They recommend further axioms to govern the iteration, or repetition of modal operators. Here are two of the most famous iteration axioms:. As a result, any string of boxes may be replaced by a single box, and the same goes for strings of diamonds.
This amounts to the idea that iteration of the modal operators is superfluous. So the acceptability of axioms for modal logic depends on which of these uses we have in mind. Unfortunately, what ought to be is not always the case.
Controversy about iteration repetition of operators arises again in deontic logic. The idea is that there are genuine differences between the obligations we actually have and the obligations we should adopt. The interaction axioms raise questions concerning asymmetries between the past and the future. A standard intuition is that the past is fixed, while the future is still open. However, possible world semantics for temporal logic reveals that this worry results from a simple confusion, and that the two interaction axioms are equally acceptable.
Depending on which assumptions one makes about the structure of time, further axioms must be added to temporal logics. A list of axioms commonly adopted in temporal logics follows. An account of how they depend on the structure of time will be found in the section Possible Worlds Semantics.
It is interesting to note that certain combinations of past tense and future tense operators may be used to express complex tenses in English. For a more detailed discussion, see the entry on temporal logic. The founder of modal logic, C. These systems require revision of the standard systems of propositional logic. See Mares and the entry on relevance logic. Kvart is another good source on the topic.
Counterfactual logics differ from those based on strict implication because the former reject while the latter accept contraposition. The purpose of logic is to characterize the difference between valid and invalid arguments. A logical system for a language is a set of axioms and rules designed to prove exactly the valid arguments statable in the language. Creating such a logic may be a difficult task. The logician must make sure that the system is sound , i.
Furthermore, the system should be complete , meaning that every valid argument has a proof in the system. Such a demonstration cannot get underway until the concept of validity is defined rigorously. Formal semantics for a logic provides a definition of validity by characterizing the truth behavior of the sentences of the system. In propositional logic, validity can be defined using truth tables. A valid argument is simply one where every truth table row that makes its premises true also makes its conclusion true. Nevertheless, semantics for modal logics can be defined by introducing possible worlds.
Then we will explain how the same strategy may be adapted to other logics in the modal family. Then the truth values of the complex sentences are calculated with truth tables. A definition of validity is now just around the corner. In deontic logic, temporal logic, and others, the analog of the truth condition 5 is clearly not appropriate; furthermore there are even conceptions of necessity where 5 should be rejected as well.
The point is easiest to see in the case of temporal logic. Then the correct clause can be formulated as follows. Validity for this brand of temporal logic can now be defined. One condition which is only mildly controversial is that there is no last moment of time, i. This condition on frames is called seriality. Density would be false if time were atomic, i.
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Each of the modal logic axioms we have discussed corresponds to a condition on frames in the same way. The relationship between conditions on frames and corresponding axioms is one of the central topics in the study of modal logics. This, in turn, allows us to select the right set of axioms for that logic.
Under such a reading, it should be clear that the relevant frames should obey seriality, the condition that requires that each possible world have a morally acceptable variant. For example, I might say that it is necessary for me to pay my bills, even though I know full well that there is a possible world where I fail to pay them. Depending on exactly how the accessibility relation is understood, symmetry and transitivity may also be desired. A list of some of the more commonly discussed conditions on frames and their corresponding axioms along with a map showing the relationship between the various modal logics can be found in the next section.
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A list of these and other axioms along with their corresponding frame conditions can be found below the diagram. In this chart, systems are given by the list of their axioms. In boldface, we have indicated traditional names of some systems. The notion of correspondence between axioms and frame conditions that is at issue here was explained in the previous section. Several stronger notions of correspondence between axioms and frame conditions have emerged in research on modal logic. The correspondence between axioms and conditions on frames may seem something of a mystery.
A beautiful result of Lemmon and Scott goes a long way towards explaining those relationships. Their theorem concerned axioms which have the following form:. In order to do so, we will need a definition. A relation may be composed with itself. We may now state the Scott-Lemmon result. For example, consider 5.
Teach Yourself Logic, #7. A bit of deviance
So the corresponding condition is. So the corresponding condition on frames is. The Scott-Lemmon results provides a quick method for establishing results about the relationship between axioms and their corresponding frame conditions. Sahlqvist has discovered important generalizations of the Scott-Lemmon result covering a much wider range of axiom types.
The reader should be warned, however, that the neat correspondence between axioms and conditions on frames is atypical. There are condtions on frames that correspond to no axioms, and there are even conditions on frames for which no system is adequate. For an example see Boolos, , pp.
Two dimensional semantics is a variant of possible world semantics that uses two or more kinds of parameters in truth evaluation, rather than possible worlds alone. However, indexicals bring in a second dimension — so we need to generalize again. Kaplan defines the character of a sentence B to be a function from the set of linguistic contexts to the content or intension of B, where the content, in turn, is simply the intension of B, that is a function from possible worlds to truth-values.
Here, truth evaluation is doubly dependent — on both linguistic contexts and possible worlds. An example is 1. This suggests that the context dimension is apt for tracking analytic knowledge obtained from the mastery of our language. On the other hand, the possible-worlds dimension keeps track of what is necessary.
Holding the context fixed, there there are possible worlds where 1 is false. Therefore, two-dimensional semantics can handle situations where necessity and analyticity come apart. Another example where bringing in two dimension is useful is in the logic for an open future Thomason, ; Belnap, et al. Here one employs a temporal structure where many possible future histories extend from a given time.
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Consider 2. If 2 is contingent, then there is a possible history where the battle occurs the day after the time of evaluation, and another one where it does not occur then. So to evaluate 2 you need to know two things: what is the time t of evaluation, and which of the histories h that run through t is the one to be considered.
So we would have the following truth condition:. Then the truth condition Now is revised to 2DNow. To properly evaluate 4 we need to keep track of which world is taken to be the actual or real world as well as which one is taken to the world of evaluation. The idea of distinguishing different possible world dimensions in semantics has had useful applications in philosophy.
For example, Chalmers has presented arguments from the conceivability of say zombies to dualist conclusions in the philosophy of mind. Chalmers has deployed two-dimensional semantics to help identify an a priori aspect of meaning that would support such conclusions. The idea has also been deployed in the philosophy of language.
On the other hand, there is a strong intuition that had the real world been somewhat different from what it is, the odorless liquid that falls from the sky as rain, fills our lakes and rivers, etc. So in some sense it is conceivable that water is not H Two dimensional semantics makes room for these intuitions by providing a separate dimension that tracks a conception of water that lays aside the chemical nature of what water actually is. Modal logic has been useful in clarifying our understanding of central results concerning provability in the foundations of mathematics Boolos, Using this notation, sentences of provability logic express facts about provability.
Furthermore, the box may be iterated. In provability logic, provability is not to be treated as a brand of necessity. The applications of modal logic to mathematics and computer science have become increasingly important.