereas sentences are true or false in a secondary sense, insofar as they express propositions that are true or false in the primary sense. It is often unproblematic to advocate one theory of truth for bearers of one kind and another theory for bearers of a different kind e.
Different theories of truth applied to bearers of different kinds do not automatically compete. The standard segregation of truth theories into competing camps found in textbooks, handbooks, and dictionaries proceeds under the assumption—really a pretense—that they are intended for primary truthbearers of the same kind.
Confusingly, there is little agreement as to which entities are properly taken to be primary truthbearers.
Nowadays, the main contenders are public language sentences, sentences of the language of thought sentential mental representationsand propositions. Popular earlier contenders—beliefs, judgments, statements, and assertions—have fallen out of favor, mainly for two reasons: The problem of logically complex truthbearers. A subject, S, may hold a disjunctive belief the baby will be a boy or the baby will be a girlwhile believing only one, or neither, of the disjuncts.
Also, S may hold a conditional belief if whales are fish, then some fish are mammals without believing the antecedent or the consequent. Also, S will usually hold a negative belief not everyone is lucky without believing what is negated. This means that a view according to which beliefs are primary truthbearers seems unable to account for how the truth-values of complex beliefs are connected to the truth-values of their simpler constituents—to do this one needs to be able to apply truth and falsehood to belief-constituents even when they are not believed.
This point, which is equally fundamental for a proper understanding of logic, was made by all early advocates of propositions cf. The problem arises in much the same form for views that would take judgments, statements, or assertions as primary truthbearers. The problem is not easily evaded. Talk of unbelieved beliefs unjudged judgments, unstated statements, unasserted assertions is either absurd or simply amounts to talk of unbelieved unjudged, unstated, unasserted propositions or sentences.
It is noteworthy, incidentally, that quite a few philosophical proposals concerning truth as well as other matters run afoul of the simple observation that there are unasserted and unbelieved truthbearers cf. If the former, the state of believing, can be said to be true or false at all, which is highly questionable, then only insofar as the latter, what is believed, is true or false.
Mental sentences were the preferred primary truthbearers throughout the medieval period. They were neglected in the first half of the 20th century, but made a comeback in the second half through the revival of the representational theory of the mind especially in the form of the language-of-thought hypothesis, cf. Some time after that, e. A truthmaker is anything that makes some truthbearer true. Different versions of the correspondence theory will have different, and often competing, views about what sort of items true truthbearers correspond to facts, states of affairs, events, things, tropes, properties.
It is convenient to talk of truthmakers whenever one wants to stay neutral between these choices. Four points should be kept in mind: The notion of a truthmaker is tightly connected with, and dependent on, the relational notion of truthmaking: For illustration, consider a classical correspondence theory on which x is true if and only if x corresponds to some fact.
One can say a that x is made true by a fact, namely the fact or a fact that x corresponds to. But they are importantly different and must be distinguished. Note that anyone proposing a definition or account of truth can avail themselves of the notion of truthmaking in the b -sense; e. Talk of truthmaking and truthmakers goes well with the basic idea underlying the correspondence theory; hence, it might seem natural to describe a traditional fact-based correspondence theory as maintaining that the truthmakers are facts and that the correspondence relation is the truthmaking relation.
However, the assumption that the correspondence relation can be regarded as a species of the truthmaking relation is dubious. Correspondence appears to be a symmetric relation if x corresponds to y, then y corresponds to xwhereas it is usually taken for granted that truthmaking is an asymmetric relation, or at least not a symmetric one. It is hard to see how a symmetric relation could be a species of an asymmetric or non-symmetric relation cf.
Talk of truthmaking and truthmakers is frequently employed during informal discussions involving truth but tends to be dropped when a more formal or official formulation of a theory of truth is produced one reason being that it seems circular to define or explain truth in terms of truthmakers or truthmaking.
However, in recent years, the informal talk has been turned into an official doctrine: This theory should be distinguished from informal truthmaker talk: Moreover, truthmaker theory should not simply be assumed to be a version of the correspondence theory; indeed, some advocates present it as a competitor to the correspondence theory see below, Section 8. Some authors do not distinguish between concept and property; others do, or should: Simple Versions of the Correspondence Theory The traditional centerpiece of any correspondence theory is a definition of truth.
