A Note on Empty Sets and Quantifiers

We talk about many things that do not exist. Dragons, unicorns, perhaps the modern maritime megalopolis of Atlantis, and so on. More practically, we may posit and discuss theoretical scientific entities — entities whose existence we have no evidence for apart from their theoretical or explanatory virtue. For instance, we can talk of the inclement weather as a result of Zeus’s wrath. A set of non-existent entities is empty. Consider the set of all dragons and count its elements. 0. Same for unicorns and modern maritime megalopoleis. That is, they are all equivalent to the null set.

Still, we want to be able to make nontrivial assertions about non-existent entities. We should be able to say things about them, for instance, “Kronos was a titan.”  Suppose someone said “No Kronos was a titan”.  On a certain level this seems contradictory.  But on the following logical construal, we can make sense of it.

Take the proposition, (1) “all dragons are green”. Because the set of dragons is just the null set, this proposition is trivially true. For we express it as, $\forall x (Dx \rightarrow Gx)$. Which is to say: for all x, if x is a dragon, then x is green. Because no x is a dragon, the antecedent of the conditional, “if x is a dragon”, is always false. Therefore, the whole conditional is always true; so “all dragons are green” is true.

Now take the proposition (2) “no dragon is green”. (1) and (2), prima facie, are contradictory. But (2) turns out to be true, also. We express (2) as, $\neg \exists x (Dx \wedge Gx)$. Or: it is not the case that there exists an x which is both a dragon and green. Because there does not exist a dragon, (2) will always be true.

But the logical representation isn’t exactly faithful to the assertive content of the linguistic utterances (1) and (2).  Existence, or lackthereof, is not explicitly asserted in (1) or (2).  This becomes apparent when we translate the logical sentences back into natural language.  “$\neg \exists x (Dx \wedge Gx)$” translates back into, “there does not exist an object such that it is both a dragon and green.”  And “$\forall x (Dx \rightarrow Gx)$” translates back into, “for all objects, if that object is a dragon then it is also green.”   These seem a little bit different than their counterparts in (1) and (2).

A different approach.  Prima facie it seems that “no dragon” and “all dragons” denote the same object, namely the null set.  And the predicates are the same.  So we are supposed to admit that the sentences are saying the same thing.  But it seems contrary to intuition to say that these sentences “say the same thing” for lack of a better word.  They denote the same thing insofar as their subjects and predicates denote the same things.

Here’s the thought we are addressing. The set of all dragons is empty. So, prima facie, “no dragon” and “all dragons” refer to the same set (the empty set). So when we predicate “is green” to both “no dragon” or “all dragons” we are referring to the same thing, and consequently the predicate must be true of both.

But “no dragon” and “all dragons” do not denote the same set. “Dragons” denotes the (empty) set of dragons. But quantifiers do not operate on the set they are attached to (e.g. “no” [or “all”] does not qualify the subject “dragons”). Instead, quantifiers modify the denotation of the predicate; in this case, “is green”. “No” means that “is green” is not true of anything in the salient set (this case, dragons). “All” means that “is green” is true of everything in the set. So (1) says, “the predicate, ‘is green’, is true of every individual in the set of dragons”. And (2) says, “the predicate, ‘is green’, is true of no individual in the set of dragons. So the conjunction of (1) and (2) says “the predicate, ‘is green’, true of both all the elements and none of the elements in the set of dragons”. And this is more than just apparently contradictory, it is actually contradictory, for nothing, regardless of its existential status, can be both $p \wedge \neg p$.

“All” and “No” do not modify the set of dragons. If they did, then there is no contradiction between (1) and (2) because the same predicate, “is green”, is applying to all the same elements. But “all” and “no” modify predicates, not sets. Consequently, in the conjunction of (1) and (2) “is green” and “is not green” are applied to all the same elements, and therein lies the contradiction.

Is the Present King of France Bald?

What is the truth-value of the following statement?

(1) “The present king of France is bald.”

The State of Play

In this post I will explain why this kind of sentence does have a truth-value.  This is a preliminary response in that I have not read any literature specific to this question.  Rather, a while ago the question was posed to me, and it hasn’t been until now that I have taken the time to give it any real consideration.

Trivially, the sentence, “the present king of France is bald” cannot be true.  This is because, as we are all well aware in the 21st century, France has no king.  Consequently the sentence cannot be true, for it is true if and only if there is a present king of France who is in fact bald.  And so we are left with two options.  Either the sentence is false or the sentence lacks any truth-value whatsoever.

It is potentially illuminating to consider another, slightly different sentence, for comparison.  Assume that that there does not exist an escalator in South College.

(2) “John is on the escalator in South College.”

Now ask yourself, is (2) true, false, or neither?  Like (1), (2), trivially, cannot be true, for (2) is true if and only if John is in fact on the escalator in South College, which is impossible as the escalator in South College does not exist.  So again we are left with two options.  Either (2) is false or else it lacks a truth-value whatsoever.

If your intuitions are anything like mine, you will suspect that (1) lacks a truth-value (and so is not false), while (2) is, in fact, a false statement.  We will see if this is so.

On Subjects and Predicates

Before we begin, it is useful to analyze and characterize the relevant subjects and predicates, how they work, and how they relate to each other.

In (1), there is a subject, “the present king of France”, and a predicate, “is bald”.  And in (2), there is a subject, “John”, and a predicate, “is on the escalator in South College”.  Typically a subject is some kind of individual (or object).  A predicate should be thought of as a set, not an object.  The predicate, “is bald”, denotes the set of all objects that are bald.  So when we say that “the present king of France is bald”, we are asserting that the object denoted by the determinative-phrase, “the present king of France”, is an element of the set of all bald things.  Predicates, like “is on the escalator in South College”, may consist in determinative-phrases, such as, “the escalator in South College”.  This does not change their status as predicates, for a legitimate set is still defined for which any object may be tested or assessed for membership.

A Closer Look at John and the King of France

As another preliminary matter, we should make explicit the truth-conditions for both (1) and (2).

(1T) “The present king of France is bald” is true iff the present king of France is bald.

(2T) “John is on the escalator in South College” is true iff John is on the escalator in South College.

The present king of France cannot be bald, for there does not exist a referent for “the present king of France”.  And John cannot be on the escalator in South College, for that escalator does not exist.  Does it follow from this that (1) is false?  Certainly not immediately.  For if (1) is false, then its negation must be true.  The negation of (1), and its corresponding truth-condition, is:

(-1T) “It is not the case that the present king of France is bald” is true iff the present king of France is not bald.

[For the sake of continuity, we include the negation and corresponding truth-condition of (2), as well:]

(-2T) “It is not the case that John is on the escalator in South College” is true iff John is not on the escalator in South College.

Now we ask, is (-1) true?  Assuming that (1) is false, it is logically necessary that (-1) is true.  But here we run into the same quagmire as (1).  It cannot be the case that the present king of France is not bald, because there is no present king of France to which the predicate can be applied.  So (-1) is either false or lacks a truth-value.  But if (-1) is false, then (1) must we true — and (1) we know to be false.  (-1) and (1) cannot both be false, for this flouts the law of noncontradiction.  And neither one can be true.  This strongly suggests that both (-1) and (1) lack truth-values.

And again for continuity, is (-2) true?  Because there exists no escalator in South College, the set defined by “is on the escalator in South College” will necessarily have no actual elements.  So John cannot be a member of that set.  Consequently, it is true that John is not a member of that set.  So (-2) has a truth-value, namely True.  Therefore it is logically necessary that (2) is false.  Though this is consistent with our initial intuitions, it suggests a nontrivial difference between sentences of kind (1) and sentences of kind (2).  We may consider these differences later.

