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Analytic and Synthetic Statements

Updated on August 25, 2014

Sentences and Propositions

There are philosophically motivated objections to the concept of a proposition. Even when admitted, the notion can be defined in different ways. On the other hand, when propositions are not allowed, certain difficulties arise about how to talk about subjects in logic.

The easy, initial way to grasp what we mean by "proposition" is this: take a sentence that is meaningful - any sentence. Such a sentence can be asserted or put down as true in meaningful discourse. This type of sentence is called declarative sentence. Now, we need to draw a distinction between this sentence itself and its meaning. This might surprise you at first but consider this: many sentences (even, theoretically, an infinite number) can have the same meaning. For instance, "someone hit me" and "I was hit by someone" have the same meaning but they are different as sentences; one is in the active and one in the passive voice. Or, suppose that we have a sentence in another language which, translated, is found to have the meaning "someone hit me." This is a different sentence - it is even in a different language! - and yet it has the same meaning. Besides, we could write the sentence "someone hit me" again and again: we have different sentences (because the sentence is the physical, symbols-put-together object, and we have not one but many of them); but they all have the same meaning. What if we write this sentence using different font, or size of letters - and so on? Again, many sentences but they all have the same meaning. It seems that the meaning is something sentences "carry" or "express" - but metaphors are bound to be unsatisfactory in this case. The meaning of a declarative sentence is what we mean by "proposition."

A similar situation arises when you learn the definition of a concept, for instance a geometric triangle. The textbook shows many different triangles - they look different from each other in some respects but, in the decisive respect, they all triangles. They share something in common: they are all triangles. The opposition to this kind of thing stems from a fear that we might fancy ourselves to be working with some abstract, spooky, things: what is the thing all triangles have in common? What is the proposition which all the sentences examined above express? The Ancient Greek philosopher Plato insisted that there are such abstract entities as "the triangle itself" which is somehow shared by everything that is, because of this sharing, itself a triangle. This abstract thing, triangularity let's say, is a real thing, Plato claimed. This view - the view that posits as real entities that are abstract (not even in space or time!) - is called Platonism. Modern philosophic schools tend to be anti-platonic and they fret about claims that abstract entities exist as real although they are not empirically verifiable. The "proposition" we are talking about comes across as such a Platonic entity. We won't, however, let this distract us. It is crucial to understand that we definitely need to draw a distinction between a string of symbols, put together in accordance with the rules of some specified grammar, and WHAT such a string expresses. The string is the sentence but what it expresses -- the meaning, intuitively speaking -- is the proposition. Aristotle, a famous ancient philosopher, thought that a proposition is a thought (like a picture in the mind actually...) We should rather stick to the linguistic items we have been working with. So, a proposition is what is expressed, the meaning, when a declarative sentence is asserted. When a proposition is stated or uttered, it can be called a statement. We can use the words "proposition" and "statement" interchangeably for our purposes. Also, we should keep in mind that the word "sentence" itself is used in the way we have said that we should be using the word "proposition." The context tells us if "sentence" means just symbols put together (or vocalized sounds put together) according to a grammar; or if "sentence" is used to refer to the meaning made by the grammatical entity that is the sentence.

You might be wondering why we need to complicate things. The bottom line is that, as indicated from the very beginning, we have to draw a distinction between the meaning and the symbols-grammar used to express it. When we study philosophy, we deal with meanings - with propositions. The distinction between analytic and synthetic we will study here is about two different types of propositions. You can call them sentences if you will, but the understanding is that we are not talking about linguistic grammar per se; we are talking about what is being said.

When students begin to study Logic, the first system - standard logical system - is very simple. This is the so-called Propositional Logic. It works in such a way that we can make the following simplification: There are two kinds of propositions: the true and the false. All statements one can make are either true or false; they have to be one or the other, if they are meaningful; and no statement can be made of a proposition that is both true and false (no such thing admitted!) In deductive reasoning, the content of what we say is not important. This is the greatest surprise, which, one suspects, serves as a barrier to keep people on the other side of understanding how logic works. Keep in mind that this is something you need to understand and work on it.

