A Polish translation of Fefermans' Alfred Tarski: Life and Logic, translated by Joanna Golińska-Pilarek and Marian Srebrny is being published. On April 1, Warsaw University Library organizes a reception, accompanied by talks given by Jan Woleński, Mieczysław Omyła (who wrote the introduction to the Polish edition), Jan Zygmunt, Wacław Zawadowski, and THE TRANSLATORS.

## Tuesday, March 31, 2009

### A contra to Putnam's 'Twin Earth'

Recall than Putnam (The Meaning of 'Meaning') argued that if a new substance is discovered with a different chemical composition but the same phenomenal qualities as certain known substance, it will not fall under the same natural kind.

His thought experiment to support this claim is this: imagine a distant planet just like ours, except the water-like substance there has different, complicated chemical composition XYZ. It looks and tastes like water etc. Imagine you travel to that planet. On Putnam's reading, even though we would initially think this is water, once we discover the difference in chemical composition, we'll reject this view and reserve the term "water" for those substances which have the structure of H

Here's a historical example that these things aren't so clear-cut. For ages, the Chinese considered jade to be the most precious substance (pretty much like gold in the West). They also were very sensitive to its authenticity. Near the end of the 18th century, a large shipment of a very similar stone with unrelated composition, jadeite, made it to China. The Chinese could tell by its feel that it was a different material. They called it "new jade". They also started calling the real jade '"old jade". Interestingly, both substances came to be called jade and the new jade came to be used pretty much just like the old jade (now, jadeite is even more valued than nephrite, the original jade).

[More details about this stuff in LaPorte's very nice book, Natural Kinds and Conceptual Change.]

His thought experiment to support this claim is this: imagine a distant planet just like ours, except the water-like substance there has different, complicated chemical composition XYZ. It looks and tastes like water etc. Imagine you travel to that planet. On Putnam's reading, even though we would initially think this is water, once we discover the difference in chemical composition, we'll reject this view and reserve the term "water" for those substances which have the structure of H

_{2}O.Here's a historical example that these things aren't so clear-cut. For ages, the Chinese considered jade to be the most precious substance (pretty much like gold in the West). They also were very sensitive to its authenticity. Near the end of the 18th century, a large shipment of a very similar stone with unrelated composition, jadeite, made it to China. The Chinese could tell by its feel that it was a different material. They called it "new jade". They also started calling the real jade '"old jade". Interestingly, both substances came to be called jade and the new jade came to be used pretty much just like the old jade (now, jadeite is even more valued than nephrite, the original jade).

[More details about this stuff in LaPorte's very nice book, Natural Kinds and Conceptual Change.]

Labels:
Laporte,
natural kinds,
Putnam,
thought experiments

## Thursday, March 26, 2009

### A cool lecture video online

A video from a nice session on Time Travel (Anne French, Ken Perszyn and Nick Smith) is available here. The sever hosts also Colossal Squid Lectures (they have nothing to do with time travel, afaik, it's just the title is so cool I couldn't help but mention this).

Nicholas J. J. Smith's research website links to a few fun papers about time travel as well. Among others: Why Would Time Travellers Try to Kill their Younger Selves? and Bananas Enough for Time Travel?

Labels:
conditionals,
logic,
time travel

## Wednesday, March 18, 2009

### A new blog on philosophy in Bristol

Alexander Bird has started a new blog on philosophy-related stuff going on in Bristol. He has already posted a podcast on alternative medicine and taking science seriously.

Labels:
academia,
Alexander Bird,
bits and bobs,
Bristol

## Tuesday, March 17, 2009

### Norton against new logics for thought experiments

John Norton (Why Thought Experiments Do Not Transcend Empiricism) argues, among other things, that no new logical considerations are needed when we consider thought experiments in science. Here' s what he calls "evolutionary argument", in his own words (pp. 16-17 of the online version of the paper):

I think there are some reasons to believe that no new, exotic logic is called for. In outlining the general notion of logic above, I recalled the evolutionary character of the logic literature in recent times. New inferential practices create new niches and new logics evolve to fill them. Now the activity of thought experimenting in science was identified and discussed prominently a century ago by Mach (1906) and thought experiments have been used in science actively for many centuries more. So logicians and philosophers interested in science have had ample opportunity to identify any new logic that may be introduced by thought experimentation in science. So my presumption is that any such logic has already been identified, in so far as it would be of use in the generation and justification of scientific results. I do not expect thought experiments to require logics not already in the standard repertoire. This is, of course, not a decisive argument. Perhaps the logicians have just been lazy or blind. It does suggest, however, that it will prove difficult to extract a new logic from thought experiments of relevance to their scientific outcomes -- else it would already have been done!