It should be noted that this terminology is not standardized: The question whether non-obtaining beings of the relevant sort are to be accepted is the substantive issue behind such terminological variations. The difference between 2 and 1 is akin to the difference between Platonism about properties embraces uninstantiated properties and Aristotelianism about properties rejects uninstantiated properties. Advocates of 2 hold that facts are states of affairs that obtain, i.
So disagreement turns largely on the treatment of falsehood, which 1 simply identifies with the absence of truth. The following points might be made for preferring 2 over 1: However, some worry that truthbearer categories, e. Some, though not all, will regard this as a significant advantage. Facts, on the other hand, cannot be identified with the meanings or contents of sentences or mental states, on pain of the absurd consequence that false sentences and beliefs have no meaning or content.
What are the constituents of the corresponding fact? The main point in favor of 1 over 2 is that 1 is not committed to counting non-obtaining states of affairs, like the state of affairs that snow is green, as constituents of reality. One might observe that, strictly speaking, 1 and 2being biconditionals, are not ontologically committed to anything.
Their respective commitments to facts and states of affairs arise only when they are combined with claims to the effect that there is something that is true and something that is false.
The discussion assumes some such claims as given. Both forms, 1 and 2should be distinguished from: The lure of 3 stems from the desire to offer more than a purely negative correspondence account of falsehood while avoiding commitment to non-obtaining states of affairs. It can also be found in the translation of Wittgenstein4. The translation has Wittgenstein saying that an elementary proposition is false, when the corresponding state of affairs atomic fact does not exist—but the German original of the same passage looks rather like a version of 2.
A fourth simple form of correspondence definition was popular for a time cf. Main worries about 4 are: Which fact is the one that mis-corresponds with a given falsehood? What keeps a truth, which by definition corresponds with some fact, from also mis-corresponding with some other fact, i. Arguments for the Correspondence Theory The main positive argument given by advocates of the correspondence theory of truth is its obviousness.
Even philosophers whose overall views may well lead one to expect otherwise tend to agree. Indeed, The Oxford English Dictionary tells us: In view of its claimed obviousness, it would seem interesting to learn how popular the correspondence theory actually is. There are some empirical data.
The PhilPapers Survey conducted in ; cf. Bourget and Chalmersmore specifically, the part of the survey targeting all regular faculty members in 99 leading departments of philosophy, reports the following responses to the question: The data suggest that correspondence-type theories may enjoy a weak majority among professional philosophers and that the opposition is divided. This fits with the observation that typically, discussions of the nature of truth take some version of the correspondence theory as the default view, the view to be criticized or to be defended against criticism.
Historically, the correspondence theory, usually in an object-based version, was taken for granted, so much so that it did not acquire this name until comparatively recently, and explicit arguments for the view are very hard to find.
Since the comparatively recent arrival of apparently competing approaches, correspondence theorists have developed negative arguments, defending their view against objections and attacking sometimes ridiculing competing views.
Objections to the Correspondence Theory Objection 1: Definitions like 1 or 2 are too narrow. Although they apply to truths from some domains of discourse, e.
The objection recognizes moral truths, but rejects the idea that reality contains moral facts for moral truths to correspond to. The logical positivists recognized logical truths but rejected logical facts. There are four possible responses to objections of this sort: The objection in effect maintains that there are different brands of truth of the property being true, not just different brands of truths for different domains.
On the face of it, this conflicts with the observation that there are many obviously valid arguments combining premises from flagged and unflagged domains. The observation is widely regarded as refuting non-cognitivism, once the most popular concessive response to the objection. Though it retains important elements of the correspondence theory, this view does not, strictly speaking, offer a response to the objection on behalf of the correspondence theory and should be regarded as one of its competitors see below, Section 8.