Consideration of Logical Form

But let’s take a closer look at the logical form of (1).

(1L) There exists an x such that x is the present king of France and x is bald.  Or $\exists{x}(Fx \wedge Bx)$.

And its negation:

(-1L) It is not the case that there exists an x such that x is the present king of France and x is bald. Or, $\neg \exists{x}(Fx \wedge Bx)$.

Now we ask, is (-1L) true?  Well it is true that there does not exist an object that is both bald and the present king of France.  And this seems to be exactly what (-1L) says.  So intuitively, (-1L) is true.  But then it is logically necessary that (1L) be false.

And now we’re in a real pickle.  On our first level of analysis we found that a sentence like (1) must lack truth-value.  But on our logical level of analysis, we find that sentences like (1) must have a truth-value (namely, False).  We cannot have it both ways.  And since the disjunction of both analyses exhausts the realm of possibilities, one of the two must be right.

Note we could also express (1) with a universal:

For all x, if x is the present king of France, then x is bald.  Or, $\forall{x}(Fx \rightarrow Bx)$

And on this rendition, (1) is actually vacuously true!  No x is the present king of France, therefore it all x is -Fx which means that the antecedent of the material conditional is always false and so the entire conditional is always true.

Picking up the Pieces

There must be a difference between (1) and (1L).  This difference is in what I will refer to as “the assertive content” of the statements (1) and (1L).  Here is the literal translation of 1L:

(3) Something is the present king of France and bald.

And this is a far cry from (1), “the present king of France is bald”.  Why?  Statement (3) consists in a subject, “something”, and is ascribed a predicate that is the conjunction of two sets, namely the set of present kings of France and the set of all bald things.  But (1) consists in a subject, “the present king of France”, and a single predicate, “is bald”.  Consequently, these two statements differ in assertive content.  (3) asserts the existence of an object which is an element of both the aforementioned sets.  (1) does not explicitly assert the existence of a present king of France, rather what is explicitly asserted is just that the present king of France is an element of the set of all bald things.  The existence of the present king of France is presupposed (and, consequently, not asserted).

What does it mean to presuppose the subject?  Existence is not a predicate.  Predicates stand for the properties of subjects, and no subject has the property of existence.  A subject either exists or it does not, but its existential status is not a property of the subject qua the definition of the subject.  For instance, we can define God as omniscient, omnipotent, and omnibenevolent.  But we cannot include in our specification that God, in addition to having the properties of the three O’s, also exists.  Defining God to have the property of existence does not make it such that God exists.  This is because, in ascribing properties (or predicates) to a subject, we must presuppose the existence of that subject.  In order to ascribe the three O’s to God, I must presuppose the existence of God.  So the ascription of predicates to entities of dubious existence amounts to something like a conditional statement: If so-and-so were to exist, then it would have such-and-such properties. And this especially highlights the fact existence is not a predicate, for consider: If so-and-so were to exist, then it would have the property of existence.  And this is a tautology.

Because no such object exists, contrary to (3)’s explicit assertion, we intuitively found it false.  Because the existence of the present king of France is not explicitly asserted in (1), there was no explicit phrase which could entail the falsity of (1) (for [1] seems to presuppose a present king of France prior to [1]’s assertion).  (1) and (3) differ in assertive content, this difference accounts for our differing intuitions.  But this raises a new question.  Are we to interpret (1) in the manner of (3)?  That is, if Donald Trump, in conversation with you, said “the present king of France is bald”, should you think you had been told something false or should you think that Trump has said something that is simply neither true nor false?

This question acknowledges our conclusion that the truth-value, or lackthereof, of statements like (1) is determined by the particular level of analysis we bring to bear.  But it raises a pragmatic point, on what level of analysis do we interpret the utterances of others in our day-to-day conversations?

The Pragmatic Point

This section marks a departure from our original question.  It seems we are headed toward some contemporary debates in pragmatics and the theory of meaning.  It is not my intention to wade through any of these arguments.  Instead, I will sketch out why I think that, if someone were to assert (1) in conversation, they would assert something false, rather than something sans-truth-value.  The following is primarily influenced by John Perry’s theory of meaning.  (A theory which I think has many virtues.  Though my own views are unsettled.)

Interpreting meaning in conversation is significantly distinct from assessing the meaning (that is truth-conditions [or truth-value, depending on who you ask]) of a statement in isolation.  This distinction stems from the interaction of two fields, viz. semantics and pragmatics.  Semantics aims to understand the truth-functional structure of language — that is, how each lexical item (or word) in a sentence directly contributes to the meaning of a sentence by way of truth-functional application.  Pragmatics, on the other hand, seeks to understand language (and meaning) insofar as a language is a way of doing things with words.  Conversation is an activity, not a rigid exercise in isolated truth-functional application.  In conversation we try to do things like change each others’ beliefs, get somebody to do something (like pass you the salt), or share information.  And when we share information, we do so with the understanding that our interlocutor has a unique set of pre-existing beliefs, modes of presentation, and ways of thinking about the world (that is, the relations between his or her concepts).  In light of this, meaning in pragmatics will be more dynamic than meaning in semantics.

In conversation (the domain of pragmatics), we do not interpret only the explicit, assertive content of another person’s utterance.  We are sensitive to myriad background and contextual factors in determining speaker-meaning.  For example, you exit the airport in the Basque country.  A man approaches you and utters the sound “/ninaizdjon/”.  He means something by his sounds, but before we can assess the meaning or truth-conditions we need some more information.  Suppose I think that the man is speaking English.  In that case, I take him to have said “Nina is John”.  This is a puzzling statement — it is not often that a person two names, let alone both a feminine and a masculine name.  Perhaps we were wrong to think that the man was speaking English.  Suppose we had some understanding of the Basque language.  When the man says, “/ninaizdjon/”, and I interpret him as speaking Basque, then I will take the man to have said, “Ni naiz John” (which, in English, is “I am John”).  Conversation doesn’t enjoy the luxuries of print; in print we can easily determine the language and parse sounds (for they are, more or less, already parsed for us on the page), but in conversation we are subjected to a constant stream of phonemes and so must bring some background, interpretive theory to make sense of an otherwise disorienting stream of sound.

This demonstrates our sensitivity to background conditions and contextual factors.  When we place the “is speaking English”-background condition on the man’s utterance, we end up with a sentence that is true if and only if “Nina” and “John” co-refer.  But when we place the “is speaking Basque”-background condition on the man’s utterance, we end up with a sentence that is true if an only if the man is named ‘John’.  Nothing in the isolated sound “/ninaizdjon/” can help us determine which of the two aforementioned background conditions is salient/relevant/appropriate/what-have-you.  We cannot determine the meaning without making this choice; context may affect meaning just as much as isolated semantics.  Here is another example.  Suppose that some time ago, Quine said “Cicero wrote beautiful prose”.  The utterance, “Quine said that Tully wrote beautiful prose”, will true if and only if, as a background condition, Quine believes that “Cicero” and “Tully” co-refer to the same object.  For if he did not have this belief, then we would be misattributing a Tully-belief to Quine when Quine has only Cicero-beliefs.  If we were attuned only to the explicit content of an utterance, we would miss out on or be mistaken about the actual meaning of another person’s utterance.  We would incorrectly assess the truth of the utterance “Quine said that Tully wrote beautiful prose”.

