Logical Analogies: Interpretations, Oppositions, and Probabilism

Abstract

I present two logical systems to show the “analogy of proportionality” common to several interpretations: modality (necessity and possibility), quantification, truth-functional relations, moral attitudes (deontic logic), states of knowledge (epistemic logic), and states of belief (doxastic logic). To display the two underlying analogical relations, I call upon the originally Scholastic convention, recently put to use again, of using squares, hexagons, and octagons “of opposition”. A combined epistemic–deontic logic happens to be found in the traditional “probabilist” theory of the “good conscience”, and I shall then briefly explain how this is so.

I present two logical systems that display an “analogy of proportionality” among six areas of interpretation discussed by logicians today and by the Schoolmen of the past: necessity and possibility, quantification, propositional connectives, moral attitudes (deontic logic), and states of knowing and believing (epistemic and doxastic logics). To illustrate the underlying relations, I call upon the convention—Scholastic in origin but recently revived—of using figures “of opposition”: squares, hexagons, and octagons. A combined epistemic–deontic logic happens to be found in the traditional “probabilist” debate over the good conscience, and I shall briefly explain how this is so1.

1. The Logical Systems

The two systems, “Σ1 and “Σ2”, can be represented by a hexagon and an octagon, respectively. Σ2 contains (logically implies) Σ1. Each system has the same two monadic operators, one strong (S) and one weak (W), governing propositional expressions like “p” or “q” (“Sp”, “Wp”). The operators are interdefinable:

Sp = df ~W~p

Wp = df ~S~p

The Scholastic “square of opposition” displays the basic logical relations among the propositions (Figure 1, the traditional names are italics):

Since the operators are interdefined, the square may be expressed with either operator:

Sp S~p		~W~p ~Wp
~S~p ~Sp		Wp W~p

The first logical system, Σ1, is illustrated by a hexagon, which incorporates the square but adds the disjunction of the contraries Sp∨~Wp, and the conjunction of the subcontraries Wp∨~Sp (Figure 2). The second logical system, Σ2, contains the first, but adds implications to and from propositions not governed by the operators S and W (“p” and “~p” alone), which constitute, as it were, an “outside link” to the network of propositions affected by the operators:

Sp⊃p		~Wp⊃~p
p⊃Wp		~p⊃~Sp

2. Modality

A basic interpretation of these systems is (alethic) modality, which includes the notions of necessity, possibility, contingency, actuality, and their negations. Here, the strong operator indicates necessity: □p (“it is necessary that p”) and the weak operator possibility: ◊p (“it is possible that p”). Contingency is the conjunction of the subcontraries: ◊p∧◊~p (“it is possible that p and it is possible that not-p”) or equivalently ◊p∧~□p (“p is possible but not necessary”). The disjunction of the contraries, □p∨~◊p, represents non-contingency (“p is either necessary or impossible”). Also, since necessity implies actuality and actuality implies possibility,

□p⊃p		□~p⊃~p
p⊃◊p		~p⊃◊~p,

this “outside” link to actuality is expressed in an octagon representing system Σ2 (Figure 3):

This system, often called “T”, is a basic modal system. Because the operators are interdefinable, it can be expressed in terms of necessity or of possibility,

□p	□~p		~◊~p	~◊p
~□~p	~□p		◊p	◊~p.2

3. Propositional Relations

The Schoolmen also used a square of opposition to illustrate the relations among propositions connected by “and” (∧) and “or” (∨). The strong operator S here represents conjunction: p∧q (“p and q”) and the weak operator W represents (inclusive) disjunction p∨q (“p or q”). The relations are between the “molecular” propositions p∧q and p∨q. Σ2 is required, since there is an outside link to “atomic” propositions, those not joined by these connectives. The following implications indicate the outside link:

[p∧q]⊃p and [p∧q]⊃q		[~p∧~q]⊃~p and [~p∧~q]⊃~q
p⊃[p∨q] and q⊃[p∨q]		~p⊃[~p∨~q] and ~q⊃[~p∨~q]

The propositional octagon (Figure 4):

4. Quantification

The square of opposition has also been applied to quantified propositions. Here, we may read the logical operators as quantifiers; the strong operator S is the universal quantifier: “∀x” and the weak operator W is the existential quantifier; “∃x”. The letter “a” stands for a “thing” and the Greek letter phi (ϕ) indicates a proposition containing a (ϕa; say, “Alice is good”); the letter “x” is a variable in the proposition (ϕx). Again, Σ2 is needed, since the rules of quantificational logic,

∀xϕx⊃ϕa		∀x~ϕx⊃~ϕa
ϕa⊃∃xϕx		~ϕa⊃∃x~ϕx,

constitute an outside link. Here, connections to and from propositions are not governed by a quantifier3.

