That Logic Blog

July 07, 2006

Meaning via Proofs

Verificationism is the idea, popular amongst the logical positivists, that the meaning of a sentence is to be equated with the method used to establish it. That is, a statement is true if and only if we can, in principle, verify its truth or if it is analytic, which is to say that it is true by definition.

Now, suppose that we are working in classical logic and wish to assert some statement φ and pretend for the moment that we are all happy little verificationists. This means that in order to assert φ, we first need to give a method for determining whether or not it is true. That is, we need to determine whether or not it is present in our model. Being logicians first and foremost, we proceed via structural induction. If φ is an analytic statement, otherwise known as an atomic proposition, then we are done. If φ is of the form "ψ and δ" then we are done if both ψ and δ are in our model. Proceeding in this manner, we can verify whether or not φ is in our model by verifying that a certain collection of atomic propositions and negated atomic propositions are present in our model.

This sort of explanation seems to give the impression that semantics precedes syntax. That is, the meaning of a syntactic expression is determined by its semantic interpretation. Or is it? Let's have a closer look at what is going on here. Returning to our assertion "ψ and δ", we reduced the verification of its presence in the model to the verification that ψ is in the model and the verification that δ is in the model. We can be somewhat facetious and invert this procedure. That is, suppose we already know that both ψ and δ are present in the model. Then we may assert "ψ and δ". Carrying on this inversion process, we can say that a statement is in the model if it is either analytic or can be constructed from the analytic statements via certain inference rules. In other words, what we have discovered is that our model of classical logic is nothing but the free boolean algebra generated by the analytic statements. But we can summarise the situation in a far more snappy manner:

Verification is dual to construction

An even more catchy war cry is:

Models are dual to proofs.

So, the inversion process that we followed allows us to say that syntax precedes semantics. This latter view is encompassed in what is known as proof theoretic semantics whereas the former view is, naturally, known as model theoretic semantics. Since, to a large extent, these theories are dual to one another both warrant serious attention. The latter has, however, had the lion's share of attention historically. Nevertheless, an intrepid group of researchers has been flying the flag of proof theoretic semantics. Its modern culmination is evident in both computational and mathematical logic in such places as interactive theorem proving and categorical semantics of linear logic. On the philosophical front, there has been a resurgence of interest lately, including a special issue of Synthese devoted to the subject. A good discussion of the sorts of things mentioned in this post is provided by the following paper from the volume (unfortunately, I cannot find a free version of the paper):

Meaning approached via proofs; Dag Prawitz

In the next post, I'll write a little about what categorical algebra has to do with this stuff.


Anonymous Nick said...

Out of curiosities sake, how would pseudo-synthethic statements and meaning be described using this free boolean math?

11:30 AM  
Anonymous Jon said...

I'm not sure what a possible answer to your question is off hand, since I am not familiar with the term "pseudo-synthetic". Could you give some examples?

1:11 PM  
Anonymous Nick said...

I meant it in the sense A.J. Ayer used the term 'pseudo-synthetic'. I'm writing on this topic at the moment and figure I will post the relevant bits here. Hopefully this clarifies things and isn't too long-winded:

What I'm describing is the distinction logical positivists made between analytic, synthetic, and pseudo-synthetic statements. Analytic statements have three main features: (1) analytic statements are necessarily true and their denials are contradictions, (2) these statements are known to be true a priori, and (3) they do not convey factual information. Taking up the first feature, mathematics is one instance in which analytic statements are necessarily true. That is, a mathematical statement cannot be taken as false for the sake of argument. One can imagine that the statement “The Golden Boy on top of the Manitoba Legislative Building is 17 feet tall” is false. However, one cannot imagine 15 + 2 = 17 is false. Verbal statements can be necessarily true as well given their logical form. For example, the disjunction “The pigeon is on the roof or the pigeon is not on the roof” is necessarily true, since if there is a pigeon, all possibilities are covered by this statement.