Correspondence theories are too obvious. They are trivial, vacuous, trading in mere platitudes. Such common turns of phrase should not be taken to indicate commitment to a correspondence theory in any serious sense. In response, one could point out: This makes it rather difficult to explain why some thinkers emphatically reject all correspondence formulations. Correspondence theories are too obscure. The objections can be divided into objections primarily aimed at the correspondence relation and its relatives 3.
C2and objections primarily aimed at the notions of fact or state of affairs 3. The correspondence relation must be some sort of resemblance relation. The correspondence relation is very mysterious: How could such a relation possibly be accounted for within a naturalistic framework? What physical relation could it possibly be? Negative, disjunctive, conditional, universal, probabilistic, subjunctive, and counterfactual facts have all given cause for complaint on this score.
All facts, even the most simple ones, are disreputable. Fact-talk, being wedded to that-clauses, is entirely parasitic on truth-talk. Facts are too much like truthbearers. Correspondence as Isomorphism Some correspondence theories of truth are two-liner mini-theories, consisting of little more than a specific version of 1 or 2.
Normally, one would expect a bit more, even from a philosophical theory though mini-theories are quite common in philosophy. One would expect a correspondence theory to go beyond a mere definition like 1 or 2 and discharge a triple task: One can approach this by considering some general principles a correspondence theory might want to add to its central principle to flesh out her theory. No truth is identical with a fact correspondence to which is sufficient for its being a truth.
It would be much simpler to say that no truth is identical with a fact. However, some authors, e. Wittgensteinhold that a proposition Satz, his truthbearer is itself a fact, though not the same fact as the one that makes the proposition true see also King Nonidentity is usually taken for granted by correspondence theorists as constitutive of the very idea of a correspondence theory—authors who advance contrary arguments to the effect that correspondence must collapse into identity regard their arguments as objections to any form of correspondence theory cf.
Concerning the correspondence relation, two aspects can be distinguished: Pitcher ; Kirkhamchap. Pertaining to the first aspect, familiar from mathematical contexts, a correspondence theorist is likely to adopt claim aand some may in addition adopt claim bof: Together, a and b say that correspondence is a one-one relation.
This seems needlessly strong, and it is not easy to find real-life correspondence theorists who explicitly embrace part b: Explicit commitment to a is also quite rare. However, correspondence theorists tend to move comfortably from talk about a given truth to talk about the fact it corresponds to—a move that signals commitment to a.
Correlation does not imply anything about the inner nature of the corresponding items. Contrast this with correspondence as isomorphism, which requires the corresponding items to have the same, or sufficiently similar, constituent structure. This aspect of correspondence, which is more prominent and more notorious than the previous one, is also much more difficult to make precise. Let us say, roughly, that a correspondence theorist may want to add a claim to her theory committing her to something like the following: If an item of kind K corresponds to a certain fact, then they have the same or sufficiently similar structure: The basic idea is that truthbearers and facts are both complex structured entities: The aim is to show how the correspondence relation is generated from underlying relations between the ultimate constituents of truthbearers, on the one hand, and the ultimate constituents of their corresponding facts, on the other.
One part of the project will be concerned with these correspondence-generating relations: The other part of the project, the specifically ontological part, will have to provide identity criteria for facts and explain how their simple constituents combine into complex wholes.
Putting all this together should yield an account of the conditions determining which truthbearers correspond to which facts. Correlation and Structure reflect distinct aspects of correspondence.
One might want to endorse the former without the latter, though it is hard to see how one could endorse the latter without embracing at least part a of the former. The isomorphism approach offers an answer to objection 3. This is not a qualitative resemblance; it is a more abstract, structural resemblance.
The approach also puts objection 3. C2 in some perspective. The correspondence relation is supposed to reduce to underlying relations between words, or concepts, and reality. This reminds us that, as a relation, correspondence is no more—but also no less—mysterious than semantic relations in general.
Such relations have some curious features, and they raise a host of puzzles and difficult questions—most notoriously: Can they be explained in terms of natural causal relations, or do they have to be regarded as irreducibly non-natural aspects of reality? Some philosophers have claimed that semantic relations are too mysterious to be taken seriously, usually on the grounds that they are not explainable in naturalistic terms. But one should bear in mind that this is a very general and extremely radical attack on semantics as a whole, on the very idea that words and concepts can be about things.