Let's recapitulate: declarative sentences - sentences that are meaningful and can be asserted in meaningful discourse - express propositions (the meanings presented by such sentences.) A proposition can only be true or false; it has to be one or the other; it cannot be both. This is called the Extensional view of what a proposition is.

There is a better view - richer, more sufficient for the study of language, and of philosophic problems for sure - the Intensional view. According to this view, we should treat a proposition as a dynamically sensitive entity, which can vary between true-false across different situations. Mark this: the situations we are talking about are themselves abstract things - they are not stories like when we use our imagination; they are all the logically possible states-of-affairs that are definable. When you study math, wonderful things like this happen: you DON'T have to actually fill in stories; it is sufficient that you lay out the plan like (you MODEL your subject.)

Here is an example. Take the sentence of English, "Socrates is asleep." It is meaningful; it can be asserted. The standard or extensional view is that this has to be true or false. Fine. You might think that the proposition expressed by this sentence is true when he sleeps and false otherwise. This would make the proposition context-sensitive. Extensionally, though, we cannot have that. This has to be treated as something that can be true or false, not both and not variable. The way to do this is interesting: we have to pack into the sentence "Socrates is asleep" all the dynamic elements - place, time, etc... Then, can you see what happens? The sentence really is "Socrates is asleep at x place and y time." Now see that you understand that the proposition so made is TRUE or FALSE, not both, and not context-variable.

Intensionally, we can do something else. We can take the proposition expressed by "Socrates is asleep" to be the set of all the logically definable situations in which Socrates is indeed asleep. This is an elegant approach and allows for context sensitivity.


Analytic and Synthetic Propositions

A proposition is the only thing that can be true or false. It may cause confusion to use "true" or "false" in referring to other kinds of things. Of course, we say "this is not true gold" but the semantic (meaning-related) terms "true" and "false" should be thought of as applying only to propositions. "True" and "false" are called truth-values.

A proposition is called analytic if its truth-value can be determined without checking information about facts. Consider first a proposition that is not analytic. "We have class on Mondays" cannot be determined for its truth-value without checking facts. This proposition is synthetic, it is not analytic. The informative propositions are synthetic. There are, however, propositions that have their truth-values necessarily and not because of what happens. You might wonder how this can be. The analytic propositions can be determined as to whether they are true or false because of the meanings of the words in them. An example is "a triangle has three angles." The meanings of the words "triangle," "three", and "angle" make it so that the proposition expressed by this sentence cannot possibly be false; it is neecessarily true. Try, for instance, to ask someone, to check, or to seek information as to whether it is true whether a triangle has three angles. This sounds rather amusing - the reason is that the statement "a triangle has three angles" is trivially true: the meanings of its words make it so that it has to be true no matter what happens. This is an analytic true statement. The statement made by the sentence "a triangle has four angles" is also analytic - and false: the meanings of the words, again, make it so that this statement cannot possibly be true.

A writer who writes "on that far away planet, triangles actually had two angles" is confused. He puts down as true, under some conditions, a statement that is analytic and false: this means that this statement is logically-necessarily false; it is false regardless of circumstances or under any circumstances you wish.

Understand that the meanings of words are considered settled. We couldn't communicate if linguistic expressions were volatile or open to arbitrary revision. In Alice in Wonderland, a character who is made deliberately to speak nonsense says "words mean exactly what I want them to mean." In logical analysis, we assume a competent user of the language, one who knows the meanings of the words (like in geometry we assume someone who understands the proof when we present a proof.) Given adequate linguistic knowledge, it is automatic that, for instance, the proposition made by the sentence "an elephant is an animal" is analytic and true while the proposition "a stone is an organic being" is also analytic and false. But, "there are animals" is actually synthetic - the meanings of the words do not fix its truth-value as true or false. We can know what "there is" and "animal" mean without being able, on that basis, to determine if the statement is true or false. Of course, in actuality, this statement is true; but it COULD be false: you can logically conceive a world in which there are no animals. But you cannot conceive a world in which what we define as "elephant" is not an animal.