Pertinent to the second feature, the mentioned mathematical and verbal statements are known to be true a priori. This refers to the fact that analytic statements are known to be true independent from observation. One need only reflect on the meaning of the terms and the rules of the language used to determine the truth of an analytic statement. To illustrate, the statement “Nick’s pet is a bird” is not analytic, as an observation is needed to determine whether or not Nick’s pet is a bird, a cat, or some other animal. However, the statement “All pigeons are birds” is known to be true, independent of observing all pigeons. This results from the usage of the word “bird” and “pigeon” within the English language and the logical form of the statement. This hints at the third feature: that analytic statements are not factually informative.

Analytic statements are not factually informative given that they are necessarily true or contradictory and therefore false. For example, “There is a pigeon on the roof or there is not a pigeon on the roof” is known to be true before one climbs the ladder to observe if a pigeon is indeed perched on the roof. However, this truth is not specific to any thing in the world, for it deals with the logical form of the statement. The word “pigeon” in the statement may be replaced with “dove” or “cat” or “hippopotamus” and remain true. Thus, the statement “There is a pigeon on the roof or there is not a pigeon on the roof” is true and addresses the subject matter of a pigeon on a rooftop, yet is not factually informative, as it does not indicate which of the cases is being fulfilled. This is not to say that analytic statements are trivial. While an analytic statement is not factually informative, it remains informative about the rules of mathematics and logic. Moreover, an analytic statement can be informative in the sense that it provides information that was previously unknown. For example, one may know what a sphere is without knowing the surface area of a sphere is equal to 4πr2. This is, however, information related to the defining characteristics of a sphere. For contrast, compare the statement “A sphere’s surface area is equal to 4πr2” with “The bloated pigeon on the roof is of the shape of a sphere”. The truth of the former statement can be determined by reflecting on the defining characteristics of a sphere, whereas the truth of the latter statement requires the observer to measure the bloated pigeon. Analytic statements thus deal with the syntactical relations of words and numbers. That is to say, an analytic statement deals with the way words and numbers necessarily relate to one another if they are to be used meaningfully. An analytic statement, however, does not deal with the way words and numbers are semantically used to refer to states of affairs in the world. That is, analytic statements are not factually informative.

Synthetic statements have three main features as well: (1) it is possible for a synthetic statement to be either true or false, (2) the truth-value is determined a posteriori, and (3) synthetic statements relate factual information. Expanding on the first property, it should be noted that a statement may be modified and still be possible to be true or false. Borrowing from the previous example, “There is a pigeon on the roof” may be modified to its negation “There is not a pigeon on the roof”. Both statements serve as synthetic statements with different truth-values. In the analytic statement, the two statements were combined in the disjunctive proposition “There is a bird on the roof or there is not a bird on the roof” and was determined to be true. However, isolating “There is a bird on the roof” and the contrary “There is not a bird on the roof” results in one statement being true and the other false. As opposed to the analytic case, the person must not reflect on the statements but must now verify which statement is true (and hence which is false). The truth or falsity of a synthetic statement is therefore determined a posteriori as only an observation will be able to determine the truth-value of the statement after it is propounded. This shows that the synthetic statement is factually informative, for it relates to something that cannot be arrived at by reflection, but requires one to observe what is out in the world.

Now, as for pseudo-synthetic statements, I refer to a statement that according to logical positivists cannot be true. Not only this though, logical positivists claim that a pseudo-synthetic statement does not have truth-capacity. Since positivists insist that there are only two procedures that determine the truth of a statement--a priori procedures for analytic statements and a posteriori proecedures for synthetic--a statement is said to be pseudo-synthetic in that it says something about factual information (or lapsing into relating to factual information), yet this rules out the a priori procedure for determing the truth of the statement. At the same time, those making pseudo-synthetic statements refuse to specify observations that would be relevant to determining the truth of the statement, and hence ruling out the a posteriori method.

So, a statement like 'God loves bloggers' would be considered pseudo-synthetic in that it says something about the world yet does not give rise to any observation that could determine the truth of the statement.

And according to Ayer's verification principle is said to be factually meaningless.

Did that help?

3:19 PM  
Anonymous Jon said...