The common practice to aim this attack specifically at the correspondence theory seems misleading. As far as the intelligibility of the correspondence relation is concerned, the correspondence theory will stand, or fall, with the general theory of reference and intentionality. We can express this point by saying that these inferences are instances of the following form: The Stoics discussed several patterns of this kind, using ordinal numbers instead of letters to capture abstract forms like the ones shown below.
If the first then the second, and the first; so the second. If the first then the second, but not the second; so not the first.
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Either the first or the second, but not the second; so the first. Not both the first and the second, but the first; so not the second. These schematic formulations require variables. They can be endorsed or rejected, and they exhibit containment relations of some kind. So presumably, propositions are abstract things that can be evaluated for truth or falsity. This leaves it open what propositions are: But let's assume that declarative sentences can be used to express propositions.
For discussion, see Cartwright and the essay on structured propositions. A significant complication is that in ordinary conversation, the context matters with regard to which proposition is expressed with a given sentence. What counts as being tired can also vary across conversations. Context sensitivity, of various kinds, is ubiquitous in ordinary discourse. To be sure, ordinary conversation differs from theoretical discourse in mathematics. But the distinction between impeccable and risky inferences is not limited to special contexts in which we try to think especially clearly about especially abstract matters.
So when focusing on the phenomenon of valid inference, we can try to simplify the initial discussion by abstracting away from the context sensitivity of language use.
Another complication is that in speaking of an inference, one might be talking about i a process in which a thinker draws a conclusion from some premises, or ii some propositions, one of which is designated as an alleged consequence of the others; see, e. But we can describe a risky thought process as one in which a thinker who accepts certain propositions—perhaps tentatively or hypothetically—comes to accept, on that basis, a proposition that does not follow from the initial premises.
It isn't obvious that all impeccable inferences are instances of a more general valid form, much less inferences whose impeccability is due to the forms of the relevant propositions. But this thought has served as an ideal for the study of valid inference, at least since Aristotle's treatment of examples like 2.
Again, the first premise seems to have several parts, each of which is a part of the second premise or the conclusion. Aristotle, predating the Stoics, noted that conditional claims like the following are sure to be true: Correspondingly, the inference pattern below is valid. For example, the syllogistic patterns below are also valid. But however the inferences are represented, the important point is that the variables—represented here in italics—range over certain parts of propositions.
And many propositions apparently contain correspondingly general elements. For example, the proposition that every senator is deceitful contains two such elements, both relevant to the validity of inferences involving this proposition. That is, even simple propositions have logical form. And as Aristotle noted, pairs of such propositions can be related in interesting ways.
If every S is P, then some S is P. For these purposes, assume there is at least one S. If no S is P, then some S is not P. It is certain that either every S is P or some S is not P; and whichever of these propositions is true, the other is false. Similarly, the following propositions cannot both be true: But it isn't certain that either every S is P, or no S is P.
Perhaps some S is P, and some S is not P. This network of logical relations strongly suggests that the propositions in question contain a quantificational element and two general elements—and in some cases, an element of negation. This raises the question of whether other propositions have a similar structure. Consider the proposition that Vega is a star, which can figure in inferences like 8. Aristotle's logic focused on quantificational propositions; and as we shall see, this was prescient.
But on his view, propositions like the conclusion of 8 still exemplify a subject-predicate structure that is shared by at least many of the sentences we used to express propositions. Typically, a declarative sentence can be divided into a subject and a predicate: Until quite recently, it was widely held that this grammatical division reflects a corresponding kind of logical structure: Aristotle would have said that in the premises of 8being purple is predicated of every star, and being a star is predicated of Vega.
But despite the complications, it seemed clear that many propositions have the following canonical form: Subject-copula-Predicate; where a copula links a subject, which may consist of a quantifier and a general term, to a general term. Such examples invite the hypothesis that all propositions are composed of terms along with a relatively small number of syncategorematic elements, and that complex propositions can be reduced to canonical propositions that are governed by Aristotelian logic.