Well, here's a question. What's wrong with the following argument, and if it isn't wrong, how does its form differ from the form of the evolutionary argument?

Norton, among other things, works on philosophy of relativity. Now, relativity theory has been around, pretty much, since the same time when Mach wrote about thought experiments. So philosophers interested in science have had ample opportunity to identify and solve any philosophical issue that may be introduced by relativity theory. So, in this field, any philosophically interesting claim has already been made and any philosophically argument has been given, and Norton's work in philosophy or relativity is redundant. Unless, of course, philosophers of science since the discovery of relativity theory have just been lazy or blind.

Labels:
logic,
Norton,
thought experiments

## Thursday, March 12, 2009

### A volume on thought experiments online

This is kinda old news, but if you haven't noticed, a volume on Thought Experiments in Science and Philosophy edited by Horowitz & Massey is freely available online here.

Labels:
thought experiments

### A neat example of an a posteriori necessity

Recall Kripke's examples of a posteriori necessities in science were of the sort:

Gold = the element with atomic number 79LaPorte (Natural Kinds and Conceptual Change) provides a neat example that is more analogous to the well-known 'Hesperus=Phosphorus':

Water = the substance composed of two atoms of hydrogen and one atom of oxygen

The statement 'Brontosaurus = Apatosaurus' has a history quite like that of 'Hesperus=Phosphorus'. 'Brontosaurus' was coined as a genus-term in 1874 by O.C. Marsh who thought he had discovered a new genus of dinosaur in Wyoming. As it turns out, the fossils he discovered were fossils of a dinosaur genus that he himself had already discovered and named 'Apatosaurus'. Marsh supposed that the specimen he associated with the name 'Brontosaurus' and the specimen he associated with the name 'Apatosaurus' could not be from the same genus because there was such a difference in size between the two specimens. He did not realize that the reason for the difference in size was only that one of his specimens was not fully grown. later, another scientist, Elmer Riggs, straightened out the matter, determining that Marsh had applied two names to the one genus. Riggs discovered empirically that 'Brontosaurus=Apatosaurus' is true. (p. 37)The story about Brontosaurus comes from Gould's Bully for Brontosaurus: Reflections in Natural History, 1991, pp. 79ff.

Labels:
a priori,
identity,
Kripke,
modal logic,
necessity

## Wednesday, March 11, 2009

### Mereology and species

Laporte on species-kinds and species-individuals

I'm reading Laporte's Natural Kinds and Conceptual Change. On p. 15-16 he talks about how species don't have to be viewed as individuals, and that they can be interpreted as natural kinds. For some reason, he thinks mereology is relevant. Here is the original passage:

I'm reading Laporte's Natural Kinds and Conceptual Change. On p. 15-16 he talks about how species don't have to be viewed as individuals, and that they can be interpreted as natural kinds. For some reason, he thinks mereology is relevant. Here is the original passage:

Suppose that the organisms of any species make up an individual, or something else that is not a kind. Call such an object a species-individual. Suppose, further, that talk about the species could satisfactorily be interpreted as talk about the species-individual. In that case, I will argue, such talk about the species could also be satisfactorily interpreted as talk about a kind. Here is why: Although the species-individual is not a kind but rather an individual, there is a property, for any such individual, of being part of that individual. For that property, just as for any other property, there is a corresponding kind, such that possession of the property is the essential mark of the kind. Thus, for any species, there is a kind the members of which are preciesely the parts of its species-individual. Call such a kind a species-kind. Because, for any species, there is both a species-individual and a species-kind, there is no reason that discourse about the species would have to be interpreted as being about the species-individual rather than the species-kind.A few things come to my mind:

