$ways to use the modal logic playground$

Ways to use the modal logic playground how to#

(For more details, I would recommend looking at the very nice exposition of this in the relevant chapter of Fitting and Mendelsohn's First-Order Modal Logic textbook.) In other words, the addition of the machinery isn't so much just to create new logical systems, but to give us the needed expressive power to avoid some paradoxes. We're going to need some quantifiers, and we're going to need 'property abstraction', which is the modal analogue to the definite descriptor in ordinary first-order logic. Notice now that there isn't really any way to do this using just propositional modal logic. So clearly, in order to avoid making fallacious modal arguments we are going to have to have some formal machinery to distinguish between cases like 1* and 1**. But (3) would only follow only from 1** and 2, not from 1* and 2. It is true of the President, that he is necessarily President.ġ* is a harmless conceptual truth, but 1** is a patent falsehood. It is necessarily true of the President, that he is the President.ġ**. It appears that we need a way to distinguish two different ways to ascribe modal properties, we need to distinguish, in other words:ġ*. It turns out that the problem is a kind of ambiguity in (1). Therefore, necessarily Barack Obama is the President of the United States.ģ would appear to follow from 1 and 2 by a substitution rule.Barack Obama is the President of the United States.De nition A truth assignment is a function v : P fT Fg. Necessarily, the President of the United States is the President of the United States. to Modal Logic W.Gunther Propositional Logic Our Language Semantics Syntax Results Modal Logic Our language Semantics Relations Soundness Results Semantics Note, we desire a way of deciding the truth of a statement.(There may be some that I haven't come across, of course.) The point of adding quantification, and other devices that belong to first-order modal logical such as the property abstraction operator, is to increase the expressive power of the modal logic in order to use modal logic to formalize a broader range of natural language arguments that turn upon possibility and necessity.Ĭonsider, for instance, the following 'argument': I don't recall reading any discussion of a formal system where a quantified modal formula was an axiom of the system. Something TRUE in all worlds is said to be believed by the agent (It is TRUE that the agent has the Ace of Spades).I'm not an expert on modal logic, so take my answer with a grain of salt. What is left over are the epistemic alternatives (worlds possible given ones beliefs). Then eliminate those worlds that are not possible given what the agent knows. First compute all the possible ways the cards could be dealt to the opponent. The ability to play is determined partially by the agent's belief in the opponents hand. Complete knowledge of the opponents hand is impossible to determine.

From the point of view of an agent A (we will not represent A to keep notations simple), f will be read as: 'A believes f'.Įxample: An agent is playing poker. ex: temporal logic, deontic logic, epistemic logic.įor our purpose, we will use modal operators in the context of epistemic logic (logic of knowledge). The modal operators can be used in various contexts and take specific meanings.

The two modal operators are duals of each other:.The formula f is TRUE if f is TRUE for in at least one world accessible from the current world.The formula f is then TRUE if f is TRUE for all worlds accessible from the current world.In situations where classical quantification is desired, one may simply add Et as an axiom to FS, so that the classical principles become derivable rules. The semantics of modal connectives define what worlds are considered accessible from other worlds. The most general way to formulate quantified modal logic is to create FS by adding the rules of FL to a given propositional modal logic S.Classical propositional logic extended by 2 operators.Then introduce the accessibility relation that produces an 'overlay' on this 'clique'). If all goes well, we end up lifting the Stone.

(Teaching method: before introducing full-blown semantics, start with example where each interpretation corresponds to a world, and where all worlds are accessible. It would be handy at this point if Id learned how to draw suitable diagrams. Janine believing p is not dependent on the truth of p.Bel(Janine, Father (Jupiter, Cronos)) ?.ModalLogic motivation: classic logic limitations: (example is from Wooldridge)

Ways to use the modal logic playground how to#