There is another way for a proposition to be analytic. It can be the logical form of the proposition that makes it possible to determine its truth-value (true or false) right away and without relying on factual information. Every proposition that has the logical form, for instance, "p and not-p" has to be false. This is a contradiction. Consider the propositions made by the sentences: "it is raining and it is not raining," "we have class on Mondays and we don't have class on Mondays," "it will rain tomorrow and it will not rain tomorrow," etc. They all have the same logical form: if we symbolize an individual proposition by "p", the logical form shared by all the above is "p and not-p." So, every proposition that has this logical form is necessarily false; it cannot possibly be true. It is analytically false (since we don't need reference to any extra-linguistic information to determine the truth-value.)

Deep down, we think that, in both cases, it is the meaning of the words that settles the truth-value. In the logical form "p and not-p" the words that matter, whose meaning makes the proposition analytically false, are "and" and "not." Notice that these are the only two fixed words - the rest are variables since ANY proposition can be plugged in for "p." Such special words (like "not," "and," "either-or," "if-then," "all," "some," etc.) are key to the logic of a language. Their meanings fix the logic of a language. Given the meanings of "and" and "not," it is logically impossible for any proposition of the form "p and not-p" to be true! So, such propositions are analytic and false. Examples of propositions that are analytic because of logical form and true are: "it is Monday today or it is not Monday today" (the logical form, notice, is "either p or not-p"); "if Schmuck is lazy, then at least one person is lazy" (logical form is "if 'a' Fs', then at least one thing Fs'"); "assuming that if you fail the exam you fail the class, then if you haven't failed you can't have failed the exam" ("if (if p then q), then (if not-q, then not-p"));...

It is absurd to dispute about the truth-value of analytic propositions. People who are ignorant of the meanings of words might but this is a matter of having a defect. Very often, it is not obvious even to a knowledgeable person that what is at stake may be an analytic proposition. Imagine the waste in arguing about mis-identified analytic propositions. For instance, consider the following statement: "if it is true that either you study or you fail, and you failed, you couldn't have studied." Does this proposition (form "if p or q, and q, then not-p") have a logical form that makes it always true, or always false? Or is the form synthetic? This is not easy for anyone.

The informative statements in any language have to be synthetic. Analytic statements are true/false by virtue of the meanings of their words (including the special logical words like "and" and "not" that make logical forms necessarily true or false if that's the case.) Now consider some harder examples of statements about which it might not be easy to tell if they are analytic or synthetic.

-- "Every object is in space." Do the words "object" and "space" settle the truth-value or not?

-- "If I think, then I must exist."

-- "The causes of the battle, whatever they were, must have preceded the event of the battle itself."

-- "Every event takes place in time."

-- "Time flows in one and only direction, from the past and through the present and toward the future."

-- "If an act is morally obligatory, then it must be morally permissible in the first place."

-- "If wishing makes it so, then it cannot be so."

-- "Either there will be a battle tomorrow or there won't be a battle tomorrow."

-- "A bad student is a student."

-- "No bad student is really a student." (Is this nonsense? Is it a denial of an analytic true statement, "every bad student is a student"? Caution: the word "student" might have two different meanings in the two occurrences in the sentence!)

-- "Yellow is not the same as blue."

These are all contradictions. They are conjunctions (compound statements with two members joined by "and"; and one of the conjuncts is the negation of the other. They all have the same form - "p and not-p." They are analytic and false statements.
These are all contradictions. They are conjunctions (compound statements with two members joined by "and"; and one of the conjuncts is the negation of the other. They all have the same form - "p and not-p." They are analytic and false statements.
The common logical form of all these disjunctions is "either p or not-p." They are analytic and true statements.
The common logical form of all these disjunctions is "either p or not-p." They are analytic and true statements.
w@ is our actual world and w1, w2, w3 are logically possible worlds. "T" symbolizes "a triangle has three angles." "R" symbolizes "a famous monument is shaped like a triangle." The first is analytic (true everywhere); but the second is synthetic.
w@ is our actual world and w1, w2, w3 are logically possible worlds. "T" symbolizes "a triangle has three angles." "R" symbolizes "a famous monument is shaped like a triangle." The first is analytic (true everywhere); but the second is synthetic.