An example may help to clarify the reasoning. The organisms belonging to Raphanus sativus (and their parts) are precisely the parts of a species-individual. Those parts are precisely the members of a species-kind. Thus the organisms belonging to Raphanus sativus are precisely the members of a species-kind. And it would therefore seem that discourse about the species Rahanus sativus could satisfactorily be interpreted as discourse about that species-kind, whose members are just the organisms of Raphanus sativus. (pp. 15-16)

- On Laporte's account, species-individuals seem to be mereological fusions of the species representatives. This is indicated by the claim that the "organisms of any species make up an individual," and by the suggestion that in case of the species-individual of the species Raphanus sativus, "the organisms belonging to Raphanus sativus (and their parts) are precisely the parts of a species-individual".
- Laporte insists that if it makes sense to speak of species-individuals, it also makes sense of species-kinds. The first problem is, the belief that species are individuals, by itself, does not entail that species-individuals are to be construed as mereological fusions of the species. So, in fact, Laporte's argument, if it works, shows only that (given certain background assumptions about properties) if it makes sense to speak of species-individuals-construed-mereologically, then also it makes sense to speak of species-kinds.
- The more important problem, however, is that if one really construes species-individuals as mereological fusions, then (at least in classical mereology) it is not the case that being a member of a species is coextensive with beign a part of the corresponding species-individual.
- Laporte says: "The organisms belonging to Raphanus sativus (and their parts) are precisely the parts of a species-individual" - so far, so good - this means that if you take the fusion of a species, then every representative of this species and every part of such a representative (we might add, and every fusion of arbitrary selection of parts of the species members) is a part of the species-individual.
- Then he continues: "Those parts are precisely the members of a species-kind." Okay , let's say so (given that we accept the addition in the brackets above). So, for instance, if we take the species homo sapiens, the species-individual, as Laporte construes it, would be the mereological fusion of all humans. Now take the property of being a part of this fusion. This gives you the species-kind. What's important, not only every human being will be a member of this species-kind. Also, every part of a human being will be part of the species-individual, and as such, a member of the species-kind thus construed. Moreover, even the fusion of my left leg and your, dear reader, right hand (assuming you're a human being) will be a member of such species-kind.
- Now, we can see why it doesn't follow that "the organisms belonging to are precisely the members of a species-kind.". Just like the fusion of my left leg and your right hand is not an organism belonging to homo sapiens, there are fusions of parts of organisms belonging to Raphanus sativus (radish), which are not organisms belonging to Raphanus sativus. Similarly, just like my left foot is not a homo sapiens, even though it is a part of the Laportesque homo-sapiens-individual, there are also parts of the Raphanus sativus-individual, (say a leaf) which are not, buy themselves, organisms belonging to Raphanus sativus.
- The bottom-line: if you construct species-indvidual as a mereological fusion, you'd better steer clear off classical mereology. Perhaps, a variant of non-classical mereology can be used to make more sense of this, but this requires a lot more work, and prima facie, I don't think there's much hope in this project (I might be wrong though).

Labels:
Laporte,
mereology,
natural kinds,
species

## Tuesday, March 10, 2009

### Ajdukiewicz and some other classics, online

If you read German, Ajdukiewicz's stuff on syntactic connectivity is available here. His paper on the notion of existence (in English) is here.

Mostowski's classification of logical systems is available here.

Tarski's paper on the notion of truth is here (in German). His "Semantic conception of truth..." in English is available here.

Reichenbach's Elements of symbolic logic are here.

Principia Mathematica can be found here.

Carnap's Empiricism, Semantics and Ontology is here, and his Meaning and Necessity is here.

Ayer's Language, Truth and Logic is here.

The Need for Abstract Entities (Church) is here.

Austin's plea for excuses is here.

Soles' paper on Russell's causal theory of meaning is located here.

Angelelli's paper on disputation in the history of logic is here.

Mostowski's classification of logical systems is available here.