An utterance like (1) presupposes the existence of the subject which takes on a predicate. But in ordinary conversation, that presupposition places a background truth-condition on the utterance. The background truth-conditions of an utterance directly contribute to the judgment of the truth-value of that utterance.  So (1) is subject to the following truth conditions.

The explicit (1T): “The present king of France is bald” is true iff the present king of France is bald.

The background: AND iff there exists a present king of France (or, alternatively, there exists an individual uniquely denoted by the determinative-phrase “the present king of France”).

AND iff (1) is a sentence in English

(1) is true iff both the explicit and the background are true.  Because the existential-background truth-condition is not satisfied, (1) is not satisfied, and so (1) must be false.

To reiterate, one last time, the findings of this post: In an isolated context a sentence like (1) will not have a truth-value.  But language is used in varying contexts, and in its use, there must be some additional background truth-conditions which are not explicitly contained in the assertive content of the sentence.  An example of such a background condition will be the existence of the presupposed subject, the present king of France.  I hope that my response to our initial question doesn’t come across as a sort of “Well, it does and it doesn’t”.  I am more inclined toward pragmatic accounts of meaning rather than semantic; so to be unequivocal: if Trump says, “the present king of France is bald”, he would be asserting something false.

As a sort of post-script, I would like to note that when I began writing this, I intended to argue that sentences like (1) have no truth-values.  When I almost finished, I changed my mind and came to the here-written conclusions.

I should also note that there are conversational contexts where one could say, “the present king of France is bald” without necessitating the existential-background truth-condition.  If I am in a conversation with Trump, and we have both already acknowledged the fact that there is no present king of France, then later in our conversation, when Trump says “the present king of France is bald”, I will know that he does not mean to assert or imply the existence of the present king of France.  (This is because I know that he knows there the present king doesn’t exist, and so will not mean to say as much.)  In light of this, one possible interpretation of Trump might be, “if there were a king of France in the present day, he would be bald”.  (Perhaps Trump is making a point about the dangers of contemporary aristocratic French diet, or perhaps he means to say that the genetic line of French kings is prone to premature baldness.)

The Paradox of Mediate Knowledge

This post is intended to briefly describe Drestke’s characterization of the paradox of mediate knowledge in his paper, “Perception and Other Minds”, and to apply the concept to three different cases.  (1) Knowledge of the external world, (2) knowledge of the unobserved, (3) knowledge of other minds.  This is not intended to be a defense of anyone’s argument, but I will link to other posts containing relevant arguments.

…either P is the sort of thing that can be known immediately (non-inferentially, directly, with no evidential basis) or it is the sort of thing that can’t be known at all.  For suppose P is the sort of thing that is known indirectly – on the basis, say, of Q-states.  If is to be known on the basis of a Q-state, then a Q-state must be a more or less reliable sign, indicator, or symptom of a P-state, and this fact must itself be known or, at least, reasonably well-established.  But the fact that a Q-state is a reliable sign of a P-state can never be known.  We can never know this because we can never determine whether P-state exists when a Q-state exists since a P-state is not (by hypothesis) directly knowable, and to know that it exists indirectly is to know that it exists on the basis of a Q-state which assumes the validity of the very correlation that is in question.  Since we cannot tell whether P exists except via Q, we have no way of telling that the correlation between P and Q is as we must assume it to be in knowing that P is the case on the basis of Q.  Therefore, P cannot be known even on the basis of Q.

So it doesn’t seem like indirect knowledge is possible.  Because there is always some sign or somesuch x that stands between us and direct knowledge of what is so, but is somehow supposed to be an indicator of what is so.  But we can never reliably establish the fact that the sign truly indicates the corresponding object.  And all we can ever have knowledge of is the sign, so we can never really know about the object for which it stands.  This sort of situation happens in many areas of perceptual knowledge, and I will discuss three.

1. Knowledge of the external world.

Descartes fashioned his evil genius illusion scenario, and Russell posited his sense-data.  Both of these are the Q-states, which are purportedly signs of things in the mind-independent world around us.  We infer from sense-data to presence of an actual object.  But this inference is not wholly justified, for we have no reason to suppose that the mind-independent object faithfully corresponds to the appearance in our sense-data.  And so the skeptical paradox arises: how can we have any knowledge of the external world?

2. Knowledge of the unobserved.

Hume is the most famous skeptic of our knowledge of the unobserved.  All we ever get in experience is constant conjunction of some A followed by some B.  Some examples: My hand feels hot when it is over the fire | there is constant conjunction between my hand over the fire and my sensation of heat. One billiard ball moves toward another and causes the other to move | there is a constant conjunction between the striking of two balls and the movement thereafter. The sun rises every morning | there is a constant conjunction between the sun rising and certain interval of time.

Here, constant conjunction will be the Q-state.  On the basis of our experience and direct knowledge of constant conjunction, we make an inference to there being some genuine causal relation.  But we cannot establish a definitive link between constant conjunction and causal connection, and so our inference cannot be justified.  And when our belief in causal connection is unjustified, then we get the skeptical paradox: how can we have any knowledge of unobserved events?

3. Knowledge of other minds.

How do we know that another person has a mind?  Presumably we watch them perform intentional actions, like going for a walk or weaving a basket underwater.  We hear them say “ouch!” when they’re in pain.  And so on.  These are the things for which we have direct knowledge.  But on one level, these seem no more than mere behavioral signs (the Q-states).  On the basis of these observations of behavioral signs, we infer that some agent has genuine mental states.  To know that “ouch” is a sign of genuine pain, we need to know that there is a connection between an other’s saying “ouch” and their being in a genuine state of pain.  But it is this connection that we cannot know (and indeed are trying to establish).  The inference from behavioral signs to mental states will be unjustified.  So how can we ever have any knowledge of other minds?

A way forward.

So we see how the paradox of mediate knowledge generates skepticism in three domains of perceptual knowledge.  One solution might be to simply eliminate Q-states all together.  We do not directly perceive sense-data and indirect perceive mind-independent objects; we simply directly perceive the mind-independent objects.  We do not see only constant conjunction and so have indirect knowledge of cause; we simply directly perceive the relevant causal relations.  We do not see behavioral signs from which we infer the presence of a mind; we simply directly perceive the minded-ness of the other individual.  I do not mean to argue for this thesis presently, but just offer it as something to think about in response to the paradox of mediate knowledge.

Davidson, Indirect Discourse, and “That”

In this post I will explain Davidson’s analysis of sentences containing indirect discourse and how we ought to treat their logical form.  This will pay close attention to the role of samesaying with an utterance.  The analysis here will reveal how sentences containing indirect discourse are a type of performative utterance.  Bringing these observations to bear, we will explain how and when we can substitute co-referring terms in that-clauses on Davidson’s account.  Finally, we will assess the adequacy of Davidson’s analysis by considering the similarity between samesaying and sense and reference, in order to show that Davidson’s prima facie anti-intensionalist stance is, in fact, intensional; and discuss how Davidson might reply.

The problem with sentences containing indirect discourse is that surface grammar does not adequately account for their meaning.  In “Scott said that Venus is an inferior planet”, we can substitute “is an inferior planet” for “is identical with Venus or Mercury” and not affect the truth of the sentence (for the former is co-extensive with the latter) (204).  But intuitively, this seems illegitimate because it no longer seems to represent what it is that Scott said, and so the meaning of the whole sentence has changed (204).  An adequate theory of meaning for utterances with that-clauses will specify how the meaning of the utterance depends upon the meanings of its finite component elements and syntactic structure (205) (so that we can construct a finite set of truth-conditions).  And the theory must also explain when the substitution of co-referring terms in a that-clause is permissible.