This is not to say that all propositions were, or could be, successfully analyzed in this manner. But via this strategy, medieval logicians were able to describe many impeccable infererences as instances of valid forms. And this informed their discussions of how logic is related to grammar. Many viewed their project as an attempt to uncover principles of a mental language common to all thinkers.
Ockham also held that a mental language would have no need for Latin's declensions, and that logicians could ignore such aspects of spoken language. The ancient Greeks were aware of sophisms like the following: This bad inference cannot share its form with the superficially parallel but impeccable variant: See Plato, Euthydemus d-e.
So the superficial features of sentences are not infallible guides to the logical forms of propositions. Still, the divergence was held to be relatively minor. Spoken sentences have structure; they are composed, in systematic ways, of words.
And the assumption was that sentences reflect the major aspects of propositional form, including a subject-predicate division. So while there is a distinction between the study of valid inference and the study of sentences used in spoken language, the connection between logic and grammar was thought to run deep.
This suggested that the logical form of a proposition just is the grammatical form of some perhaps mental sentence. Indeed, some of the real successes highlighted known problems. Some valid schemata are reducible to others, in that any inference of the reducible form can be revealed as valid with a little work given other schemata. Then 9 is an instance of the following valid form: But we can treat this as a derived form, by showing that any instance of this form is valid given two intuitively more basic Stoic inference forms: For suppose we are given the following premises: A; and if A, then either not-A or not-B.
We can safely infer that either not-A or not-B; and since we were given that A, we can safely infer that not-B. Similarly, the syllogistic schema 10 can be treated as a derived form. But if some S is not P, then as we saw above, not every S is P.
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This reasoning shows how 10 can be reduced to inferential patterns that seem more basic—raising the question of how much reduction is possible. Euclid's geometry had provided a model for how to present a body of knowledge as a network of propositions that follow from a few basic axioms.
Aristotle himself indicated how to reduce all the valid syllogistic schemata to four basic patterns, given a few principles that govern how the basic patterns can be used to derive others; see Parsons for discussion. And further reduction is possible given insights from the medieval period. Consider the following pair of valid inferences: Fido is a brown dog, so Fido is a dog; Fido is not a dog, so Fido is not a brown dog. Plausibly, the first pattern reflects the default direction of valid replacement: Suppose we take it as given that poodles are dogs of a particular sort, and hence that every poodle is a dog.
But the validity of this inference form can also be viewed as symptom of a basic principle that came be called dictum de omni: Or as Aristotle might have put it, if the property of being a dog belongs to every poodle, then it belongs to any poodle. In which case, Fido is a dog if Fido is a poodle. And since the property of being a dog surely belongs to every brown dog, any brown dog is a dog. The flip side of this point is that negation inverts the default direction of inference.
Anything that isn't a dog isn't a brown dog; and similarly, if Fido isn't a dog, Fido isn't a poodle. If some brown dog is a clever mutt, it follows that some dog is a clever mutt, and hence that some dog is a mutt. If no dog is a mutt, it follows that no dog is a clever mutt, and hence that no brown dog is a clever mutt. The corresponding principle, dictum de nullo, encodes this pattern: If every dog is clever, it follows that every brown dog is clever; but if every dog is a clever mutt, it follows that every dog is a mutt.
So when the universal quantifier combines with a general term S to form a subject, S is governed by the inverted rule of replacement. But when a universally quantified subject combines with a second general term to form a proposition, this second term is governed by the default rule of replacement.
The first principle reflects the sense in which universal quantification is transitive. In this sense, classical logic exhibits a striking unity and simplicity, at least with regard to inferences involving the Aristotelian quantifiers and predication; see Sommers and Ludlowdrawing on Sanchezfor further discussion.
Alas, matters become more complicated once we consider relations. A quantifier can be part of a complex predicate. But classical logic did not capture the validity of inferences involving predicates that have quantificational constituents. But this schema, which fails to reflect the quantificational structure within the predicates is not valid.