Analytic-Synthetic Statements and A Priori-A Posteriori Knowledge

We should learn two more terms that make frequent appearances in the study of philosophy. Next, we will see how, if at all, these terms combine with analytic-synthetic.

A proposition is knowable a priori if it can be known without depending on experience. Plato famously based a big part of his philosophy on the claim that there are such propositions. Other philosophers, like John Locke, denied it. Today, evolutionary theory has no problem with the notion that we might be pre-wired to have a priori knowledge (some advantage for survival and reproduction may make such knowledge be picked up by natural selection.) A proposition is knowable only a posteriori if it is not knowable a priori: if it has to be learned through checking experience, facts, and such.

An example of a priori knowledge, you might want to know? The whole of Mathematics! Consider this: "2 = 2 = 4" Take this to be a string of symbols expressing a proposition. Now, imagine that you have experiences, again and again, that seem to deny this. Two and another two things of the same kind put together - each time, again and again, you are left with three, not four! Would this repeated experience "teach" you that "2 = 2 = 4" does not express a true proposition? Not really. The events you are experiencing are weird. Why are they weird? Maybe it is a matter of experience, after all, if two and two make four or not. Could we perhaps teach a baby by messing with its mind this way? (Unethical experiment.) Experiments have shown, by the way, that babies cry when the arithmetic is messed up this way, or in some way; they don't cry otherwise (when two and another two toys leave them with exactly four toys.)

The point is that we have knowledge - propositions that our minds pick out as necessarily true - that does not depend on experience - it is a priori. Whether the sun rises in the east we DON'T know a priori! But that it is true that "the sun rises in the east or does not rise in the east" has been claimed to be a priori. That a triangle (given what we mean by it) has to have a total of 180 degrees for the sum of its angles, is knowable a priori. I realize that you didn't always know this - so why is it knowable a priori? Here is the reason: suppose you are completely cut off from the world out there - you are in solitary. You are given a geometry book. This defines certain terms for you ("triangle," etc.) and lays down a game you play. You prove theorems, as you know geometry does. Maybe, you can't do it but that doesn't matter. You could; you might be a genius for that matter. One day, you prove that the triangle's angles add up necessarily to 180 degrees. Notice that you have done all this a priori - no access to the world of experience from which you are cut off as it is. Nor do you have to wait until you get out of jail to confirm if this is true. If the world doesn't confirm it that would mean that the triangles out there are not like the ones you thought about (which, by the way, happens to be the case in a spherical universe like ours, whereas the geometrical triangles are presumed flat.) Once again, you did not depend on factual information and facts-experience-etc. cannot correct the knowledge of geometry you accumulated while thinking and playing the geometry-game. This is a priori knowledge.

You might think that a priori knowledge goes fixedly together with analytic statements; and a posteriori goes tightly together with synthetic statements. This is a good start and traditionally this was the view. It has turned out to be more complicated, though, and this is one of the exciting areas of philosophy - no one pursues such matters outside of philosophy. The complication of the neat match-up (analytic-apriori / synthetic-aposteriori) started even in the early 19th century with Immanuel Kant, a famous German philosopher, who claimed that there are propositions which, though synthetic, are known a priori. An example is: "every event has a cause". This is not analytic, notice. ("Every effect has a cause" is analytic - why?) Yet, even though "every event has a cause" is synthetic, Kant claimed that any creature with a mind that can do what the human mind does must be working in such a way that this sentence, although synthetic, is known a priori. Not everyone agrees with Kant of course.

Other combinations are possible. This is an advancanced subject, however, and we should not pursue it further at this point. Hopefully, your appetite is whetted for more.

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