## Final Report Summary - USHP (UNITY OF STRUCTURED PROPOSITIONS)

This project was carried out at LOGOS, University of Barcelona. The topic was the unity of structured hyperpropositions. The LOGOS counterpart (Scientist in Charge) was Professor Manuel García-Carpintero.

The objectives were answering these four questions:

• What is propositional structure? (1)

• How are propositions structured? (2)

• How are propositions composed from their parts? (3)

• What would be an outline of a mereology of structured hyperpropositions? (4)

The theoretical point of departure was that propositions are identified with abstract logical procedures, or logical flow-charts, of which there are six different ones according to the inductive definition of logical structures (so-called constructions) of my background theory, Transparent Intensional Logic. Therefore, structure is procedural structure.

The answer to (1) is that a proposition is a procedure whose product (depending on whether the proposition is empirical or mathematical/logical) is either a truth-condition or a truth-value.

The answer to (2) is that atomic propositions are structured thanks to the logical operation of predication, which I construe as an instance of the operation of functional application. Predication is not a stand-alone part of any proposition. Rather it is the formal structure of the proposition. (Thus the unifier is on the side of form and not of content.) Compound propositions are structured thanks to predication within each constituent proposition together with whatever logical connectives combine any two atomic propositions. Account for molecular propositions is straightforward, once the atomic ones have been sorted out. The structure of atomic empirical propositions is more complicated than that of atomic non-empirical ones, because the former must include also empirical parameters such as possible worlds (for modal variability) and instants of time (for temporal variability).

The answer to (3) is that the parts of propositions are themselves (sub-) procedures that produce either functions (mappings) or functional arguments which, upon application (predication) yield values, which are themselves either functions or functional arguments. Thus propositions are composed from their parts in virtue of interlocking procedures.

The details of the answer to (4) are still being investigated, but the outline is clear enough. Some of the axioms defining classical mereology are unsuitable for a mereology of structured hyperpropositions. Above all, classical mereology has no room for the interaction among the constituent parts of a whole, whereas interaction is key to being structured. At least two axioms that must go are idempotency and extensionality, which is hardly surprising. What is perhaps surprising, and in my view good news, is that the emerging mereology is well-founded, thus anchoring it to much research into mereology.

The work performed toward meeting the objectives has included reading up on the relevant literature, conceiving and discussing my new theory with peers and presenting it in speech and in print. In concrete terms, the workload amounts to the following:

• 4 tutorials to advanced students;

• co-organizing a two-day conference on the topic of the project;

• giving 11 invited lectures as a Marie Curie Ambassador on three continents (Europe, North America, Asia);

• giving 4 invited talks;

• giving 2 contributed talks;

• publishing 7 papers;

• submitting a paper (containing the core of the new theory);

• writing up 5 papers for submission;

• preparing a monograph on the topic of the project.

The main results are so far the following. (1) I have identified two major pitfalls that any viable theory of structure must avoid. One is the list pitfall, which is to merely arrange target parts (whether operations or operands) in a sequence without accounting for their interaction. A list offer too little. The other is the regress pitfall, which arises whenever one of the target parts is the designated unifier. Regress arises because that part must hook up with other parts via intermediaries, which in turn require intermediaries, and so on. A regress offers too much, as it proliferates intermediary entities. I have shown in one paper (under review) how to solve the list problem in a principled way. I am currently developing, in another paper (abstract submitted to conference), a solution to the regress pitfall. It is absolutely critical for any theory of structured entities to offer principled solutions to both problems, because otherwise the unity problem will not be addressed satisfactorily, as I have argued at length in (under review). (2) My methodology has consisted in stressing the procedural character of the constructions of TIL, which it shares with the programming languages likewise based on the (typed) lambda-calculus, and bringing the resulting formal semantics to bear on the unity problem. I have shown, first of all, that TIL affords a solution to the two fundamental problems (list and regress). I have also argued, secondly, why my approach is conceptually superior to the prevalent contemporary theory, namely, the act-theoretic conception of propositions as speech acts of predication. My own objection is that the key notion of predication remains underdeveloped, lacking a logical underpinning and being inherently problematic, whether acts of predication come with assertoric force or not. One comparison with act theories has already appeared in print, the other (a co-authored paper with García-Carpintero) is being written up. (3) I have shown how to manipulate parts of propositions, as when quantifying into structured propositions. (4) TIL comes with two atomic procedures that provide input as operands. I have improved the philosophical understanding of what both kinds of procedures amount to. This is central when distinguishing between whether a procedure occurs executed (thus providing the entity it is designed to produce) or displayed (as a procedure in its own right in disregard of its product), because we want to be able to predicate properties of procedures themselves as well. The potential impact and use is this. The research provides the procedural conception of structured propositions with firmer foundations, which shows it to be on the right track. We are entitled to some confidence that the theory is reliable. Therefore, there are at least two audiences – computer scientists-cum-formal semanticists and philosophers of language-cum-philosophical logicians – who can take the theory and start applying it to a string of cases that demand hyperintensional treatment. This includes not least such attitudes as are controlled by very restrictive closure principles, whether the attitudes in question are entertained by solitary agents or are held by groups or exchanged within a multi-agent system.