Its instances include bad inferences like the following: One can also formulate the following schema: But the problem remains. Quantifiers can appear in complex predicates that figure in valid inferences like And many inferences of this form are invalid.
Again, one can abstract a valid schema that covers 12letting parentheses indicate a relative clause that restricts the adjacent predicate.
But no matter how complex the schema, the relevant predicates can exhibit further quantificational structure. Consider the proposition that every patient who met some doctor who saw no lawyer respects some lawyer who saw no patient who met every doctor. Moreover, schemata like the one above are poor candidates for basic inference patterns. As medieval logicians knew, propositions expressed with relative clauses also pose other difficulties; see the entry on medieval syllogism.
If every doctor is healthy, it follows that every young doctor is healthy. But consider 13 and But one wants a systematic account of propositional structure that explains the net effect; see Ludlow for further discussion. Sommers offers a strategy for recoding and extending classical logic, in part by exploiting an idea suggested by Leibniz and arguably Panini: But if impeccability is to be revealed as a matter of form, then one way or another, quantifiers need to characterized in a way that captures their general logical role—and not just their role as potential subjects of Aristotelian propsitions.
Frege and Formal Language Frege showed how to resolve these difficulties for classical logic in one fell swoop. His system of logic, published in and still in use with notational modificationswas arguably the single greatest contribution to the subject.
So it is significant that on Frege's view, propositions do not have subject-predicate form. His account required a substantial distinction between logical form and grammatical form as traditionally conceived.
It is hard to overemphasize the impact of this point on subsequent discussions of thought and its relation to language. Though for Frege, functions are not abstract objects. In particular, while a function maps each entity in some domain onto exactly one entity in some range, Frege does not identify functions with sets of ordered pairs. And in this respect functions differ fundamentally from numbers p.
This function maps zero onto one, one onto two, and so on. We can specify a corresponding object—e. But according to Frege, any particular argument e. Functions need not be unary. For example, arithmetic division can be represented as a function from ordered pairs of numbers onto quotients: Mappings can also be conditional. Consider the function that maps every even integer onto itself, and every odd integer onto its successor: But we could also index argument places, as shown below.
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But the idea, however we encode it, is that a proposition has at least one constituent that is saturated by the requisite number of arguments. If it helps, think of an unsaturated proposition-part as the result of abstracting away from one or more arguments in a complete proposition.
Frege was here influenced by Kant's discussion of judgment, and the ancient observation that merely combining two things does not make the combination truth-evaluable. The proposition can be represented as follows: Frege thought of the relevant function as a conditional mapping from individuals to truth values: According to Frege, the proposition that John admires Mary combines an ordered pair of arguments with a functional component indicated by the transitive verb: Likewise, Frege did not distinguish the proposition that three precedes four from the proposition that four is preceded by three.
More importantly, Frege's treatment of quantified propositions departs radically from the traditional idea that the grammatical structure of sentence reflects the logical structure of the indicated proposition. Likewise, someone sang iff: For now, assume that the domain contains only people.
In this last example, the quantifier combines with a complex predicate that formed by conjoining two simpler predicates. But on a Fregean view, grammar masks the logical division between the existential quantifier and the rest: With regard to the proposition that every politician is deceitful, Frege also stresses the logical division between the quantifier and its scope: Here too, the quantifier combines with a complex predicate, albeit a conditional rather than conjunctive predicate.
This captures the idea that every politician is deceitful iff no individual is both a politician and not deceitful. If this conception of logical form is correct, then grammar is misleading in several respects.
Second, grammar masks a difference between existential and universally quantified propositions; predicates are related conjunctively in the former, and conditionally in the latter. More importantly, Frege's account was designed to apply equally well to propositions involving relations and multiple quantifiers.
And with regard to these propositions, there seems to be a big difference between logical structure and grammatical structure. On Frege's view, a single quantifier can bind an unsaturated position that is associated with a function that takes a single argument.
But it is equally true that two quantifiers can bind two unsaturated positions associated with a function that takes a pair of arguments. And it follows from all three propositions that Romeo likes Juliet: