Interpolation in non-classical logics

We discuss the interpolation property on some important families of non classical logics, such as intuitionistic, modal, fuzzy, and linear logics. A special paragraph is devoted to a generalization of the interpolation property, uniform interpolation. Supported by PRIN project 2006/2007 ‘Large-scale development of certified mathematical proofs’.     

Interpolation and definability in abstract logics

A semantical definition of abstract logics is given. It is shown that the Craig interpolation property implies the Beth definability property, and that the Souslin-Kleene interpolation property implies the weak Beth definability property. An example is given, showing that Beth does not imply Souslin-Kleene.     

Interpolation properties of superintuitionistic logics

A family of prepositional logics is considered to be intermediate between the intuitionistic and classical ones. The generalized interpolation property is defined and proved is the following. Theorem on interpolation. For every intermediate logic L the following statements are equivalent:

1. Craig's interpolation theorem holds in L,
2. L possesses the generalized interpolation property,
3. Robinson's consistency statement is true in L.

There are just 7 intermediate logics in which Craig's theorem holds. Besides, Craig's interpolation theorem holds in L iff all the modal companions of L possess Craig's interpolation property restricted to those formulas in which every variable is proceeded by necessity symbol.     

Sentential Connectives and Translation

In the first exposition of the doctrine of indeterminacy of translation, Quine asserted that the individuation and translation of truth-functional sentential connectives like ‘and’, ‘or’, ‘not’ are not indeterminate. He changed his mind later on, conjecturing that some sentential connectives might be interpreted in different non-equivalent ways. This issue has not been debated much by Quine, or in the subsequent literature, it is, as it were, an unsolved problem, not well understood. For the sake of the argument, I will adopt Quine’s background assumption that all the semantic features of a language can be reduced to the speakers’ dispositions toward assent and dissent, as far as only the truth-conditional core of the meaning of sentences is concerned. I will put forward an argument to the effect that the speech dispositions of most, if not all, English (French, Italian, etc.) speakers constrain a unique translation of their connectives. This argument crucially relies on an empirical conjecture concerning the behaviour of these operators.     

Sentential synonymity judgment

Despite the acknowledged theoretical importance of sentential synonymity, there has been little interest in the psychological mechanisms responsible for recognizing it. Here an attempt is made to explore ways in which specific models originally devised for other particular psycholinguistic tasks (e.g., sentence-picture verification, three-term series problems, syllogistic reasoning) can be generalized to synonymity judgment and ways in which recent psychosemantic theories can be made more specific to deal with this same paradigm. Differential predictions from these models are derived for a set of spatiotemporal bidimensional comparative synonym pairs (e.g., High before Low, First above Second), and an experiment to test these predictions is reported. It is argued that, although the results are consonant with predictions from one type of model, they do not favor it definitively. This is because models of this type do not detail how bidimensional comparatives can be parsed into the same propositional representation and because other models can be subjected to further post hoc modifications to account for the data. Finally, general problems arising from attempts to adapt existing models to new tasks are discussed.     

Justification logics and hybrid logics

Hybrid logics internalize their own semantics. Members of the newer family of justification logics internalize their own proof methodology. It is an appealing goal to combine these two ideas into a single system, and in this paper we make a start. We present a hybrid/justification version of the modal logic T. We give a semantics, a proof theory, and prove a completeness theorem. In addition, we prove a Realization Theorem, something that plays a central role for justification logics generally. Since justification logics are newer and less well known than hybrid logics, we sketch their background, and give pointers to their range of applicability. We conclude with suggestions for future research. Indeed, the main goal of this paper is to encourage others to continue the investigation begun here.     

A Note on the Interpolation Property in Tense Logic

It is proved that all bimodal tense logics which contain the logic of the weak orderings and have unbounded depth do not have the interpolation property.     

Sentential context and the interpretation of unambiguous words

Two experiments were carried out in order to investigate how the linguistic context (in the form of a sentence) facilitates the interpretation of unambiguous words. Experiment I established that if a sentential context as a whole is sufficient to evoke an inference that calls to mind a particular aspect of a word's meaning, the presence of a verb with appropriate selectional restrictions does not enhance the process. Experiment II showed that when no other cues are provided within a sentence, both verbs and adjectives are effective in enhancing a specific aspect of the meaning of a word. These findings were taken to support the hypothesis that in understanding a sentence, people instantiate particular aspects of the meanings of words in order to construct specific interpretations, and linguistic context guides this process of selection.     

Programs and logics

We use the algebraic theory of programs as in Blikle [2], Mazurkiewicz [5] in order to show that the difference between programs with and without recursion is of the same kind as that between cut free Gentzen type formalizations of predicate and prepositional logics.     