For Davidson, a sentence containing indirect discourse involves (1) an utterance referring to a speaker S in a context, (2) an utterance conveying the content of a that-clause, (3) an utterance of S with the same content of (2).  I say, “Galileo said that the earth moves”.  S is Galileo in context, and “the earth moves” conveys the content of an utterance of Galileo’s.  What Davidson wants to bring out here is that there is some judgment of synonymy between (2) and (3).

Such synonymy is what Davidson calls samesaying.  Samesaying is when you use words of the same “import here and now” as someone else used them “then and there” (210).  So in indirect discourse, when I say “Galileo said that the earth moves”, I am trying to represent Galileo and I as samesayers by attributing an expression to him (“the earth moves”) that is the same in purport to what he said – that is, synonymous, or has precisely the same content.  This is the key to appropriate substitution of co-referring terms, discussed later.

In light of our new notion of samesaying, how should we think about the logical form of utterances containing indirect discourse?  Suppose Galileo uttered, “eppur si muove”, and I say that “the earth moves”.  Then Galileo and I are samesayers, for our words are of the same import relative to our respective contexts – but this is not to say that it has been asserted that we are samesayers, just that it is so.  Because we are samesayers, there must exist an utterance asserting that Galileo and I are samesayers.  The logical form of such an utterance is: ∃x(Galileo’s utterance x and my utterance y make us samesayers) (210).  So I can attribute any utterance x to Galileo, provided that an utterance of mine y, corresponds to x (is the same in import as x) (210).  So consider (210):

(1) The earth moves.

(2) ∃x(Galileo’s utterance x and my last utterance make us samesayers).  Note that ‘y’ has been substituted for my last utterance, namely “the earth moves”.

If we abbreviate the second line, we get:

(1) The earth moves.

(3) Galileo said that.

How does it get abbreviated this way?  “That” is a demonstrative singular term which refers to an utterance, viz. the utterance of Galileo’s such that it samesays with my last utterance, (1).

So how does this inform the logical form of an utterance containing indirect discourse?  Such utterances consist in (a) an expression referring to a speaker (e.g. “Galileo”), (b) the two-place predicate “said”, and (c) a demonstrative “that”, referring to an utterance of the referent of (a) which samesays with the content of the that-clause (e.g. “the earth moves).  So it should be analyzed and recognized as two semantically distinct sentences, viz. “Galileo said that” and “the earth moves” (212).  From the logical form and the semantic distinctness of clauses of utterance with indirect discourse, it follows that these kinds of utterances are performatives.

A performative is an expression which introduces an utterance in a particular kind of way..  For example, “this is a joke: knock-knock…”.  The “this is a joke” functions to introduce the following utterance, “knock-knock…”, as (importantly) something other than just an assertion, viz. that it is a joke (and perhaps not to be taken seriously) (211).  Intuitively, “Galileo said that” and “the earth moves” looks like an introducing clause and an introduced clause (211).  “Galileo said that” is the performative part of the utterance (211); it’s point is to announce a further utterance in a particular way.  In this case, it introduces my further utterance as one that conveys the content of another’s utterance (Galileo’s), and must function as such.  This is to say that I announce my utterance of “the earth moves” as an action which samesays with an utterance of Galileo’s.  Notice that performative utterances have truth-values.  Suppose I said, “Galileo said that Obama is Kenyan”; this must be false, as clearly Galileo said no such thing.  Likewise, if I said, “this is a joke: Trump won the Republican Primary”, then I said something false, for Trump’s victory is a fact and not a joke (or at least not a funny joke).  So in both cases the entire performative utterance is false.

Now that we have explained samesaying, the logical form of an utterance containing indirect discourse, and why utterances of indirect discourse are performatives, we now have the resources to explain legitimate substitution of co-referring terms in a that-clause.  Consider the utterance “Quine said that Cicero wrote beautiful prose”.  This expression (a) refers to a speaker (Quine), (b) has a two-place ‘said’ relation, and (c) ‘that’ refers to an utterance of Quine’s.  We divide the sentence into “Quine said that” and “Cicero wrote beautiful prose”.  “Cicero” and “Tully” are co-referring terms; would substituting “Cicero” for “Tully” be a legitimate substitution?  So consider “Quine said that” and “Tully wrote beautiful prose”.  “Quine said that” announces the following utterance, “Tully wrote beautiful prose” in such a way that the entire performative utterance is true iff “Tully wrote beautiful prose” samesays with an utterance of Quine’s.  So we need to look at the conditions under which “Cicero wrote beautiful prose” (which we assume is what Quine actually said) and “Tully wrote beautiful prose” samesay.  They samesay iff the “Tully” in our substitution is used with the same import as “Cicero” in Quine’s utterance.  But to know if they have the same import, we will have to know something about Quine.  Namely, we will have to know whether Quine believes that “Cicero” and “Tully” co-refer to the Roman orator or not, when he made his utterance.  Suppose he did believe that “Cicero” and “Tully” co-refer.  Then, for him, “Tully” will have all the import of “Cicero”, and consequently “Tully wrote beautiful prose” and “Cicero wrote beautiful prose” will samesay and our substitution will be legitimate.  But suppose he did not believe that “Cicero” and “Tully” co-referred.  Rather, the only “Tully” he knows is a delinquent undergraduate.  So for Quine, “Cicero” and “Tully” cannot have the same import when he uses them.  Consequently, our substitution of “Tully” for “Quine” is illegitimate, for if you asked Quine if he had said that “Tully wrote beautiful prose” he would deny it (for no delinquent undergraduate writes beautiful prose). This is how Davidson would characterize the substitution of co-referring terms in a that-clause.  If the substitution preserves the import (of Quine’s utterance) – samesays – then the substitution is successful.  But if the substitution does not samesay, then the substitution is not successful (for if the substitution does not samesay with the speaker’s utterance, then it falsely attributes an expression to the speaker).

But Davidson’s account is not immune to criticism.  His notion of samesaying is particularly suspect.  If we take samesaying to be “using words the same in import ‘here and now’ as his ‘then and there’”, then this just sounds like a matter of using some combination of words with the same sense and reference as words spoken by the attributed speaker.  It is not as though import could be the semantic value of an expression, for that’s just a truth-value.  Nor could import mean the corresponding extension, for then we would not have said anything about the problem of substitution of co-referring terms.  So it seems most natural to think of import as sense or mode of presentation.  But if this is so, then this creates a problem for Davidson.  Sense is a Fregean notion, and is the “thought” grasped in virtue of hearing the utterance.  “Tully” and “Cicero” are two different ways of thinking about the same individual, and consequently one could believe that “Cicero wrote beautiful prose and Tully did not”.  That is, the words may be of different import.  But if import just is sense, then Davidson’s account cannot be successful.  This is because Davidson is committed to Tarski’s truth criterion: the meaning of the utterance depends upon the meanings of its finite component elements and syntactic structure.  “Cicero” and “Tully” denote the same individual, so they must mean the same thing.  But if they mean the same thing, then I could not believe that “Cicero wrote beautiful prose and Tully did not”.  But if I could believe that “Cicero wrote beautiful prose and Tully did not”, then I must have different ways of thinking about “Tully” and “Cicero” – that is, the expressions must differ in their senses.  Consequently, what determines the legitimacy of samesaying between expressions is the manner in which each expression is thought of.  And this violates Tarski’s truth criterion because manner of thought is not truth-functional notion, so Davidson’s account cannot be successful.

