As well as the standard negation of Polish logic, it is possible to define two others: A strong negation, which is ordinarily expressed as “Not Possible”, and a weak negation, which is ordinarily expressed as “Not necessary”. Normally I don’t use them. However, they do have use.

I had mentioned in one of the earlier posts in this series that I had read about a number of controversies on the foundations of mathematics. A Dutch mathematician named L. E. J. Brouwer had in the early 1900s created his own philosophy of mathematics, which he termed intuitionism, in which he rejected the law of the excluded middle. His student Arend Heyting gave an axiomatic basis for this variant of mathematical logic. Although this has never gained the support of most mathematicians, it remains an area of some interest. What I found was that at least one version of the Heyting axioms of Intuitionistic Propositional Logic could be expressed as theorems of Polish logic, if I used a strong negation instead of the standard.

There is also a branch of logic termed “Fuzzy logic”, which assigns numeric degrees of truth to propositions. The truth table are constructed so that a logical “AND” has the value of the highest truth truth value of the propositions being combined, and a logical “OR” has the lowest. The principle obstacle to adoption seems to be indecision on the nature of the conditional. However, if all the degrees between true and false are expressed by the single value U, This is remarkably similar to the Polish logic. The algebra of the Polish logic and the algebra of fuzzy logic both belong to a class known as deMorgan algebras, which is similar to Boolean algebra but lacks the complementation laws, which unsurprisingly are equivalent to the law of the excluded middle and law of bivalence.

The absence of the law of the excluded middle makes the standard mathematical proof by contradiction tricky. This connected to negation introduction or negation elimination rules in natural deduction systems. A possible set of natural deduction rules can be formulated for the Polish logic, but it requires a stronger contradiction: possible and not possible, necessary and not necessary, or even necessary and impossible.

This also has connection to the field known as paraconsistent logic.

As with classical logic, this can also be extended by the use of propositional functions applied to particular objects or sets of objects to predicate logic.

In spite of the versatility of this system, it still isn’t quite enough to capture traditional modal logic. In particular, with only three values, it is not possible to capture the distinction between “contingently true” and “contingently false”. However, it may clarify the way such a system would need to be constructed.

Many of these subjects are rather technical, and I would welcome a discussion with other experts on logic. However, of all the people I know, few are interested in symbolic logic at all, and none of those interested in closely examining my results.

There are a few avenues I have yet to pursue, but for now, this will end my series of musings on logic. Perhaps I will go back to Aristotle and pick up the thread of historical investigation.

Please forgive me. This has been dancing through my head for the past 15 years or so, and I don’t think I’m going to get past it until I get it written down. By this time, the early 1990s or so, I don’t remember exactly, I obsessed with finding exactly why there was a mismatch between the Lewis systems of modal logic, and the Lukasiewicz three-valued logic, because there were so many points of similarity, and because I was getting comprehensible and consistent results from the algebra. Yes, these are considered incompatible, but exactly? And I couldn’t find out. After the 1920s, the modal logic community followed Lewis like lemmings off a cliff (except that lemmings apparently don’t really do that, making them somewhat smarter than logicians) and the multivalued logic community went into higher and higher numbers of logical values. The communities stopped speaking to each other and nobody I could find was even asking the question of why these were incompatible. I went to the original work of Luksasiewicz. He had already done everything that I had, so I was covering old ground there.

Then one day after I had moved to Utah, I looked again into a newer work on multivalued logic, specifically the three valued case, and there were several competing systems, and I noticed something peculiar about all the logical equivalence relations they were defining. None of them, except one, met the conditions for a mathematical equivalence, and the Lukasiewicz logic wasn’t one of them. (Nor convenience and to avoid misspelling the name, I’ will just call it Polish logic from now on) I thought “Easy enough, I will just define a relation that is.” I was quite pleased that I could now express all the truth tables I had worked out as proper equivalences, just like I could do in classical logic. Then I noticed that I could get this same table by applying the “Definitley” or “Necessarily” true operator to the Polish equivalence. Then it occurred to me that what was good for the equivalence, or the biconditional might be good for the conditional as well. So I tried it.

SHAZAM!!! The familiar triangular form of a mathematical ordering relationship fell out. I was struck by the most blinding, dazzling flash of hindsight I have ever experienced. OF COURSE IT MUST BE SO!!! I am calling this the strict Polish conditional. It had all the nice properties of the material conditional that make it so useful in classical logic. While the important logical laws of modus ponens and the transitive property fail with the Polish conditional, they succeed with the strict Polish conditional, and a comparison of the truth tables for them revealed why. The ordinary Polish conditional allows for doubtful conditionals (equivocal or dichotomous, take your pick it works out the same either way) and of course you cannot expect valid reasoning with those. Take “I get to the bus station on time”. If the conditional is “If I get to the bus station on time, I will get to ride the bus” is doubtful, then you cannot safely reach the conclusion “I will get to ride the bus”. However, if you can guarantee that the conditional holds, your reasoning is safe. At one stroke, the failure of the laws that bothered Lewis turned into a powerful argument for Polish logic.