Tarski's paper on the notion of truth is here (in German). His "Semantic conception of truth..." in English is available here.

Reichenbach's Elements of symbolic logic are here.

Principia Mathematica can be found here.

Carnap's Empiricism, Semantics and Ontology is here, and his Meaning and Necessity is here.

Ayer's Language, Truth and Logic is here.

The Need for Abstract Entities (Church) is here.

Austin's plea for excuses is here.

Soles' paper on Russell's causal theory of meaning is located here.

Angelelli's paper on disputation in the history of logic is here.

Labels:
Ajdukiewicz,
Austin,
Carnap,
Church,
logic,
Mostowski,
Reichenbach,
Tarski,
texts

## Friday, March 6, 2009

### Brian Ellis on the logic of natural kinds

I'm reading Brian Ellis' Natural Kinds and Natural Kind Reasoning (in Natural Kinds, Laws of Nature and Scientific Methodology). I'm looking especially at section 6, The Logic of Natural Kinds. There, he introduces some notation and puts forward a bunch of principles that are meant to be necessarily true about natural kinds (it's almost like reading an early piece presenting axiomatic approach to modal logic: here's the language, here's the intuitive reading, and here are the principles).

Anyway, one of the principles doesn't seem quite right, and honestly, I don't know what Ellis wanted to say there. Here's a brief description of the "logic".

Abbreviations. `

(1) If x=

(2) For every K, there is an intrinsic property P such that PeK

(3) If x∈K and PeK, then NPx

(4) If PeK, and K

(5) If x∈K

(6) If x, y∈K, and x=

(7) If K

(8) If K

(9) If x∈K

(10) For all x, (Nx∈K or Nx∉K)

(11) There are no two natural kinds, K

(12) The class of things defined as the intersection of the extensions of two distinct natural kinds K

Now, apart from the disadvantages of the attempt to determine a logic in a purely axiomatic manner, I'm worried especially about principle (6). Let's take a look at it again:

If x, y∈K, and x=

The first observation is that this is ambiguous between two readings:

Reading A. If x, y∈K, and x=

Reading B. If x, y∈K, and x=

But let's ingore this.

Second, both readings seem false. Just take y to be x, and take K

So, perhaps, Ellis was assuming that different variables are referring to different objects/kinds (I'm trying to be charitable here, and try this reading, event though this reading is unlikely in the light of the fact that Ellis wants to have non-identity of kinds in the consequent)??

Well, again, this won't fly. For take x and y to be two distinct, and yet intrinsically identical objects belonging to one species K. Let K

Am I missing something? Is there another natural principle that Ellis might've had in mind, but failed to state??

Anyway, one of the principles doesn't seem quite right, and honestly, I don't know what Ellis wanted to say there. Here's a brief description of the "logic".

Abbreviations. `

- 'x∈K' reads: 'x is a member of the natural kind K'
- 'PeK' reads 'P is an essential property of K'
- 'K
_{1}⊂K_{2}' reads 'K_{1}is a species of K_{2}' - 'x=
_{e}y' reads: 'x is essentially the same as y' - 'x=
_{i}y' reads: 'x and y are intrinsically identical in their causal powers, capacities and propensities.'

(1) If x=

_{i}y, then x=_{e}y(2) For every K, there is an intrinsic property P such that PeK

(3) If x∈K and PeK, then NPx

(4) If PeK, and K

_{1}⊂K_{2}, then PeK_{1}(5) If x∈K

_{1}and K_{1}⊂K_{2}, then x∈K_{2}(6) If x, y∈K, and x=

_{i}y, and there is a K_{1}and K_{2}such that x∈K_{1},y∈K_{2}, K_{1}, K_{2}⊂K and K_{1}≠K_{2}(7) If K

_{1}≠K_{2}, then there is a property P such that it is not the case that PeK_{1}≡PeK_{2}(8) If K

_{1}⊂K_{2}, and K_{2}⊂K_{3}, then K_{1}⊂K_{3}(9) If x∈K

_{1}, K_{2}, and K_{1}≠K_{2}, then either K_{1}⊂K_{2}or K_{2}⊂K_{1}, or there is a K such that K_{1},K_{2}⊂K(10) For all x, (Nx∈K or Nx∉K)