In light of this, Davidson might respond by offering a more robust characterization of “import”.  Take “import” to be the truth-conditions of an utterance and reconsider “eppur si muove” and “the earth moves”.  Each is true iff the earth moves; intuitively they have the same import.  Now consider “Cicero wrote beautiful prose” and “Tully wrote beautiful prose”.  The former is true iff Cicero wrote beautiful prose; the latter is true iff Tully wrote beautiful prose.  So substitution of “Tully” for “Cicero” is illegitimate.  In this way, Davidson preserves his account without invoking sense, and so conforms to Tarski’s truth criterion.  Samesaying relies on identity between the concrete – truth-conditions – not some abstract, like mode of presentation.

But this is not an effective reply.  For if we intend to samesay with an utterance of Quine’s, then we will need to know what Quine took the truth-conditions of his utterance to be.  But we cannot always know what the speaker to whom we’re attributing an utterance takes the truth-conditions of his utterance to be.  So we cannot always know when it is legitimate to substitute co-referring terms because we do not know if we will preserve identity of truth-conditions of the attributee’s utterance.  So we cannot determine whether our performative utterance is true or false.  For me to truly assert that “Quine said that Tully wrote beautiful prose”, I must know that Quine believed that “Tully” and “Cicero” co-refer for our utterances to samesay.  While it will often be the case that we know what the truth-conditions of the speaker’s (to whom utterance is attributed) utterance are, this is not always so (as suggested by Tully the delinquent undergraduate).

It seems that we want a way to attribute utterances to speakers when we do not have sufficiently reliable knowledge of the contents of their beliefs.  One way of accomplishing this might be to revise samesaying such that when there is not reliable knowledge of the content of the speaker to whom the utterance is attributed’s beliefs, then samesaying will instead be: when my utterance conveys the same content to the hearer as the attributed-speaker’s utterance conveyed to himself.  On this view, it is legitimate for me to substitute “Tully” for “Cicero”, even if Quine did not believe that they co-referred, because to the hearer, the utterance just means the same thing as “Cicero wrote beautiful prose”.  And now the hearer knows something true about Quine, namely that he thought that Cicero wrote beautiful prose.  He is just ignorant of the fact that Quine does not believe that “Tully” and “Cicero” co-refer.

But I think that this move starts to veer off-track.  The notion of samesaying begins to looks more broad, less clear, and more context dependent.

Works Cited

Martinich, Aloysius, and David Sosa. The Philosophy of Language. New York: Oxford UP, 2013. Print.

Goodman and the Grue Emeralds – an Induction Discussion

In this post I will talk about Goodmanized predicates and raise some concerns over their legitimacy. Subsequently, I will dismiss those concerns and argue that Goodmanized predicates are just as legitimate as normal predicates. I will conclude by discussing what Goodmanized predicates reveal about how we acquire beliefs about the unobserved.

Consider the following. Every emerald observed so far has been green. Each time we observe a new emerald, it is found to be green. Each instance of finding a green emerald (and none of any other color), is supposed [^1] to confirm the hypothesis that all emeralds are green. “Green” is a predicate that applies to every observed emerald. Now let’s define a new predicate, “grue”. The definition of “grue” is: x is grue iff it is first observed before t and is green, or else first observed after t and is blue [^2] (74). So something is grue if it is first observed to be green before some arbitrary time t, but if it is first observed after t it must be blue (and not green) to be grue. Suppose t is 7/2/17. Every emerald, then, is green. But every emerald has also been grue. So every observed emerald has (so it seems) supported the hypothesis that : all emeralds are green, and the hypothesis that all emeralds are grue. Of course, it seems that after 7/2/17 each newly observed emerald will not be grue – certainly no emerald could be both grue and green (provided that it is observed after t), for the definitions would conflict. If all we do is infer from particular instances to general claims, then we have equal evidence for two competing hypotheses [^3]. The mystery is why we are wont to believe that emeralds after t will be green and not grue.

But if “grue” is not a legitimate predicate, then the mystery disappears. Goodmanized predicates have some odd features that warrant closer inspection like (1) they are disjunctive definitions and include a time reference and (2) Goodmanized predicates are not natural terms for things actually in the world. When a predicate contains a time-dependent disjunction, it applies to one set of objects before t and a different set of objects after t. In this way, it appears poorly defined.

Another worry is that “grue” is not a term for a natural phenomenon, whereas “green” is. Because of this, we cannot legitimately predicate “grue” of emeralds. Here’s the thought. Green seems to be a natural property. Naturally existing in the world are green things, even if there is no one in the world to observe it. An emerald is naturally green – we just need to look and see. But we cannot just see that something is grue, because we must also know t and where we are in relation to it. We can’t see anything about t in the emerald. Green predicates a naturally occurring property, where grue predicates an artificial and contrived property that does not reflect our natural ontology.

But these worries can be dismissed; Goodmanized predicates are not illegitimate. It is a mistake to think that grue is poorly defined. The definition of grue is exactly the same before and after t. Which part of the disjunction is effective in the application of the predicate is determined by t, but not the definition. Objects that are grue before t are still grue after t – they do not become suddenly not-grue. Moreover, while grue is admittedly an artificial term, that does not mean that it is illegitimate to predicate it of natural objects. Consider the set of green things, e.g. some grass, the bushes in your mother’s yard, emeralds. This is easy and legitimate. Now consider the set of grue things, e.g. grass and emeralds first observed before t, and peacocks and blueberries first observed after t. Note that it is just as easy and legitimate to consider this set as it was to consider the green set. When we say X is grue, we say that X belongs to the set of grue things. If there was something wrong in the consideration of the set of grue things, then there would be something illegitimate about predicating grue of X. But there is not.

Now that we have preserved the legitimacy of Goodmanized predicates, we can ask what they reveal about how we come to get the beliefs we have about the unobserved. Let’s return to our consideration of the grue and green emeralds. Recall the state-of-play: every emerald thus far observed has been both green and grue, so each observation has (ostensibly) supported two conflicting hypotheses, viz. that all emeralds are green and that all emeralds are grue. After t, one of the two hypotheses must fail, for their predictions contradict each other [^4].

But which hypothesis fails? Presumably, the rules of induction are what enable us to project into the future – that is, to be able to make accurate predictions with regard to each subsequent, unobserved instance. The traditional view of induction works like this. ($\Phi\,\!$) Every A thus far observed has been found to be B. Each newly observed instance of an A as a B (and assuming that no A has been observed as not B) confirms a hypothesis, viz. all As are Bs. So, anyone who has observed a long positive correlation between things of any two kinds A and B in a wide variety of circumstances over a long interval of time, has reason (or will come) to believe that all As are B. The thought is that each new instance provides us with more reason to subscribe to the hypothesis – that is, it confirms the hypothesis, raising the probability of the hypothesis being true. But this way of thinking about induction cannot decide between all emeralds being grue or all emeralds being green, because an equally long positive correlation has been observed between emeralds and greenness as has been observed between emeralds and grueness.

What the competing hypotheses “all emeralds are grue” and “all emeralds are green” show us is that this way of thinking about projection into the future is wrong – that this is an inadequate way to explain how we form beliefs about unobserved cases. Why? Because intuitively we believe that after t, the emeralds will persist in being green and cease to be observed as grue.  We suppose that “green” can easily and legitimately figure in our inductive inferences.  And we suspect that “grue” cannot easily and legitimately figure in our inductive inferences.  Consequently, we believe in the green hypothesis over the grue hypothesis.  But what is it about “green”, rather than “grue”, that allows it to legitimately figure in our inductive inferences?

