Modal logic, defined most basically and simply, concerns involves the incorporation of possibility and necessity into otherwise ordinary propositional calculus. While it is common for tutorials on modal logic to assume knowledge of propositional calculus on the part of the reader, we will gradually be working our way up from the basics of propositional calculus until the audience has obtained sufficient understanding of the subject to begin to incorporate the more advanced modal notions and their symbolism.
But what does all this have to do with language? I would argue (and not everyone would agree with me on this) that language and logic are in the ultimate instance about the world. Formal logic sets forth the preconditions for sensible linguistic statements in a systematic manner. It helps to see how or whether or not or to what extent a formal sentence is valid or sound by translating it into a concrete linguistic example about the world and thinking about whether or not or to what extent the utterance makes sense. More specifically, with respect to modal logic, we use modal operators to signify modal words and utterances having to do with possibility and necessity, which are linguistically represented in every day life with adverbs like “possibly” and “necessarily.”
Take note of this set of formal notation.
A ⇒ B – “If A then B”
A → B – “If A then B”
A ⊃ B – “If A then B”
A ⇔ B – “A if and only if B”
A ≡ B – “A if and only if B”
A ↔ B – “A if and only if B”
¬A – “It is not the case that A”
˜A – “It is not the case that A”
!A – “It is not the case that A”
A ∧ B – “A and B”
A • B – “A and B”
A & B – “A and B”
A ∨ B – “A or B”
A + B – “A or B”
A ǀǀ B – “A or B”
These are the logical operators we will encounter in propositional logic. The next set of operators will not be countered in our look at propositional logic, but they will be useful to know for students of logic interested in studying more advanced forms of logic.
The turnstile indicates provability. p ⊢ q means q is provable from p.
p ⊨ q = p semantically entails q
Precedence grouping – as in algebra, parentheses simply indicate that you perform the enclosed operations first.
A ⊕ B – “The statement is true when either A or B is true, but not when both are true.”
A ⊻ B – “The statement is true when either A or B is true, but not when both are true.”
⊥ – “The statement ⊥ is unconditionally false.”
“The statement ⊤ is unconditionally true.”
∀ x: P(x) = P(x) is true for all x
∃ x: P(x) – “There is at least one P(x) such that x is true.”
∃! x: P(x) – “there is exactly one x such that P(x) is true.”
x := y – “x is another name for y”
x ≡ y – “x is another name for y”
x :⇔ y – “x is logically equivalent to y”
Keep in mind that ≡ can be used of either logical equivalence or material equivalence. Its significance depends on context.
A symbol or a set of symbols is an expression. Those with even the most minimal training in algebra ought to immediately recognize such language. The aim of the logician in writing an expression is to obtain a well-formed formula. Formation rules specify what constitutes such a well-formed formula(Cresswell & Hughes, 1996).
Formation rule 1 – a letter standing alone is a wff(Cresswell & Hughes, 1996).
Formation rule 2 – if a is a wff, so is not-a. Prefixing the negation sign to any wff itself results in a wff(Cresswell & Hughes, 1996).
Formation rule 3 – if a and b are wff, so is (a ∨ b)(Cresswell & Hughes, 1996).
Logic uses letters (p, q, r…) to represent propositions(Cresswell & Hughes, 1996). This is another respect in which logic is a formalization of the precondition for sensible linguistic utterances. Indeed, it is why we refer to the particular form of formal logic with which we’re dealing “propositional calculus.” It is simply a way of systematizing propositions. A letter that represents a proposition is called a “propositional variable”(Cresswell & Hughes, 1996).
You’ll probably recognize the term “variable” from algebra class. A variable is simply a symbol (in this case a letter) that is used to stand in for a concrete instance of something. That is why it is called a “variable”; because what it stands for varies depending upon what concrete instance it is representing. A proposition can also be referred to as a “statement” or an “assertion.” For the present purpose, a proposition must be either true or false. Nonsense statements which do not meet the precondition for truth or falsity do not constitute legitimate propositions. “Truth” and “falsity” are both what we refer to as “truth-values” of propositions. When asked what the “truth-value” of a proposition is, then, we would say that the proposition is either “true” or “false.”
Let’s talk about some of the logical operators listed near the beginning of this article. The negation operator is an example of what is known as a proposition-forming operator(Cresswell & Hughes, 1996). The reason for this is that adding this operator to an already-existing proposition forms an entirely new proposition(Cresswell & Hughes, 1996). For example, adding it to the proposition “the dog is fat” forms the new proposition “it is not the case that the dog is fat.”
Operators operate upon “arguments”(Cresswell & Hughes, 1996). “It is the case that the dog is fat” and “it is not the case that the dog is fat” look like mere assertions or propositions (and they are that too) but they also count as arguments(Cresswell & Hughes, 1996). They are not very sophisticated or interesting arguments, but they are arguments nonetheless. An operator which requires only one argument (“It is the case that…”) is a monadic argument, and one that requires two arguments (for example, an either/or statement) is known as a dyadic argument.
Let’s look at some formal, symbolic definitions of some of the operators we’ve discussed.
Keep in mind that “df” is normally written as a subscript. It means “is defined as.”
Formal definition of “conjunction” – [Def ∧] (a ∧ b) = df ˜(˜a ∨ ˜b)(Cresswell & Hughes, 1996).
Formal definition of “material implication” – [Def ⊃] = df (˜a v b)(Cresswell & Hughes, 1996).
Formal definition of “material equivalence” – [Def ≡] = df ((a ⊃ b) ∧ (b ⊃ a))(Cresswell & Hughes, 1996).
Now that we have a basic overview of the formal notation of propositional calculus, we’re going to throw some modal operators into the mix. L is a modal operator which refers to necessity. Therefore, La means “necessarily a” or “it is necessarily the case that a.” Necessity is also oftentimes represented in modal logic with □. Therefore, □A means “necessarily a” or “it is necessarily the case that a.”
Another one of the most basic operators of modal logic is M, which refers to possibility. Therefore, Ma means “It is possible that a” or “Possibly a.” Possibility is also represented in modal logic with ◊. Therefore, ◊a means “It is possible that a” or “Possibly a.”
Cresswell & Hughes. “A New Introduction to Modal Logic.” London ; New York : Routledge, 1996. 01/01/1996 1 online resource (x, 421 p.).