(11) There are no two natural kinds, K

_{1}and K_{2}, such that necessarily for all x, x∈ K_{1}or x∈K_{2}(12) The class of things defined as the intersection of the extensions of two distinct natural kinds K

_{1}and K_{2}is not necessarily the extension of a natural kind, unless K_{1}⊂K_{2}or K_{2}⊂K_{1}Now, apart from the disadvantages of the attempt to determine a logic in a purely axiomatic manner, I'm worried especially about principle (6). Let's take a look at it again:

If x, y∈K, and x=

_{i}y, and there is a K_{1}and K_{2}such that x∈K_{1},y∈K_{2}, K_{1}, K_{2}⊂K and K_{1}≠K_{2}The first observation is that this is ambiguous between two readings:

Reading A. If x, y∈K, and x=

_{i}y, and there is a K_{1}and K_{2}such that x∈K_{1},y∈K_{2}, K_{1}, then K_{2}⊂K and K_{1}≠K_{2}Reading B. If x, y∈K, and x=

_{i}y, and there is a K_{1}and K_{2}such that x∈K_{1},y∈K_{2}, then K_{1}, K_{2}⊂K and K_{1}≠K_{2}But let's ingore this.

Second, both readings seem false. Just take y to be x, and take K

_{1}and K_{2}to be K. Clearly, x belongs to K and is intrinsically identical to itself. Yet, neither K is a subspecies of K, nor is it different from K. The same substitution falsifies reading B.So, perhaps, Ellis was assuming that different variables are referring to different objects/kinds (I'm trying to be charitable here, and try this reading, event though this reading is unlikely in the light of the fact that Ellis wants to have non-identity of kinds in the consequent)??

Well, again, this won't fly. For take x and y to be two distinct, and yet intrinsically identical objects belonging to one species K. Let K

_{1}be a superspecies of K, and let K_{2}be a superspecies of K_{1}. This interpretation falsifies both readings.Am I missing something? Is there another natural principle that Ellis might've had in mind, but failed to state??

Labels:
Ellis,
identity,
logic,
meaning,
natural kind reasoning,
natural kinds,
philosophy

## Monday, March 2, 2009

### Scott & Tarski consequence (basic info)

Scott consequence relations (a.k.a. multiple-conclusion consequence relations), as oppossed to Tarski consequence operations (that have single formulas as consequences) take the consequence relation to bear one set of formulas to another. What follows is a brief explanation of the relation between these two types of consequence relations.

Let A, B, C, ... stand for sentences, a, b, c, ... for finite sets of sentences, and u, v, w, ... for arbitrary sets of sentences. Relative to a language, the complement of a set u will be denoted by u'.

A Scott sequent is of the form:

A set of Scott sequents is a Scott consequence relation if it satisfies (for any A, a, b, c, d):

Condition (i) captures the idea that if something follows if all the premises in a set are true, then the truth of some other sentences doesn't have any impact on this fact. (Nota bene, there are certain contexts where we might not want monotonicity to hold, say, for conditionals. For instance, even if it is true that if it's sunny, we'll go for a walk today, it's probably false that if it's sunny and I get hit by a car, we'll go for a walk today. Let's not worry about these things). Condition (ii), on the other hand, expresses the requirement that if the truth of all sentenes in the premise set makes at least one sentence in the conclusion set true, then the truth of all sentences in the premise set makes at least one sentence in any superset of the conclusion set true.

The Cut condition expresses the assumption that if [i] the truth of all sentences in a true entails the truth of either A, or one of the sentences in b, and [ii] the truth of all the sentences in a together with the truth of A entails the truth of at least one sentence in b, then [iii] the truth of all the sentences in a entails the truth of at least one sentence in b (i.e. entails that at least one sentence in b is true; it doesn't have to entail any specific sentence in b). For indeed, assume the [i] and [ii], and suppose that all the sentences in a are true. By [i] either A is true, or at least one sentence in b is true. In the latter case, we're done. In the former case, all the sentences in a are true, and A is true. Then, by [ii], at least one sentence in b is true.