If we form beliefs about unobserved events based solely on the flat-footed understanding of induction (marked [$\Phi\,\!$]), then we couldn’t have the belief that emeralds observed after t are going to be green (but we do), because there is an equally well-supported competing hypothesis. We have no more reason to believe that they will be grue than that they will be green, when considering just the evidence from our experience on the traditional ($\Phi\,\!$) view of induction.  For all emeralds observed thus far have been both green and grue.

According to Goodman, legitimate inductive inferences are the ones that are performed on lawlike correlations rather than accidental correlations. A lawlike inductive hypothesis is confirmed by its positive instances [^5]; a coincidental inductive hypothesis is not confirmed by its positive instances.  If the green hypothesis or the grue hypothesis is lawlike, then that correlation is confirmed by its positive instances.  Intuitively, we think that the green hypothesis will be the lawlike one.

So in order to have beliefs about the unobserved, it seems that we must have some way of determining whether a given correlation is lawlike or not – or at least there must be some way we come to believe that one hypothesis is lawlike rather than another.  Positive instances are not enough.

I will try to show that we can have a perceptual experience of a causal relation.  If this is so, then we may have a way of determining between lawlike and coincidental correlations, and an explanation of why we believe in the green hypothesis over the grue hypothesis.

Consider the fact that we have a concept of causation. How did we acquire it [^6]? Consider how you acquired a concept of a chair. You first saw the chair, and then you abstracted from it. Now consider a pot of water on a flaming stovetop. You see the pot, the water starting to bubble, and you see the fire and the stovetop. But in addition to this, I submit that you also see (or have some perceptual awareness of the fire boiling the water. You don’t just see the individual elements (e.g. the pot, the fire) isolated from one another, you see the elements interacting. That is, you have perceptual awareness of the causal relation between the fire and the boiling water. If you saw the scene as composed just of isolated elements and didn’t see the fire boiling the water, then you couldn’t abstract the causal relation and come to have a concept of causation. But you do have a concept of causation, and this suggests that it comes from our perceptual experience.

If we have a concept of causation, then we can believe two things to be connected causally and apply the concept to the situation. I can believe that A caused B and form the inductive hypothesis that: all As are Bs. Without a concept of causation, I cannot experience a causal relation between A and B and will not come to believe or hypothesize that all As are Bs. When I see that every emerald I have observed has been green, I am inclined to form the belief that something about the emerald causes it to be green. However, when I see that every emerald I have observed has been grue, I am not inclined to form the belief that there is something about the emerald that causes it to be grue. Why the difference? Because I do not have a perceptual experience of something about the emerald causing it to be grue [^7] – my experience is of the green.  Why is my experience of the green and not the grue?  Because looking at a green emerald, I believe that there is something that causes it to be green – there is a reason or an explanation of it.  But when I look at a grue emerald, I do not believe that something caused it to be grue.  For part of what it is to be grue is to be first observed at a certain time.  But the time of first observation is something entirely coincidental.  I do not believe that there is something that causes the emerald to be grue, other than when I happened to first observe it – which could have been any other time t.  Consequently, the causal relation I perceive is between green and the emerald, not grue and the emerald.

But what happens when we think we have an experience of a causal relation, but the relation turns out to be coincidental? This worry is not devastating. We are often mistaken as to the contents of our perceptions. Consider a man who thinks he sees a bird in the distance when it is in fact a small plane. His perception is mistaken, but we do not dismiss him as a poor perceiver or suddenly begin to doubt his judgment of his perception. Likewise, should some coincidences mistakenly be seen as causal, this is no reason to doubt our general ability to perceive causal relations. It just means that we will sometimes be wrong. Some inductive hypotheses with positive instances will fail, even when we thought they were lawlike, but this is how we come to develop our system of scientific law. But if we can in fact perceive causal relations, then this will be how we come to differentiate between the lawlike and coincidental and thus come to have beliefs about the unobserved.

[^1]: Whether it actually confirms the hypothesis or not is less clear. This will be discussed later.

[^2]:  “Bleen” is another example of a Goodmanized predicate. An object is bleen iff it is first observed before t and blue, or else first observed after t and green.

[^3]: Note that no emerald changes color, even after t.

[^4]: Because an emerald cannot be both blue and green. After t, if an emerald is first observed to be grue, then it is not green, because by the definition of grue it must be blue. But after t, if an emerald is first observed to be green, then it cannot be grue, because if it is green then it is not blue.

[^5]: A fair coin flipped 10 times and always lands on heads is accidental. If this happened with a loaded coin, it would be lawlike.

[^6]: Assuming that we do not innately possess concepts like causation. (Indeed, it is hard to see how we could innately possess them.)

[^7]: Furthermore, it is difficult to see how I could ever acquire a concept of grue through perception. Because grue includes a time-dependent disjunction, when I see the green/grue thing I only see the green because the time of first observation is not included in my perceptual experience of the emerald and so I wouldn’t abstract the concept of grue from it. I see the greenness, but I don’t see the grueness.

The Fregean Conjecture

“Semantic composition is functional application” – the Conjecture.

In an extensional theory of linguistic meaning, there are only three kinds of things: individuals, functions, and truth-values.  The meaning of a sentence is determined by the individual meanings of each of its words as well as its syntactic structure.  A simple example:

Jack drinks

There is “Jack”, an individual, and there is “drinks”, a function.  Drinks is a function which takes a single argument (in this case, “Jack”), and maps it to a truth-value.  The meaning, then, is a truth-condition: “Jack drinks” is true iff Jack drinks = T.  Suppose Jack does in fact drink.  Then we plug in “Jack” into the function “(x) drinks”, and the output will be the truth-value, T.  Suppose instead that Jack’s been sober almost two months.  The function “(x) drinks” will map “Jack” onto the truth-value, F.  (Probably.)  More formally:

[Jack](Let F be that function f  such that For All x in domain, f(x) = T  if and only if x drinks, otherwise f(x)=F.) = T

As sentences grow in complexity, it can be difficult to keep track the syntactic structure – that is, exactly which component of the sentence is an argument for whatever other function in the sentence.  It can be useful to see an example of a sentence broken up into its constituents.

The cowboy on the cliff rides hard into the west.

([The [cowboy]] [[on] [the [cliff]]) ([[rides] [hard]] [[into] [the [west.]]])

Note that the only individuals in this sentence are “cowboy”, “cliff”, and “west”.  This means that rest of the words are functions.

(Will update.)

Intermediate Logic – Textbook PDF

Logic and Structure

This is a resource for becoming acquainted with propositional and predicate logic, completeness, second order logic, intuitionistic logic, and Godel.  This is a good introductory/intermediate book.

Natural Language Semantics – Textbook PDF

Semantics in Generative Grammar

This is a solid textbook for learning the formal semantics of natural language.  That is, the applied logic and structure of natural language.

Modal Logic textbook pdf

Modal Logic for Open Minds

This is a great resource for those interested in learning more about modal logic – the logic of possible worlds.

The Analytic and Synthetic in Kant and Frege

In this brief and hasty post, I will present both Kant’s and Frege’s notions of the analytic/synthetic distinction.  Frege claims that he does not mean to assign new meanings to the analytic/synthetic, but rather state more precisely what earlier writers meant by them.  I will argue that if Kant had Fregean logic, rather than Aristotelian logic, in his toolkit, then he would have characterized the analytic/synthetic distinction differently.  In particular, he would have recognized the judgments of arithmetic are analytic and that analytic judgments can be ampliative.  So Frege’s version will, in fact, amount to a more precise rendition of what earlier writer meant by the distinction.