The octagon of quantification (Figure 5):

5. Deontic Logic

“Deontic logic” is the logic of τὸ δέον, debitum: the “ought” (“should”), the logic of “duty” or “obligation”4. Ways of signifying moral notions lie deeply embedded in our languages and can be formulated logically in systems compatible with various ethical views. The Schoolmen touched upon the matter, Leibniz formulated a deontic system, and more recent systems date from G. H. Von Wright [3,4]5.

For the strong operator S in deontic logic, I use the letter “O” (“ought”) in the proposition “Op”, and for the weak operator W, I use the letter “M” (“may”) in “Mp”. Either symbol can be taken to relate a free moral agent, say, Socrates, to the state-of-affairs that ought to obtain (Op: “Socrates should bring it about that p”; and Mp: “Socrates may bring it about that p”).

O and M, too, are interdefinable:

Op = df ~M~p	Socrates should do p just in case he may not leave p undone
Mp = df ~O~p	Socrates may do p just in case he is not obliged to leave p undone

“O~p” can be abbreviated as “Fp” (“Socrates is forbidden to do p”). Also, ~Op can be read “Socrates need not do p”, abbreviated “Np”6. We have, then, these equivalences:

Op = ~M~p

O~p = ~Mp = Fp

~Op = M~p = Np

~O~p = Mp,

as well as the implications Op⊃Mp: (if Socrates must, he may) and Fp⊃Np (if he is forbidden, he need not).

Scholastic moral theologians, in their “probabilist” discussions, of which we shall speak below, called the disjunction of the contraries “law (lex)”, since Socrates must either do p or not do p. They called the conjunction of the subcontraries “freedom (libertas)”, since Socrates may, but need not, do p (in other words, p is “optional”, a “work of supererogation”). The hexagon of Σ1 displays these relations, since there is no outside link to actual behavior; the fact that p ought to be the case does not mean that it will be the case (Op⊃p is obviously invalid, as is Fp⊃~p). Leibniz displayed these relations in a square of opposition, and I include his terms in italics (for him, “optional” was “indifferent”); I also include the Scholastic “law” and “freedom” (Figure 6).

Again, we can express this system with a single operator:

Op		~Op		~M~p		~Mp
~O~p		~Op		Mp		M~p.

Leibniz used his square in a legal context, but the Schoolmen were concerned with individual conscience, the “moral law”.

6. Behavior

If we introduce a link to actual behavior, we have, as it were, systems of “virtue and vice”. Socrates does his duty (Op∧p, O~p∧~p) or fails to do it (O~p∧p, Op∧~p)7. The schoolmen spoke of “veritatem manifestare”8, and we can use an octagon to express the outside link to “compliance”; that is, the implications

Op⊃p		Fp⊃~p
p⊃Mp		~p⊃Np.

The octagon of good behavior (Figure 7):

For example, Socrates pits the olives when his wife Xanthippe told him to, and he does not eat all the olives when she told him not to (Op⊃p and Fp⊃~p). And, when he does eat all the olives, Xanthippe told him he might, and when he doesn’t pit any olives, she said he doesn’t have to (p⊃Mp and ~p⊃M~p).

On the other hand, we have an “octagon bad behavior” when p and ~p are switched, and, in the case of lex, compliance would become “non-compliance” (Figure 8):

Here Socrates doesn’t pit the olives when Xanthippe told him to, and he eats all the olives when she told him not to (Op⊃~p and Fp⊃p). But, in the case of libertas, when he does not eat all the olives, Xanthippe said he might, and when he pits the olives, she told him he didn’t have to (~p⊃Mp and p⊃Np).

7. The Logic of Knowledge

Traditionally, good behavior requires that the person be able to act both knowingly and freely. If Socrates acts freely, he then meets the following condition, where “Dp” stands for “Socrates decides freely to bring about the state-of-affairs p” and lambda “λ” stands for “the physical laws of nature are in force”):

Dp∧◊[λ∧~Dp]

“Socrates decides to bring about p but it may be that while the laws of nature hold he does not so decide”9.

Besides being free, Socrates must also know or at least believe that he has a duty. We may take knowing and believing in the sense of Jaako Hintikka’s “logic of the two notions” [14]10. In fact, the logics of knowledge and belief correspond to the two logical systems Σ2 and Σ1.

In Hintikka’s epistemic logic, the operator “K” relates a knower (say, Socrates) to what he knows, the state-of-affairs p in “Kp”, meaning “Socrates knows that p”11, This means that Socrates knows that p, where the knowledge is truth-entailing, that is:

* Kp⊃p and K~p⊃~p		what Socrates knows is true
* p⊃~K~p and ~p⊃~Kp		he does not know what is false.

We have, then, an outside link to “truth”, to what is actually true or false “outside knowledge”. In the following octagon, Kp∨K~p means that Socrates knows whether p is true or false, and ~K~p∧~Kp that he does not know whether p is true or false (Figure 9).

