In other words, these axioms correspond to the reflexivity and transitivity of time order. Translate each of these to statements about times, and you get: Here are some axioms we might add to describe the eventually modality above: If we imagine the context to have an associated time, eventually moves us from a context with time t to a context with some greater time s. In the example above, we introduced eventually to keep track of times and their ordering. In each of these cases, a modal operator is introduced, and then certain axioms are assumed that govern that operator's behavior. Modal operators have also been used to express the idea that a proposition is true after a certain program is run that its truth is known by some agent that the truth is morally required that a truth is provable in a certain system or that a value of a certain type is constructible. Above, halted is true in a certain way meaning at some later time. These operators express the way in which a proposition is true.
WAYS TO USE THE MODAL LOGIC PLAYGROUND PLUS
That is the promise of modal logic.Ī modal logic is ordinary logic plus at least one modal operator: an operator that takes a proposition and returns a proposition. Using such a specification is easier checking a proof is simpler synthesizing programs to meet this requirement might be doable with complete automation. This axiom removes the need for times, ordering, arithmetic, or even first-order reasoning. Call the operator eventually, expressing the notion that at a future time the proposition will be true.
Instead of introducing clocks and comparisons, we introduce a modal operator: a unary operator on propositions. There is another way to reason about time that hides all of these details. All this means that tasks like theorem checking, proof search, or program generation are harder and require more computational resources.
Our specification of the computer now relies on reasoning about order, a tricky thing proofs require checking arithmetic. This new way of rendering the button-halt property is at least accurate, but it requires our proofs to do meticulous cycle-counting. We can imagine proving this constructively by arguing that, well, suppose the button is pushed, then in one cycle we'll be in the interrupt handler, and in ten cycles we'll halt the machine, so let s = t + 11, and then we are done. Time might be identified with clock cycles then this rule is believable. ∀ t, ∃ s, pressed(button, t) ∧ s ≥ t ∧ halted(s) We might extend the pressed and halted predicates to take a time argument, and render the rule like this: To better handle this gap between the button being pressed and the computer halting, we figure we need to add a notion of time. There is a moment when the button is pressed but the computer has not yet halted. However, after some contemplation this is false. Our first attempt might be something like For example, we might want to make statements like, “If the button is pressed, the computer will halt.” However, it's tricky to see how we can turn this into a rigorous logical statement. The book can be used as the primary text for seminars on philosophical logic focused on non-normal modal logics as a supplemental text for courses on modal logic, logic in AI, or philosophical logic or as the primary source for researchers interested in learning about the uses of neighborhood semantics in philosophical logic and game theory.Suppose we want to reason about the execution of a program over time. In addition, the book discusses a broad range of topics, including standard modal logic results bisimulations for neighborhood models and other model-theoretic constructions comparisons with other semantics for modal logic neighborhood semantics for first-order modal logic, applications in game theory applications in epistemic logic and non-normal modal logics with dynamic modalities. In addition to presenting the relevant technical background, it highlights both the pitfalls and potential uses of neighborhood models – an interesting class of mathematical structures that were originally introduced to provide a semantics for weak systems of modal logic. This book offers a state-of-the-art introduction to the basic techniques and results of neighborhood semantics for modal logic.