1. Kant’s Analytic/Synthetic Distinction.

First we will characterize Kant’s distinction.  An analytic judgment is one in which the concept of the predicate is contained within the concept of the subject. For instance, consider the judgment “all bachelors are unmarried”.  Unmarried is contained within the concept of bachelor, for to deny that all bachelors are unmarried would be a contradiction; we would say that she just didn’t understand the meanings of the terms.  A synthetic judgment, in contrast, is one in which the concept of the predicate is not contained within the concept of the subject. For instance, consider the judgment, “all bodies are heavy.”  The concept of heavy is not contained within the concept of body, so there is no contradiction in denying the judgment. In this way, two concepts are synthesized in making this judgment.

This leads Kant to the conclusion that all mathematical judgments are synthetic a priori.  They are a priori because they are necessary.  For it cannot be the case that the internal angles of a triangle do not sum to 180 degrees, nor could it be the case that 7+5 does not equal 12.  Kant admits that it may seem as though 7+5=12 is analytic (and follows from the principles of non-contradiction).  But this is not so.  An analytic judgment is a judgment in which the concept of the predicate is contained within the concept of the subject, but no where in the concept of the combination of 7 and 5 do we find the concept of 12.  This is more apparent with a very large numbers: you could be thinking of 230987 and a moment later be considering the combination of 222634 and 8353, and not realize that you had been thinking of the name number.  So these judgments synthesize two concepts.  To form the judgment, then, we must appeal to something more than just the two concepts – we appeal to a priori intuition.  Mathematics, then, is characterized by the construction and synthesis of concepts in a priori intuition.  A priori intuitions are the pure forms of space and time (the formal requirements of any sensation or experience) – they are the canvas upon which we can have any experience at all.  Arithmetic is based on the pure intuition of time, and geometry on that of space.  So in a priori intuition we construct arithmetic propositions by manipulating and arranging units of time.

1. Frege’s Analytic/Synthetic Distinction.

Now we will characterize Frege’s distinction.  For Frege, whether a judgment is analytic or synthetic is not dependent on the content of the judgment, but rather on the justification of the judgment.  For a judgment to be analytic, its truth and justification must rest on definitions and purely general logical laws.  For example, that 5+7=12 is true just in virtue of the definitions of 5, 7, and 12, and logical laws [1].  Any analytic judgment is traceable back to only definitions and purely logical laws.  If one of these definitions or laws is denied then anything could follow – an unfortunate situation.

A definition is a stipulation, not an assertion, so it cannot be denied.  But once a definition is stipulated, it is immediately transformed into an analytic judgment.  Consider the following. I stipulate that a “sachelor is an unmarried man”.  Immediately, I can judge sachelor to be an unmarried man, or even say that “a sachelor is a bachelor” (provided I am familiar with the concept of bachelor). These would be analytically true judgments, for I have defined the terms in the right way, and the rest relies on the law of identity.

A judgment is synthetic when the truth of the judgment appeals to, in addition to logical laws and definitions, a “special science” – that is, when there is appeal to some fact that is neither a law nor a definition.  So the truth of the proposition “most swans are white” is synthetic because it is not in the definition of a swan that it is white (to be a swan is not to be white).  Rather, the truth and justification of the proposition relies on at least partly on experience, which is not in the domain of definition or logical law.

The justification for an arithmetic judgment rests on its proof.  So for Frege, arithmetic judgments will be analytic.  This is because their proof rests only on definitions and general logical laws.

[1] Frege assumed that all mathematics was reducible to logic.  So mathematical operators like “+” will have their logical counterparts.

1. Frege: why arithmetic judgments are analytic

Because I briefly mentioned Kant’s reasons for supposing that all analytic judgments are synthetic, I will quickly describe why Frege thinks that all arithmetic judgments are analytic.  I will not offer all of the arguments that Frege provides, because this post is more concerned with comparing the notions of analytic/synthetic and speculating how their respective logics influenced their characterizations.

If arithmetic judgments are to rely on a priori intuition, we must be able to conceive of Number in a priori intuition. In what way could we do this? We might think of Number as magnitudes of some kind, either spatial or temporal. Therefore, we must examine what it is to have an a priori intuition of magnitude. It is unclear what an a priori intuition of magnitude could possibly be. What things have magnitudes? Lines, numbers, areas, forces, e.g.. We could conceive of a line of a particular magnitude in a priori intuition, but we ought not confuse this with conceiving of magnitude. A line of some length can be brought under the concept of magnitude, as can numbers, areas, and so on. But that these all belong to the concept of magnitude is not to say that any one of them is an intuition of magnitude in general. In intuition we can consider things which belong to the concept of magnitude, but we cannot intuit magnitude itself. So what would an a priori intuition of magnitude or number be? What is an a priori intuition of 100,000? It cannot be “100,000”, for that just denotes 100,000; it is the name of the number, not the number. Again it cannot be a line or any other thing. We have no a priori intuition of 100,000. Worse yet, consider what your intuition of 0 could be – that is, what is your sensation of 0, what would it be to experience 0? Of 1? When we ask these questions no intuition of the number itself strikes us. Yet Kant claims that intuition must be sensible. Thus we have shown that there can be no a priori intuition of number. Because no intuition, empirical or a priori, can give us Number, arithmetic judgments cannot be synthetic.

Now for Frege’s positive argument for the analyticity of arithmetic.  While Frege does not disagree with Kant that geometric judgments are synthetic a priori, he does disagree that there is a substantive analogy between geometric and arithmetic judgments. Geometry is a special science whose domain is all that is spatially intuitable. Arithmetic, however, has a much wider domain – so much so such that it is intimately connected with the laws of logic in a way that geometry is not. The domain of geometry is all that can be represented in three-dimensions. Now consider a four-dimensional object – such a thing cannot be represented in spatial intuition. We can think – that is, conceptualize – a four-dimensional object, but should we represent it in intuition as an aid, then we are still working within three-dimensional space. Our representation of a four-dimensional object must still be three-dimensional. Our intuition merely represents or symbolizes the four-dimensional thing, but does not directly display it. It is this same notion that allows us to intuit a curved line as straight. Calling the line straight does not involve any contradiction. This is to say that we can still go on to prove things within that system despite our taking the curved thing as straight. This means that we can reject a geometric axiom without ending in contradiction, for the axioms are independent of each other. This makes geometry a special science.

In this way, geometry is unlike logic, for if you were to deny a logical law, you would wind up in contradiction. E.g. deny the law of non-contradiction, and you truly arrive in a contradiction, namely that a is not a. And it seems nothing intelligible can be thought after this. Recall that analytic truths are truths that can be traced back to just definitions and logical laws, and that if one of these were to be denied then nothing can properly follow. Logic is an analytic enterprise, then. But geometry relies on logical laws, definitions, and axioms (which can be denied and are what make it a special science). This is why geometry is synthetic.

But we cannot say the same for arithmetic as we can for geometry. One cannot assume the contrary of a fundamental proposition of arithmetic [2] and proceed to reason successfully – we cannot perform the same conceptual magic tricks as we do in geometry. This shows that arithmetic must be intimately connected with the laws of logic. For Frege, logic is the most general domain – the laws of logic are the laws of thought, and so apply to every domain of thought, even the special sciences. Language has a truth-functional, logical structure in which the truth conditions of an utterance are built up out of the truth-conditions of its component parts.  The reason that language has this structure is that the thought requires this structure. And, moreover, the domain of arithmetic is much wider than that of geometry, governing all that is numerable – and all that is numerable is thinkable, vice versa (for you can map one number to every unique [item of] thought). Because arithmetic is connected to logic in this way, arithmetic must be analytic, for to deny its fundamental propositions would lead to contradiction in the same way as for logic. And so arithmetic must proceed only from general logical laws and definitions. And so arithmetic must be analytic.

[2] It is unclear exactly what Frege means by this phrase.  My guess would be the Peano axioms.

1. Comparing Kant and Frege’s distinctions.

Having shown both Kant’s and Frege’s conceptions of the analytic/synthetic distinction, we are now in a position to ask whether their conceptions of the distinction are equivalent.  Here’s what’s at stake.  If their conceptions differ, then Frege’s arguments against Kant that arithmetic judgments are analytic become empty, as Frege is using “analytic” in a special sense and so is merely “talking past” Kant, as it were.  This is because he takes himself to be arguing against the synthetic-ness of arithmetic judgments in Kant’s sense; so his arguments must pertain to Kant’s distinction.  If their conceptions are the same, however, then arithmetic judgments are analytic a priori (if Frege’s arguments are successful), contrary to Kant’s claim that they are synthetic a priori.

The most apparent discrepancy between Frege’s and Kant’s conceptions is that one conception concerns the justification of the judgment (Frege), whereas the other conception concerns the conceptual content of the judgment (Kant).  So it may appear that their senses of analyticity are distinct.  But this is not so.

It is important to consider the fact that Kant was working with Aristotelian logic which is based on subject and predicate.  Frege was, unsurprisingly, working with Fregean logic, which is based on function and argument and is thus much more powerful.  When Kant asserts that a judgment is synthetic, he is saying that the judgment must appeal to something outside the meanings of the terms.  When Frege asserts that a judgment is synthetic, he is saying that the judgment must appeal to some special science (or particular fact).

That a judgment appeals to something outside the meanings of the terms, however, is just to say that a judgment is not based solely on definitions and logical laws, relying on some particular fact or other.  When we are talking about concepts that are not contained within their subjects, we are appealing to some particular fact or special science.  For example, to justify “most swans are white” is to observe the matter of fact that most swans are white, with some exception – it is not to reason from the terms.  This is wholly consistent with Frege’s account.

However, Kant also maintains that analytic judgments are explicative, whereas synthetic judgments are ampliative.  To be explicative is just to explicate knowledge that we already have – to make explicit things known; to be ampliative is just to amplify knowledge – that is, extend our knowledge about the concept.  But if Frege is in fact just stating more precisely Kant’s distinction, then analytic judgments can be ampliative.  Complex formal logical judgments can lead to conclusions that are not apparent, but surprising, and thus amplify our knowledge of concepts.  For instance,

If you look at (1) and (2), then at the formulae in (13), your knowledge has been amplified, for prima facie you cannot see the relation between them.  One cannot see just from looking at at certain numerical propositions all the derivations that proceed from them, even though those derivations follow just from general logical laws and definitions, so those conclusions must be ampliative.  We learn important new relations regarding relevant concepts.  Kant cannot maintain that analytic judgments all are explicative and that synthetic judgments are all ampliative, if his analytic/synthetic distinction is the same as Frege’s.  So we must show why Kant would and should give up his claim that analytic judgments are ampliative.

Consider the fact that Kant does not present his distinction in the context of logical examples, like Frege, but rather in the context of concepts in natural language.  E.g. he provides the example that “all bodies are extended”.  Clearly this is analytic and does not amplify our concept of body.  In contrast, “all bodies have weight” is synthetic and does amplify our concept of body by attaching a new, distinct concept of weight to it.  These examples are reminiscent of Aristotelian logic or syllogism.  All men are mortal, Socrates is a man; therefore, Socrates is mortal.  An Aristotelian syllogism has a concept/subject structure – concept B holds of subject A.  This kind of structure is amenable to the analysis of concepts, subjects, and arguments in and of natural language, items like, “body”, “weight”, “extension”, “mortal”, “man”, “green”, etc…  And these are the kinds of items that Kant has in mind when he first presents the distinction – items familiar to natural language.  And these are the kinds of items that will turn out to be explicative in analytic judgments (A holds of A) and ampliative in synthetic judgments (A holds of B).

But we can see that Frege’s notion of analyticity still captures all of the natural language examples that Kant has in mind.  To see how, consider a Fregean treatment of the judgment that “all bodies are extended”.  Is this a matter of just laws and definitions?  Let’s see.  Define a body as a thing which occupies space.  Define to-be-extended as to occupy space.  Now substitute in our definitions and we get the judgment, “all things which occupy space, occupy space”.  And this seems trivial – so trivial that the only way to deny it is to deny the law of identity.  This is to say that one could only reasonably hold this judgment as false if things that occupy space didn’t occupy space!  A flat-out contradiction.  So in virtue of only (1) the definition of body, (2) the definition of extension, (3) the law of identity.  So this judgment will be analytic in Frege’s sense.  We can see that Frege’s sense of the distinction captures the natural language examples that Kant had in mind in first presenting his distinction.  It is also explicative as opposed to ampliative, for no new knowledge was about any of the relevant concepts was obtained.  What about Kant’s synthetic proposition, “all bodies have weight”?  Define weight as the downward force an object exerts.  Substitute, and: “all things that occupy space, exert a downward force”.  This is synthetic.  Logical laws, like the law of identity, and definitions do not suffice to justify this judgment (for downward force is not identical to occupation of space); empirical facts about the world must come into the justification.  Moreover, this will be ampliative, for we extend our knowledge of bodies when we learn they have weight.  So Frege’s conception of the analytic/synthetic distinction capture the natural language cases Kant primarily considered.

1. Were Kant to have Fregean logic

While Kant may have presented his distinction with natural language in mind, Frege presents his distinction of the analytic/synthetic with logic and arithmetic in mind.  Analytic natural language judgments will continue to be explicative for both Kant and Frege.  But it’s in the realm of logic and arithmetic that analytic judgments are sometimes going to be ampliative – that is, that they are going to extend our knowledge of the relevant concepts (recall the complexity of formal logical arguments).  Because Kant was more concerned with judgments in natural language, it seems plausible that he would abandon his ampliative/explicative distinction in light of the ampliative nature of analytic mathematical judgments.  For if Kant had been working the more powerful Fregean logic, rather than the Aristotelian, he would have recognized some arithmetic and logical judgments as analytic and ampliative.  Recall that Aristotelian logic was based on subject/concept arguments.  These were basic in structure, so its conclusions seemed rather explicative.  Recall our earlier syllogism, and notice that each premise is a stipulation and conclusion seems rather explicative.  We have two definitions, (1) all men are mortal and (2) Socrates is a man.  The law of identity will allow to instantiate Socrates in (1) to get: Socrates is mortal.  On Frege’s picture, this is analytic and ampliative.  And it is just a containment (logical) relation between concepts, and so conforms to Kant’s conception of the analytic/synthetic distinction.

References

Frege, Gottlob. The Foundations of Arithmetic; a Logico-Mathematical Enquiry into the Concept of Number. Evanston, IL: Northwestern UP, 1968. Print.

Kant, Immanuel. Critique of Pure Reason. Trans. Norman Kemp Smith. New York: St. Martin’s, 1965. Print.