The consequence operation is extended to infinite sets of sentences by requiring:

Observe:

A few more words about the relation between Tarski and Scott consequence are due. Recall, that a Tarski consequence is characterized by a set of sequents a>A, where A is a sentence, satisfying reflexivity (A>A), monotonicity (if a>A and b is a superset of a, then b>A), and cut (if a>A and a, A>B, then a>B). The operation is extended to infinite sets of sentences by requiring compactness. A Tarski theory > generates a provability operator Cn: Cn(u)={A| u>A}. A Tarski theory is a set u such that u=Cn(u).

Now, even though intersections of Tarski theories (wrt. a Tarski consequence operation) are Tarski theories, this doesn't hold for Scott consequence operations. Not every intersection of a Scott theory is a Scott theory.

Off the top of my head, here's an example of two Scott theories whose intersection is not a Scott theory (although, I'm sloppy, so don't trust me and double-check if this is correct).

Second, {Q,R} is also a theory: the complement of this set is {P}, and Q&R does not (even on our assumption) entail P.

Third, their intersection is {R}. But this actually isn't a theory, because, by assumption, R>{P,Q}.

There's a restricted version of the claim, though.

Say a set of theories is downward directed if for any two theories in that set, there is a theory in that set which is included in both theories. We have the following:

Say a Scott consequence is singular iff for any a, b, if a>b, then a>B for some B in b.

Another fun fact. Every set of sets of sentences S determines a Scott consequence >

Finally, the representation theorem for Scott consquences says something like this: take a Scott consequence >. Take the set of all theories of >, call it Con. Take the consequence relation generated according to the above instructions from Con. It turns out, this consequence relation will be the same as the original >. This means, Scott consequence relations are uniquely determined by their theories.

A Scott sequent is of the form:

a⇒bwhich can be read: if all sentences in a hold (=are true/are accepted/whatnot), so does at least one sentence from b [in short: a secures b]. For typographical reasons, this will also sometimes be written as "a>b".

A set of Scott sequents is a Scott consequence relation if it satisfies (for any A, a, b, c, d):

- reflexivity: A⇒ A
- Monotonicity: If a⇒b, a⊆c, b⊆d, then c⊆d.
- Cut: If a⇒b, A and a,A⇒b, then a⇒b.

Condition (i) captures the idea that if something follows if all the premises in a set are true, then the truth of some other sentences doesn't have any impact on this fact. (Nota bene, there are certain contexts where we might not want monotonicity to hold, say, for conditionals. For instance, even if it is true that if it's sunny, we'll go for a walk today, it's probably false that if it's sunny and I get hit by a car, we'll go for a walk today. Let's not worry about these things). Condition (ii), on the other hand, expresses the requirement that if the truth of all sentenes in the premise set makes at least one sentence in the conclusion set true, then the truth of all sentences in the premise set makes at least one sentence in any superset of the conclusion set true.

The Cut condition expresses the assumption that if [i] the truth of all sentences in a true entails the truth of either A, or one of the sentences in b, and [ii] the truth of all the sentences in a together with the truth of A entails the truth of at least one sentence in b, then [iii] the truth of all the sentences in a entails the truth of at least one sentence in b (i.e. entails that at least one sentence in b is true; it doesn't have to entail any specific sentence in b). For indeed, assume the [i] and [ii], and suppose that all the sentences in a are true. By [i] either A is true, or at least one sentence in b is true. In the latter case, we're done. In the former case, all the sentences in a are true, and A is true. Then, by [ii], at least one sentence in b is true.