The objectives were answering these four questions:

• What is propositional structure? (1)

• How are propositions structured? (2)

• How are propositions composed from their parts? (3)

• What would be an outline of a mereology of structured hyperpropositions? (4)

The theoretical point of departure was that propositions are identified with abstract logical procedures, or logical flow-charts, of which there are six different ones according to the inductive definition of logical structures (so-called constructions) of my background theory, Transparent Intensional Logic. Therefore, structure is procedural structure.

The answer to (1) is that a proposition is a procedure whose product (depending on whether the proposition is empirical or mathematical/logical) is either a truth-condition or a truth-value.

The answer to (2) is that atomic propositions are structured thanks to the logical operation of predication, which I construe as an instance of the operation of functional application. Predication is not a stand-alone part of any proposition. Rather it is the formal structure of the proposition. (Thus the unifier is on the side of form and not of content.) Compound propositions are structured thanks to predication within each constituent proposition together with whatever logical connectives combine any two atomic propositions. Account for molecular propositions is straightforward, once the atomic ones have been sorted out. The structure of atomic empirical propositions is more complicated than that of atomic non-empirical ones, because the former must include also empirical parameters such as possible worlds (for modal variability) and instants of time (for temporal variability).

The answer to (3) is that the parts of propositions are themselves (sub-) procedures that produce either functions (mappings) or functional arguments which, upon application (predication) yield values, which are themselves either functions or functional arguments. Thus propositions are composed from their parts in virtue of interlocking procedures.

The details of the answer to (4) are still being investigated, but the outline is clear enough. Some of the axioms defining classical mereology are unsuitable for a mereology of structured hyperpropositions. Above all, classical mereology has no room for the interaction among the constituent parts of a whole, whereas interaction is key to being structured. At least two axioms that must go are idempotency and extensionality, which is hardly surprising. What is perhaps surprising, and in my view good news, is that the emerging mereology is well-founded, thus anchoring it to much research into mereology.

The work performed toward meeting the objectives has included reading up on the relevant literature, conceiving and discussing my new theory with peers and presenting it in speech and in print. In concrete terms, the workload amounts to the following:

• 4 tutorials to advanced students;

• co-organizing a two-day conference on the topic of the project;

• giving 11 invited lectures as a Marie Curie Ambassador on three continents (Europe, North America, Asia);

• giving 4 invited talks;

• giving 2 contributed talks;

• publishing 7 papers;

• submitting a paper (containing the core of the new theory);

• writing up 5 papers for submission;

• preparing a monograph on the topic of the project.

The main results are so far the following. (1) I have identified two major pitfalls that any viable theory of structure must avoid. One is the list pitfall, which is to merely arrange target parts (whether operations or operands) in a sequence without accounting for their interaction. A list offer too little. The other is the regress pitfall, which arises whenever one of the target parts is the designated unifier. Regress arises because that part must hook up with other parts via intermediaries, which in turn require intermediaries, and so on. A regress offers too much, as it proliferates intermediary entities. I have shown in one paper (under review) how to solve the list problem in a principled way. I am currently developing, in another paper (abstract submitted to conference), a solution to the regress pitfall. It is absolutely critical for any theory of structured entities to offer principled solutions to both problems, because otherwise the unity problem will not be addressed satisfactorily, as I have argued at length in (under review). (2) My methodology has consisted in stressing the procedural character of the constructions of TIL, which it shares with the programming languages likewise based on the (typed) lambda-calculus, and bringing the resulting formal semantics to bear on the unity problem. I have shown, first of all, that TIL affords a solution to the two fundamental problems (list and regress). I have also argued, secondly, why my approach is conceptually superior to the prevalent contemporary theory, namely, the act-theoretic conception of propositions as speech acts of predication. My own objection is that the key notion of predication remains underdeveloped, lacking a logical underpinning and being inherently problematic, whether acts of predication come with assertoric force or not. One comparison with act theories has already appeared in print, the other (a co-authored paper with García-Carpintero) is being written up. (3) I have shown how to manipulate parts of propositions, as when quantifying into structured propositions. (4) TIL comes with two atomic procedures that provide input as operands. I have improved the philosophical understanding of what both kinds of procedures amount to. This is central when distinguishing between whether a procedure occurs executed (thus providing the entity it is designed to produce) or displayed (as a procedure in its own right in disregard of its product), because we want to be able to predicate properties of procedures themselves as well. The potential impact and use is this. The research provides the procedural conception of structured propositions with firmer foundations, which shows it to be on the right track. We are entitled to some confidence that the theory is reliable. Therefore, there are at least two audiences – computer scientists-cum-formal semanticists and philosophers of language-cum-philosophical logicians – who can take the theory and start applying it to a string of cases that demand hyperintensional treatment. This includes not least such attitudes as are controlled by very restrictive closure principles, whether the attitudes in question are entertained by solitary agents or are held by groups or exchanged within a multi-agent system.