Nick: Sorry for the delay replying, it's been rather hectic over here :-)

I'm not sure that algebraic semantics shed much light on pseudo-synthetic statements. The main point of algebraic semantics is that by translating logical problems into algebraic ones, we hope to be able to prove some nice general theorems about various classes of logics. I see traditional algebraic semantics as more of a mathematical tool than a philosophical tool.

As for the example of "God loves bloggers", unless you admit God as an existing entity, this statement would not even be present in whatever algebra you are working in. I can imagine setting up an ad-hoc mathematical analysis of meaning along the lines you suggest by using, say, partial functions. However, once again, this would just be translating the problem into a different language.

3:08 PM  
Anonymous nick said...

No worries about the delay. If you're still in Melbourne, consider taking the boat to Tassie. Despite what the mainlanders say, they are not two-headed creatures out there and there are some awesome views, like this one I took when I hiked to the top of the hazards.

I suppose 'Two-headed creatures love bloggers' would also require an ad-hoc mathematical analysis. Sigh. Could you recommend a good place to start as far as algebraic semantics is concerned?

9:03 AM  
Anonymous Jon said...

Nick: Where to start depends on your mathematical background. Introduction to lattices and order gives a nice introduction to some of the mathematics involved. However, the book is slanted towards applications in computer science and mathematics, though it does contain some stuff on logic. Most importantly, it proves the Stone duality theorem, and the more general Priestley duality theorem.

For a more general introduction to universal algebra, you may like to have a look at this book, which is both good and free.

Finally, if you have access to JSTOR, the following article gives a nice (if somewhat parochial) introduction to the whole business:

The basic concepts of algebraic logic. Paul Halmos. American Mathematical Monthly vol. 63 (1956) pp 363--387.

9:31 AM  
Anonymous Jon said...

As far as Tassie goes, I have yet to make it out there, though it is certainly on the list.

9:32 AM  
Blogger J said...

"For example, the disjunction “The pigeon is on the roof or the pigeon is not on the roof” is necessarily true, since if there is a pigeon, all possibilities are covered by this statement."

Yes, but there occasionally seems to be some linguistic issues regarding analytic statements that logicians routinely overlook: "He is bald or he is not bald." Does baldness
mean completely hairless? At least in many instances, a person would say tho' X has a bit of hair around his ears, he is bald.

"He is ambitious, OR he is not ambitious."

"He loves his neighbors, or he doesn't love his neighbors." T v F?

Are these tautologies? Perhaps, but it seems there are some statements--in ordinary language, --which are sort of definition dependent. Methinks that is sort of what Quine was getting at in Two Dogmas, though I disagree with his conclusions and do feel analyticity definitely holds in formal contexts (but perhaps not ordinary language).

The other more epistemogical issue which you touched on is the status of the law of the excluded middle itself (LOTEM); most would say it's necessary and known a priori, right. But justifying any sort of a priori knowledge is hardly an easy task. Was the LOTEM inferred? You mentioned induction--yet it seems very strange to say the LOTEM was established inductively. I admit my dilletante status in these matters, but at some point the skeptic/naturalist does seem to raise a decent point: yes there seems to be a distinction between analytic/synthetic statements, but isn't the very premise of logic (LOTEM) itself inductive, and/or a matter of definition, and thus any axioms are themselves dependent on those initial empirical, inductively-established axioms (including the LOTEM)? Perhaps that is nominalism, but
how else does one PROVE the LOTEM is binding in all analytic contexts? The LOTEM may be a somewhat Kantian given, and yet there are (as the ordinary language examples show) instances where it seems to be challenged (a few of the strange results of quantum physics are also somewhat contra-LOTEM, as Quine realized).

Causality and probability, of course, are other matters of concern for the logician who would hold to the purity of analyticity, tho' I realize that involves all sorts of difficult issues.

Consider this somewhat bizarre example: X eats twinkies for breakfast and then at the office, kills Y before lunch.

Is "eating twinkies for breakfast led X to kill Y OR eating twinkies for breakfast did not lead X to kill Y" analytic?
I do not think so.

3:34 AM  

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