The consequence operation is extended to infinite sets of sentences by requiring:

Compactness u⇒v iff a⇒b for some finite a⊆u, b⊆v.The first neat thing to observe about Scott consequence operation in comparison to Tarski consequence operation is that dual notions can be easily defined. For instance, for any Scott consequence, a dual consquence operation can be defined:

a⇒It turns out, this dual operation is also a Scott consequence (convince yourself!). In general, amy notion definable for Scott consequence operations has a dual notion._{d}b iff b⇒a

Observe:

Fact 1. Any set of Scott sequents is contained in a least Scott consequence relation.A set of sentences u is called a Scott theory of a consequence relation iff it is not the case that:

u⇒u'The basic idea is that a theory is a set of sentences closed under consequence relation: that is, a set that doesn't make any sentence that doesn't already belong to it true. This is captured by the following fact:

Fact 2. A set of sentences u is a theory iff for any a, b:The basic reason why this holds is that if b has no common elements with u, this means that b is a subset of u', the complement of u. This means that if a secures b, then (by monotonicity) a also secures u'. But we also assume that a is a subset of u. Thus, by applying monotonicity again, we reach the conclusion that u secures u', which means that u is not a theory, which contradicts the assumption.

if a⇒b and a⊆u, then u∩b is non-empty.

A few more words about the relation between Tarski and Scott consequence are due. Recall, that a Tarski consequence is characterized by a set of sequents a>A, where A is a sentence, satisfying reflexivity (A>A), monotonicity (if a>A and b is a superset of a, then b>A), and cut (if a>A and a, A>B, then a>B). The operation is extended to infinite sets of sentences by requiring compactness. A Tarski theory > generates a provability operator Cn: Cn(u)={A| u>A}. A Tarski theory is a set u such that u=Cn(u).

Now, even though intersections of Tarski theories (wrt. a Tarski consequence operation) are Tarski theories, this doesn't hold for Scott consequence operations. Not every intersection of a Scott theory is a Scott theory.

Off the top of my head, here's an example of two Scott theories whose intersection is not a Scott theory (although, I'm sloppy, so don't trust me and double-check if this is correct).

- Language L: say we have only three sentences P, Q, R, no connectives.
- Consequence operation: Start with extending the language with conjunction, disjunction, and material implication thus obtaining language L'. Then, for any two sets of sentences of L, a and b, a>b iff in L', the conjunction of the members of a tautologically entails (we're using classical propositional logic) the disjunction of the elements of b, on the assumption that R -> (P or Q). [convince yourself this actually is a Scott consequence operation]
- Theories: {P, R} and {Q, R}.

Second, {Q,R} is also a theory: the complement of this set is {P}, and Q&R does not (even on our assumption) entail P.

Third, their intersection is {R}. But this actually isn't a theory, because, by assumption, R>{P,Q}.

There's a restricted version of the claim, though.

Say a set of theories is downward directed if for any two theories in that set, there is a theory in that set which is included in both theories. We have the following:

Fact 3. The intersection of a downward directed set of Scott theories is always a Scott theory.Also, one can prove the existence of inclusion minimal theories containing some set of sentences and inclusion maximal theories disjoint from a set of sentences. Also:

Fact 4. For any Scott consequence >, the restriction of > to sequents whose second members are singletons is a Tarski relation. It's called a Tarski subrelation of >. For any >, the set of theories of the Tarski subrelation of > is the intersection of all theories of > plus the trivial theory.So, why exactly does a difference shows up when we talk about intersections of Scott theories not being in general intersections? Well maybe this helps to understand why this happens:

Say a Scott consequence is singular iff for any a, b, if a>b, then a>B for some B in b.

Fact 5. Any intersection of theories of a singular Scott consequence is also a theory of that singular Scott consequence relation.So, the difference arises basically because a finite set of sentences may be Sott-secured by another set, without any particular member of this set being secured.

Another fun fact. Every set of sets of sentences S determines a Scott consequence >

_{S}, defined by:a⇒_{S}b iff for any u∈S, if a⊆u, then u∩b is non-empty.

Fact 6. Any set in S is a theory of the consequence operation generated by S.

Finally, the representation theorem for Scott consquences says something like this: take a Scott consequence >. Take the set of all theories of >, call it Con. Take the consequence relation generated according to the above instructions from Con. It turns out, this consequence relation will be the same as the original >. This means, Scott consequence relations are uniquely determined by their theories.

Labels:
consequence,
logic,
Scott,
set theory,
Tarski

Subscribe to:
Posts (Atom)