I now had four conditionals to work with, The ordinary material conditional in two valued logic, the strict conditional as defined by Lewis, the Polish conditional, and the strict Polish conditional. I already knew that the ordinary material conditional didn’t work for 3-valued logic, and the strict Lewis conditional didn’t either, for the same reason: Neither accounts for a middle truth value. The ordinary Polish conditional was better, but still not quite up to the job. But the strict Polish conditional…ah, now that was safe. I could safely guarantee valid conclusions with it. I could block introduction of error by preventing true assumptions from reaching doubtful conclusions, and doubtful assumptions from reaching false conclusions.

So, I tried casting the axioms of the Lewis system S5 into Polish logic, with the exception that I used the strict Polish conditional instead of the strict Lewis conditional. Success!! Theorems, not just axioms! Furthermore, if the strict Lewis conditional is is true, then so is the strict Polish conditional. (Not conversely). So, I had a system of modal logic that is truth functional and can be done with truth tables, not the cumbersome axiomatic approach. Anything you can do in S5 I can do better!

This turns the lightning bug into lightning. It turns Billy Batson into Captain Marvel. It turns humble, lovable shoeshine boy into the mighty Underdog.

At the risk of sounding like a commercial from Ronco; “But wait! There’s more!”

About this time I transferred from Phoenix College to ASU, and while I was there, I took the opportunity to do some extracurricular library digging. I found a bibliography of the history of symbolic logic in the Journal “Symbolic Logic” and troubled myself to look up references. One of them that I found was an early attempt at a logic of indecision. This didn’t work because it was inconsistent: a calculation done one way and a calculation done another way came up with different results. I believe I know why, now, but It would take some digging (on the order of an archaeological research with a grant and a team of undergraduate laborers) to find the references again. I also looked up the work of C. I. Lewis as published in Symbolic Logic, by Lewis and Langford, 1932, and found the axioms for his systems. I also found his definition for the “Strict implication” he used in his axiomatic systems, and that the had examined the same Lukasiewicz three valued logic I had been working with, and found it deficient because a a couple of important logical laws were missing. I will return to this subject later.

I also took a course on mathematical structure, which considered how mathematical object can be defined, the relations among them and the properties of those relations, and the mathematical operations that can be performed on them. I didn’t think to apply this to mathematical logic at the time, that came later, but it appears that it has not occurred to many logicians or philosophers, either. But this subject lay fallow for a while.

I believe it was after I left ASU that I began seriously flogging the algebra to discover why it was yielding some seriously paradoxical results. Let us take, for example “Charlie is a boy” versus “Charlie is a girl”, and what happens if we give this a truth value of U, for undecided. The algebra gives “Charlie is a boy and Charlie is a girl” a truth value of undecided. But what if these are mutually exclusive condition? Should not the expression be false? This point was raised by Lewis and it was his decision to incorporate the premise that they would be into his logic. As it turns out, the reason is that the “law” of the excluded middle is so easily and often taken for granted that we try to sneak it in anyway, even when the logic explicitly rejects it as a law.

As it turns out, the concept “The fribbles are thurgish” is undecided (to use an absurdity for illustrative purposes) can be mean two different things. It could mean that the thribbles are thurish is either true or false, but we don’t know which, or it could mean that there is an intermediate state between thurgish and not thurgish. Another example? “The door is open”. We could mean that the door is either definitely open or it is definitely closed, but we don’t know which is the case, or it could mean that the door is ajar, an intermediate state between fully open and fully closed. Another one. “it is raining”. We may not know which is the case, or there may be states of the weather that make it difficult to determine which is this the case.

The logic makes it clear that these are two different cases. One of them, which I call the “Dichotomous” sense, can be symbolized, but does not have a single truth value. The other sense, which I call the equivocal sense, can be symbolized and does have the single truth value U. What makes it tricky is that the negation behaves the same in both sases. The negation of a dichotomous statement “It is raining” and evaluated as true or false, would be naturally be “it is not raining”, and evaluated as false or true. If “The door is open” is considered equivocal because the door is ajar, “the door is not open” is also equivocal for the same reason. Without the symbolism to make the distinction clear, these two different kinds of uncertainty can be easily confused. “if it is possible that some proposition is true and it is possible that that proposition is not true then that proposition is equivocal. ” This is a theorem of the logic I use and I can prove it.

