Meaning and Partiality Revised

Muskens presents in Meaning and Partiality a semantics of possibly contradictory beliefs and other propositional attitudes. We propose a different partial logic based on a few key equalities for the connectives and with four values (truth, falsehood, and undefinedness with negative and positive polarity; only the first truth value is designated). Aim and Scope In logical semantics the grammar and meaning of natural language sentences are defined and the logical consequences of the sentences are tested against our intuition [5,8]. A set of sentences provides a model of the world as observed by a person or, more generally, an agent, and the agent is part of the world as are other agents. In such cases it is important to be able to reason about the knowledge, beliefs, assertions and other propositional attitudes of agents. For instance, (1) is a consequence of (2), but (3) is not a consequence of (1) or (2): Mary believes that John cheats. (1) Mary believes that John cheats and smiles. (2) John cheats. (3) One approach is to store the beliefs syntactically for each agent and use the axioms and rules of a given logic, but it is not clear how to cope with sentences expressing quantification over agents, quantification into embedded sentences, etc. Another way is to model beliefs as a set of so-called possible worlds, not necessarily including the actual world. Since tautologies in classical logic are true in all possible worlds, they are always believed, and (4) holds: Mary believes that John cheats or does not cheat. (4) This might be acceptable for beliefs but hardly so for knowledge and surely not for assertions. Even worse: since contradictions in classical logic are not true in any possible worlds there is an “explosion” of consequences, hence (1) and (2) are consequences of (5): Mary believes that John smiles and does not smile. (5) This is problematic since we can usually not expect beliefs and assertions to be free from contradictions (even if contradictions can be spotted, and excluded from knowledge, they must be modelled when they occur as beliefs and assertions). To sum up: we do not even want (4) to be a consequence of (5). Muskens presents a solution [7] by replacing classical logic with a partial logic, that is, a logic with “undefined” truth values [4]. We think that although the semantics developed by Muskens makes the correct predictions for sentences like (1) – (5), it does it for the wrong reasons and for more complicated fragments of natural language it might not work either, since it is based on a special implication connective such that we do not have φ ; φ for all formulas (Muskens does not define the symbol ; explicitly). We propose a different partial logic such that φ⇒ φ and based on a few key equalities. 164 J. Villadsen / Meaning and Partiality Revised A Partial Logic Classical logic has two truth values, namely T and F (truth and falsehood). The designated truth value T with symbol> yields the logical truths. We propose to base the semantic clauses for the connectives on a few key equalities (listed with symbol . = below): [[¬φ]] =  T if [[φ]] = F > . = ¬⊥ The semantic clauses and the common F if [[φ]] = T ⊥ . = ¬> definition > ≡ ¬⊥ work for classical [[φ]] otherwise logic as well as for our partial logic. [[φ∧ψ]] =  [[φ]] if [[φ]] = [[ψ]] φ . = φ∧φ φ∨ψ ≡ ¬(¬φ∧¬ψ) [[ψ]] if [[φ]] = T ψ . = >∧ψ [[φ]] if [[ψ]] = T φ . = φ∧> φ→ ψ ≡ ¬φ∨ψ F otherwise φ↔ ψ ≡ (φ→ ψ)∧ (ψ→ φ) [[φ⇔ ψ]] =  T if [[φ]] = [[ψ]] > . = φ⇔ φ In the semantic clauses [[ψ]] if [[φ]] = T ψ . = >⇔ ψ several cases may apply [[φ]] if [[ψ]] = T φ . = φ⇔> if and only if they agree [[¬ψ]] if [[φ]] = F ¬ψ . = ⊥⇔ ψ on the truth value result. [[¬φ]] if [[ψ]] = F ¬φ . = φ⇔⊥ F otherwise φ⇒ ψ ≡ φ⇔ (φ∧ψ) We now consider additional truth values. First we add just [[†]] = N for the “undefined” truth value. We do not have φ∨¬φ, hence not φ→ φ, but φ⇒ φ of course. Unfortunately we also have (φ∧¬φ)⇒ (ψ∨¬ψ) (try with the truth values T, F and N). The reason is that in a sense there is not only a single “undefined” truth value, but a unique one for each basic formula. However, only two “undefined” truth values are ever needed, corresponding to the left and right hand side of the implication (in other words, the negative and positive occurrences suggest undefinedness with negative and positive polarity). Hence we add [[‡]] = P for the alternative “undefined” truth value. We then define ⊥ ≡ †∧‡ as seen from the truth tables: