by Corry Shores
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[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own.]
Summary of
John Nolt
Logics
Part 4: Extensions of Classical Logic
Chapter 11: Leibnizian Modal Logic
11.1 Modal Operators
Brief summary:
There are certain operators called alethic modifiers. They include deontic (or ethical) modalities (e.g. ‘it ought to be the case that’, ‘it is forbidden that’); propositional attitudes (‘believes that’, ‘knows that’, ‘hopes that’, ‘wonders whether’); and tenses (‘was’, ‘is’ , and ‘will be’). Propositional attitude modifiers are binary, but all the rest are monadic. ‘It is necessary that Φ’ is written □Φ and ‘it is possible that Φ’ is written ◊Φ. Alethic modifiers are often duals meaning that one can be converted into another by adding negations around the symbols, as for example:
□Φ↔~◊~Φ
◊Φ↔~□~Φ
Summary
There are operators that are inexpressible in predicate logic but nonetheless are still very important called alethic modifiers, which includes ‘must,’ ‘might,’ ‘could,’ ‘can,’ ‘have to’, ‘possibly,’ ‘contingently,’ ‘necessarily’ (Nolt 307). These words express modes of truth or alethic modalities. (The Greek word for truth is alethea, hence their name alethic modifiers). Modal logic studies the syntax and the semantics of alethic modalities (Nolt 307).
Modal logic also studies other kinds of propositional modalities, namely, {1} deontic (or ethical) modalities, which are “expressed by such constructions as ‘it ought to be the case that’, ‘it is forbidden that’, etc.” (Nolt 307); {2} propositional attitudes, which are “relations between sentient beings and propositions, expressed by such terms as ‘believes that’, ‘knows that’, ‘hopes that’, ‘wonders whether’, and so on”; and {3} tenses, which include past, present, and future tenses as expressed by the various modifications of the verb ‘to be’: ‘was’, ‘is’ , and ‘will be’ ” (Nolt 307).
All these alethic modalities can be understood as operators on propositions (Nolt 307). [We will see how the alethic modalities can be placed upon a proposition just like other logical operators can. The following is quotation.]
Consider, for | example, these sentences, all of which involve the application of modal operators (in the broad sense) to the single proposition ‘People communicate’:
Alethic Operators
It is possible that people communicate.
It must be the case that people communicate.
It is contingently the case that people communicate.
It could be the case that people communicate.
It is necessarily the case that people communicate.
Deontic Operators
It is obligatory that people communicate.
It is permissible that people communicate.
It is not allowed that people communicate.
It should be the case that people communicate.
Operators Expressing Propositional Attitudes
Ann knows that people communicate.
Bill believes that people communicate.
Cynthia fears that people communicate.
Don supposes that people communicate.
Everyone understands that people communicate.
Fred doubts that people communicate.
Operators Expressing Tenses
It was (at some time) the case that people communicated.
It was always the case that people communicated.
It will (at some time) be the case that people communicate.
(Nolt 307-308)
This is not even an exhaustive list of the operators in each category. Nolt also notes that except for the propositional attitude operators, the rest here are monadic.
With the exception of the operators expressing propositional attitudes, all of those listed here are monadic; they function syntactically just like the negation operator ‘it is not the case that’, prefixing a sentence to produce a new sentence. Thus, for example, the operators ‘it is necessary that’, usually symbolized by the box ‘□’ and ‘it is possible that’, usually | symbolized by the diamond sign ‘◊’, are introduced by adding this clause to the formation rules:
If Φ is a formula, then so are □Φ and ◊Φ.
(Nolt 308-309)
The operators for propositional attitudes are binary operators. However, he continues, “unlike such binary operators as conjunction or disjunction, which unite a pair of sentences into a compound sentence, propositional attitude operators take a name and a sentence to make a sentence. The place for this name may be quantified, as in ‘Everyone understands that people communicate’ ” (Nolt 309).
Many operators can be converted into one another through negations that flank the signs.
Many modal operators have duals – operators which, when flanked by negation signs, form their equivalents. The operators ‘□’ and ‘◊’, for example, are duals, as the following sentences assert:
□Φ↔~◊~Φ
◊Φ↔~□~Φ
That is, it is necessary that Φ if and only if it is not possible that not-Φ, and it is possible that Φ if and only if it is not necessary that not-Φ.
There are other duals among these operators as well. Consider the deontic operator ‘it is obligatory that’, which we shall symbolize as ‘O’, and the operator ‘it is permissible that’, which we shall write as ‘P’. These are similarly related:
OΦ↔~P~Φ
PΦ↔~O~Φ
That ‘O’ and ‘P’ should thus mimic ‘□’ and ‘◊’ is understandable, since obligation is a kind of moral necessity and permission a kind of moral possibility.
There are also epistemic (knowledge-related) duals. The operator ‘knows that’ is dual with the operator ‘it is epistemically possible, for ... that’ – the former representing epistemic necessity (knowledge) and the latter epistemic possibility. (Something is epistemically possible for a person if so far as that person knows it might be the case.) Symbolizing ‘knows that’ by ‘K’ and ‘it is epistemically possible for ... that’ by ‘E’, we have:
pKΦ↔~pE~Φ
pEΦ↔~pK~Φ
In English: p knows that Φ if and only if it is not epistemically possible for p that not-Φ; and it is epistemically possible for p that Φ if and only if p does not know that not-Φ (‘p’, of course, stands for a person).
There are temporal duals as well. Let ‘P’ mean ‘it was (at some time) the case that’ and ‘H’ mean ‘it has always been the case that’. Then:
HΦ↔~P~Φ
PΦ↔~H~Φ
| Here ‘H’ represents a kind of past tense temporal necessity and ‘P’ a kind of past tense temporal possibility. A similar relationship holds between ‘it always will be the case that’ and ‘it sometimes will be the case that and between other pairs of temporal operators.
(Nolt 309-310)
Nolt then notices how these duals remind us of two laws in predicate logic
∀Φ↔~∃~Φ
∃Φ↔~∀~Φ
(Nolt 310)
And he wonders if the duals are analogous to quantifiers (Nolt 310).
Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.
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