For practical purposes, the algebra works the same for either interpretation. What you cannot do, however, is used “undecided” as a logical value on one hand, and then turn around and try to invoke the law of the excluded middle on the other. BLAATTT! REEE!. Violation! If you’re going to say “Charlie is a girl” is undecided, you must also admit Charlie is a girl and Charlie is not a girl (i.e. a boy)” to also be undecided. You don’t have to accept such a contradiction as true. You just have to refrain from claiming it to be impossible. There is no problem with setting up a condition “Charlie is a girl” if and only if “Charlie is not a boy”. Once you know one, you know the other, and the case where they are equivocal slides through because they have the same truth value, even though they mean different things. But if you do insist ahead of time that it is impossible to say both “Charlie is a girl” and “Charlie is not a girl” then you have no business using three valued logic at all and the ordinary two valued case is sufficient.

The logic admits no compromise. Either you accept the law of the excluded middle as a general law, or do not. No, rejection of the law of the excluded middle doesn’t mean that it is never true. It only means that you can’t count on for all kinds of statement. In this particular case and a few others, it bites down hard.

“Mother may I go out to swim? Yes, my darling daughter. Hang your clothes on a hickory limb, but don’t go near the water. “

Your intuition may lead you astray until you have properly trained it. The algebra is consistent.

At about this stage in my research, I became interested in the connections with modal logic, the logic of necessity and possibility. Here, my primary source was the symbolic version developed by C. I. Lewis. I adopted his box-and-diamond notation, but I also noted that his versions were developed on an axiomatic basis and did not use truth tables. I took a look at his axioms and tried to fit them with theorems or tautologies of the three valued logic, and they didn’t quit fit. They were close, but didn’t quite match. There was even a mathematical proof that the Lewis systems could not be reduced to a finite number of truth values. There was enough similarity, and I had produced enough theorems to match most of the laws of Boolean Algebra, that I wondered what the difference was. I also took a course in logic at Phoenix College, using “The Logic Book” by Bergmann, Moor, and Nelson, which covered the natural deduction method of logical proof. We only covered propositional logic, but it gave me a better insight into the subject. I also made contact with people who were discussing the possibilities of using three-valued logic in Database programming, particularly as it was implemented in the language SQL. I didn’t quite have the full satisfactory theory yet, but I notice that there was a fierce debate over “Practically Useful” versus “Theoretically Sound” going on in articles and letters of Data Base Programming and Design Journal. The editors called a moratorium and the journal went into electronic publication only, so I lost track of the debate, and the journal has now ceased publication. It did raise some theoretical issues, which bedeviled me for a time.

One of the things things that logical arithmetic is good for is in proving logical formulas. An arbitrary expression in Boolean alebra, is usually conditional. It depends on the truth values of the variables it uses. However, for a certain class of formulas, the overall expression can be evaluated as true in every case. These are called truth-functionally true, or tautologies, and are the theorems, laws, or rules of boolean algebra. For instance, (A AND B) is equivalent to (B AND A). This is always true, regardless of whether A and B are true or false.

There are other ways of proving logical formulas. One of them is the axiomatic approach, in which one starts with a few axioms and a few rules of inference, and derives all the others. But wait, you may ask. Isn’t this a bit circular? Well, yes, it is, when applied to logic itself, and I much prefer the truth table approach when it can be used. For classical two-valued propositional logic, these give the same laws.

There are variants of standard logic which are examined for experimental purposes, in which some of the standard rules do not apply, and which do not or cannot use truth tables. These commonly use an axiomatic approach. It isn’t to my personal taste, but that’s just me.

Another topic that needs examination is the concept of a valid argument. Traditionally, rhetorical logic from the Greeks on was concerned with arguments that were logically valid. The goal is to start from true assumptions and reach true conclusions without introducing error in the reasoning itself. There are some forms of argument, commonly known as fallacies, which are considered invalid because they may introduce an error. By the time you boil things down to propositional logic, there is only one criterion that matters: Does the material conditional hold? If it does, your argument or proof is valid. If it doesn’t, then it isn’t. It’s that simple: Easy peasy. If you want to talk about classes of things, or talk about subjects where statements can be vague or uncertain, or fiddle with the laws of logic, it gets increasingly more complicated.

The other thing that needs to be considered is that from the time of Aristotle onward, logic was concerned with statements that were either true or false. No others were to be considered. This is easiest to do in mathematics, where we can control the definitions and decree certain things by convention. It also means that logic works less well in the real world. Take, for example “It is raining”. Aside from being dependent on time and place, there are certain states of the weather that make it hard to give that a definite true or false answer. If you are careful with definitions, you can control or limit the uncertainty…but then you have the problem of getting other people to agree with your definitions. Again, this is easiest to do in mathematics, unless perhaps you have engineers getting into two-fisted fights over Bessel functions or Laplace transforms.