Truth,Sentential Non-Compositionalit,and Ontology

The paper attempts to clarify some fundamental aspects of an explanationof the concept of truth which is neither deflationary nor substantive.The main aspect examined in detail concerns the ontological dimension of truth, the mind/language-world connection traditionally associated with the concept of truth. It is claimed that it does not make sense to defend or reject a relatedness of truth to the ontological dimension so long as the kind of presupposed or envisaged ontology is not made explicit and critically examined. In particular, it is shown that generally an objectual ontology is – often only implicitly – presupposed, i.e., an ontology admitting objects (substances), properties, relations, sometimes also facts, events, and the like. The paper demonstrates that such an ontology derives from the Principle of Semantic Sentential Compositionality and that this principle should be rejected. It introduces instead the Principle of Semantic Sentential Contextuality (or Context Principle) as the semantic basis of a new ontology, an ontology of primary states of affairs. After sketching such an ontology, it is shown that the relatedness of truth to the ontological dimension becomes intelligible.     

Protoalgebraic logics

There exist important deductive systems, such as the non-normal modal logics, that are not proper subjects of classical algebraic logic in the sense that their metatheory cannot be reduced to the equational metatheory of any particular class of algebras. Nevertheless, most of these systems are amenable to the methods of universal algebra when applied to the matrix models of the system. In the present paper we consider a wide class of deductive systems of this kind called protoalgebraic logics. These include almost all (non-pathological) systems of prepositional logic that have occurred in the literature. The relationship between the metatheory of a protoalgebraic logic and its matrix models is studied. The following results are obtained for any finite matrix model U of a filter-distributive protoalgebraic logic : (I) The extension _U of is finitely axiomatized (provided has only finitely many inference rules); (II) _U has only finitely many extensions.     

Embodied Simulations Are Modulated by Sentential Perspective

There is considerable evidence that language comprehenders derive lexical‐semantic meaning by mentally simulating perceptual and motor attributes of described events. However, the nature of these simulations—including the level of detail that is incorporated and contexts under which simulations occur—is not well understood. Here, we examine the effects of first‐ versus third‐person perspective on mental simulations during sentence comprehension. First‐person sentences describing physical transfer towards or away from the body (e.g., “You threw the microphone,” “You caught the microphone”) modulated response latencies when responses were made along a front‐back axis, consistent with the action‐sentence compatibility effect (ACE). This effect was not observed for third‐person sentences (“He threw the microphone,” “He caught the microphone”). The ACE was observed when making responses along a left‐right axis for third‐person, but not first‐person sentences. Abstract sentences (e.g., “He heard the message”) did not show an ACE along either axis. These results show that perspective is a detail that is simulated during action sentence comprehension, and that motoric activations are flexible and affected by the pronominal perspective used in the sentence.     

Quantum-like logics and schizophrenia

Many researchers in different disciplines have independently concluded that brains are, possibly among other things, vector processing devices. In this paper we offer support for this hypothesis coming from a new perspective. Namely, we test it against some known anomalies in the processing by schizophrenic patients of certain logical tasks: they perform better at them than normal controls, despite the observation that they do not generally employ “normal” or “commonsense” logic. On the assumption that they are compelled to use the intrinsic logic of the brain instead of commonsense logic, and that this logic is linear or quantum-like, we are able to resolve these and other anomalies. Our conclusions support the idea that human brains (at least) perform intrinsic logical operations according to the dictates of a linear (or Grassmannian, or quantum-like) logic rather than “classical” or Aristotelian logic (which seems not to be intrinsic to brains, these having evolved under the pressure of different constraints). If this is the case, then commonsense logic must be acquired through experience and the construction of contexts, an ability schizophrenic patients seem to lack, and who are consequently compelled to rely on the intrinsic logic, which is quantum-like and more efficient at certain tasks. Moreover, the proclivity toward errors of von Domarus type (namely the inference that shared attributes imply identity), which seems to be endemic to human thinking and has been discussed in connection with schizophrenia, is also explained on this basis.     

Sentential constants in systems near R

An Ackermann constant is a formula of sentential logic built up from the sentential constant t by closing under connectives. It is known that there are only finitely many non-equivalent Ackermann constants in the relevant logic R. In this paper it is shown that the most natural systems close to R but weaker than it-in particular the non-distributive system LR and the modalised system NR-allow infinitely many Ackermann constants to be distinguished. The argument in each case proceeds by construction of an algebraic model, infinite in the case of LR and of arbitrary finite size in the case of NR. The search for these models was aided by the computer program MaGIC (Matrix Generator for Implication Connectives) developed by the author at the Australian National University.     

Equivalential and algebraizable logics

The notion of an algebraizable logic in the sense of Blok and Pigozzi [3] is generalized to that of a possibly infinitely algebraizable, for short, p.i.-algebraizable logic by admitting infinite sets of equivalence formulas and defining equations. An example of the new class is given. Many ideas of this paper have been present in [3] and [4]. By a consequent matrix semantics approach the theory of algebraizable and p.i.-algebraizable logics is developed in a different way. It is related to the theory of equivalential logics in the sense of Prucnal and Wroski [18], and it is extended to nonfinitary logics. The main result states that a logic is algebraizable (p.i.-algebraizable) iff it is finitely equivalential (equivalential) and the truth predicate in the reduced matrix models is equationally definable.Most of the results of the present and a forthcoming paper originally appeared in [13].Presented by Wolfgang Rautenberg