The psychology of inferring conditionals from disjunctions: A probabilistic study

There is a new probabilistic paradigm in the psychology of reasoning that is, in part, based on results showing that people judge the probability of the natural language conditional, if $A$then $B$, $P\left(ifAthenB\right)$, to be the conditional probability, $P(B\mid A)$. We apply this new approach to the study of a very common inference form in ordinary reasoning: inferring the conditional if not-$A$then $B$ from the disjunction $A$ or $B$. We show how this inference can be strong, with $P$(if not-$A$then $B$) “close to” $P\left(AorB\right)$, when $A$ or $B$ is non-constructively justified. When $A$ or $B$ is constructively justified, the inference can be very weak. We also define suitable measures of “closeness” and “constructivity”, by providing a probabilistic analysis of these notions.     

Logic programming as classical inference

We propose a denotational semantics for logic programming based on a classical notion of logical consequence which is apt to capture the main proposed semantics of logic programs. In other words, we show that any of those semantics can be viewed as a relation of the form $T{\models }^{\star }X$ where T is a theory which naturally represents the logic program under consideration together with a set of formulas playing the role of “hypotheses”, in a way which is dictated by that semantics, ${\models }^{\star }$ is a notion of logical consequence which is classical because negation, disjunction and existential quantification receive their classical meaning, and X represents what can be inferred from the logic program, or an intended interpretation of that logic program (such as an answer-set, its well-founded model, etc.). The logical setting we propose extends the language of classical modal logic as it deals with modal operators indexed by ordinals. We make use of two kinds of basic modal formulas: ${\square }_{\alpha }\phi$ which intuitively means that the logical program can generate φ by stage α of the generation process, and ${◇}_{\alpha }{\square }_{\beta }\phi$ with $\alpha >\beta$, which intuitively means that φ can be used as a hypothesis from stage β of the generation process onwards, possibly expecting to confirm φ by stage α (so expecting ${\square }_{\alpha }\phi$ to be generated). This allows us to capture Rondogiannis and Wadge's version of the well-founded semantics [27] where a member of the well-founded model is a closed atom which receives an ordinal truth value of ${\mathtt{true}}_{\alpha }$ or ${\mathtt{false}}_{\alpha }$ for some ordinal α: in our framework, this corresponds to having $T{\models }^{\star }{\square }_{\alpha }\phi$ or $T{\models }^{\star }{\square }_{\alpha }\neg \phi$, respectively, with T being the natural representation of the logic program under consideration and the right set of “hypotheses” as dictated by the well-founded semantics. The framework we present goes much beyond the proposed traditional semantics for logic programming, as it can for instance let us investigate under which conditions a set of hypotheses can be minimal, with each hypothesis being activated as late as possible and confirmed as soon as possible, setting the theoretical foundation to sophisticated ways of making local use of hypotheses in knowledge-based systems, while still being theoretically grounded in a classical notion of logical consequence.     

Bridging Curry and Church's typing style

There are two versions of type assignment in the λ-calculus: Church-style, in which the type of each variable is fixed, and Curry-style (also called “domain free”), in which it is not. As an example, in Church-style typing, ${\lambda }_{x:A}.x$ is the identity function on type A, and it has type $A\to A$ but not $B\to B$ for a type B different from A. In Curry-style typing, ${\lambda }_x.x$ is a general identity function with type $C\to C$ for every type C. In this paper, we will show how to interpret in a Curry-style system every Pure Type System (PTS) in the Church-style without losing any typing information. We will also prove a kind of conservative extension result for this interpretation, a result which implies that for most consistent PTSs of the Church-style, the corresponding Curry-style system is consistent. We will then show how to interpret in a system of the Church-style (a modified PTS, stronger than a PTS) every PTS-like system in the Curry style.     

Reasoning about negligibility and proximity in the set of all hyperreals

We consider the binary relations of negligibility, comparability and proximity in the set of all hyperreals. Associating with negligibility, comparability and proximity the binary predicates N, C and P and the connectives $[N]$, $[C]$ and $[P]$, we consider a first-order theory based on these predicates and a modal logic based on these connectives. We investigate the axiomatization/completeness and the decidability/complexity of this first-order theory and this modal logic.     

Necessitarian propositions

The eternalist holds that all propositions specify the needed time information, and so are eternally true if true at all. The necessitarian holds the parallel view for worlds: she holds that all propositions specify the needed world information, and so are necessarily true if true at all. I will argue that the considerations for both views run parallel: the necessitarian can mimic the whole case for eternalism.     

Knowledge as ‘True Belief Plus Individuation’ in Plato

In Republic V, Plato distinguishes two different cognitive powers, knowledge and belief, which operate differently on different types of object. I argue that in Republic VI Plato modifies this account, and claims that there is a single cognitive power, which under different circumstances behaves either as knowledge or as belief. I show that the circumstances which turn true belief into knowledge are the provision of an individuation account of the object of belief, which reveals the ontological status and the nature of the object. Plato explores many alternative candidates of individuation accounts of objects of true belief, which he discards. I conclude with a Platonic sketch of a teleological account of individuation which would satisfy his requirements of turning true belief into knowledge.