The octagon with the deontic operator O (Figure 10, in terms of Socrates’s duty in regard to p):

8. The Logic of Belief

In Hintikka’s doxastic logic, the operator “B” relates Socrates to his opinion, that is, the state-of-affairs p in “Bp” (“Socrates believes that p”). “Belief”, in this sense, presupposes basic rationality and does not have the negative connotations that the word may have in a Platonic tradition (δόξα, opinio). Nevertheless, belief, although it follows from knowledge (Kp⊃Bp), is not truth-entailing (Bp⊃p and Bp⊃Kp are invalid). Therefore, there is no outside link, and Σ1 must be used to display the relation (Figure 11):

9. The Logic of Knowledge and Belief Combined

Since epistemic logic contains the doxastic logic, we may combine the octagon of knowledge with the hexagon of belief. The double lines are implications linking the two systems:

Kp⊃Bp		K~p⊃B~p
~B~p⊃~K~p		~Bp⊃~Kp.

The diagram (Figure 12):

This network of implications runs from “certitude” to “quandary”12.

10. Probabilism

The Schoolmen around the 17th century passionately discussed how we are to judge when the moral agent is bound by law or has the liberty to act as he pleases. The purpose of this metaethical controversy, called “probabilism”13, was to describe the “good conscience”, particularly in the face of moral dilemma. A number of theories or systemata (“systems”) developed, and they all relied upon a calculus of thinking about duty—combining, in effect, an epistemic logic with a deontic logic.

Obviously, we may do something when we are sure, that is, when we know14 that we may (KMp⊃Mp; also, KFp⊃Fp) or even if we are unsure but believe that we may (BMp∧~KMp). But what do we do if we “waver”: neither believing that the act is forbidden nor that it is allowed (~BFp∧~BMp)? Or how are we to act when ethicists themselves have sharply different views on what is right?

Participants in the discussion distinguished several “states of mind” according to the degree of rationality of these states15. They took this rationality as a “probability of grounds” of the reasons or arguments brought to defend their positions. “Probable” here does not mean “likelihood” (say, of heads or tails turning up at the toss of a coin); a “probable” argument is rather one that “can be defended by sound arguments (rationes graves), and is quite close to Hintikka’s notion of “defensibility”16.

These are the chief states of moral thinking17:

*

Certainty: Socrates is sure that he must or may effect the state-of-affairs p if he has a “sound grounds” for his attitude (KOp, KMp or their negative forms) and no reason, or only “slight (levis)” ones, for the opposite. In this state truth would be “manifested” (KOp⊃Op, KMp⊃Mp). He then knows whether he may or may not act:

KMp∨KFp

*

Opinion: Socrates believes that he should or may effect p when his reasons are “sound enough” or “sounder than the opposite”. “And so, although the truth is not “manifested, still there is a notable appearance of the truth”18. Hence, he opines that some act is licit or forbidden:

BMp∨BFp

and he is then said to be “afraid of being wrong”.

*

Wavering19: Socrates wavers “positively” when he suspends assent because the reasons for both positions (Fp and Mp) are “equally sound”. Hence in wavering, he does not believe that he may not, nor does he believe that he may:

~B~Mp∧~BMp.

*

Ignorance or “negative wavering”: Socrates suspends his assent when his reasons for both sides (“parts”) are totally lacking. He therefore does not know whether he may or may not act:

~K~Mp∧~KMp.

The similarity between the four Scholastic and the “epistemic states” is obvious and can be compared in this diagram, which combines epistemic, doxastic, and deontic logics (Figure 14):

The double lines connect knowledge with belief:

*

that Socrates knows that he may do p implies that he believes that he may: KMp⊃BMp (also KFp⊃Bfp);

*

that he does not believe that he may not do implies that he does not know that he may not: ~BFp⊃~Kfp (also ~Mp⊃~KMp).

These are the (epistemic) truth-entailing relations:

*

that Socrates knows that he may do p implies that he may: KMp⊃Mp (or KFp⊃Fp);

*

that he may do p implies that he does not know that he is forbidden: Mp⊃~KFp (or Fp⊃~KMp).

The various “systems” lie on a spectrum between the extreme positions called rigorism or tutiorism (a moral act demands certitude) and laxism (any opinion, even a doubtfully sound one, may be followed)20. Other systems are mitigated tutiorism (the soundest view is always to be followed), probabiliorism (the sounder view is to be followed), aequiprobabilism (a position at least as sound as its opposite may be followed), and moderate probabilism (a solidly probable viewpoint may be followed).

11. Analogy

In summary, in an “analogy of proportionality”, there are a number of interpretations of the systems Σ1 and Σ2 according to their strong and weak operators (Figure 15):

All these “ratios” embody Σ1, and Σ2 also applies to those having an “outside link”: necessary (the link to actuality), all (to existence), and (to atomic propositions), and knowing (to truth). There is also an analogy between the epistemic, doxastic, and deontic logics described and those found in Scholastic probabilism.