As it turns out, uncertainty is a very slippery concept to grasp if all you have is words, and it’s easy to get mired in contradiction and confusion.

I don’t remember why I starting playing with three valued logic. I began with extending the truth tables for “or” and “and”, with a new value I called “Undecided”. I thought it reasonable to begun with “true OR undecided (true or false, don’t know which) equals true”; “undecided OR true equals true”, “false OR undecided equals undecided”, “undecided OR false equals undecided”, and “undecided OR undecided is undecided”.

For the logical and, “True AND undecided is undecided”, “undecided AND true is undecided”, “False AND undecided is false”, “undecided AND false is False”, “undecided AND undecided is undecided.

The negation of “undecided is undecided.

As I came to discover later, this is not quite right, but it will do for a start. I should also mention here that “undecided” is a very tricky and slippery concept to work with. This is one of the advantages of using symbols: By using this symbol I meant exactly THIS with this meaning, and not THAT with that meaning. But I’m getting a little ahead of myself.

Another logical connective that is usually defined is called the material conditional, and it is taken to mean “if… then…. It is called the material conditional and it has a specific meaning and truth table. ” ‘if true then true’ equals true.” ” ‘if true then false’ equals false.” ” ‘if false then true’ equals true”. ” ‘if false then false’ equals true.

This one confuses a lot of beginning logic students: especially, the last two entries, but since it is the standard definition, I’ll take it as given for now. The reason it works is one of they keys to properly understanding logic, both two-valued and multivalued versions There are some astonishingly eminent logicians who don’t quite get it. I will explain later.

The extension I chose was ” ‘if true then undecided’ equals undecided. ” “‘if false then undecided’ equals true. ” ‘if undecided then true’ equals false”. “‘ ‘if undecided then false’ equals undecided. “‘ ‘if undecided then undecided’ equals undecided. “

These additional five entries in each truth table give me 3×3 or nine, which is appropriate.

I so was pleased with myself that I betook myself to the ASU Hayden Library to find out if anyone had discovered this before. Alas, and to my dismay, someone had. There was entire book, “An Introduction to Many Valued Logics” by Robert Ackerman. I found a useful discussion of the reasons for considering many valued logics in the introduction, but I had basically rediscovered Stephen Cole Kleene’s strong three valued logic….and it didn’t work. Oh, logicians can get some use out of it by ‘designating’ truth values, but for practical purposes it is useless, because you get no useful theorems or laws of logic out of this and can do no useful deduction. Even so simple a statement as “if the bus is on time then the bus is on time”, which ought to be true, is labeled undecided when ‘the bus is on time ” is undecided. That’s no good. There were at least half a dozen other systems presented. The most interesting was the three-valued logic of Jan Lukasiewicz. I took note of it and promptly forgot his name (because it’s Polish).

At this point I need to back up and talk about logical theorems and valid deduction.

I promised that I would continue with my discoveries, but as I sketched out the post ahead of time, I realized that it would work better if I backtrack and do a little history. In 1847, the British mathematician George Boole published a pamphlet on the Mathematical Analysis of Logic, and in 1854, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. At a later point in my independent study, I went back and read them, and found, as with many pioneering tracts, hardly anybody reads them anymore, and for good reason. Boole had the bright and revolutionary idea of using algebraic variables for whole classes of objects instead of just numbers, using the number 0 to represent falsehood and 1 to represent truth, and the addition operation to mean “or” and multiplication to mean “and”. His original methods were quite cumbersome and have been vastly improved and simplified, but modern symbolic formal logic can be traced to his work. Symbolic, because it uses symbols instead of words, and formal, because it is more concerned with form than content.

Boole dealt with statements about classes of objects, such as “All men are mortal”, “Some roses are red”, and “No king is a fink”. The terms “all”, “some”, and “none” are nowadays referred to as quantifiers, and the rest of the sentence is the predicate, so this branch of the subject is now referred to as predicate logic. Mathematics employs this branch of the subject.

A simpler version, which deals with particular statements, such as “Socrates is mortal, “This rose is red”, or “The King of Id is a fink”, is called propositional logic. This was developed somewhat after Boole’s work.

It was discovered fairly early after Boole, that it vastly simplified the calculation if “or” were taken in the inclusive sense, so that she “She wore jeans or she wore a dress” would be true if she wore both (jeans under the dress, for instance), instead of the exclusive sense. (She wore either jeans or she wore a dress, but not both.) The inclusive sense has become conventional in logic, while English uses the exclusive one. We won’t talk about the logically literate having fun with using the inclusive sense when the exclusive is intended, by answering such questions as “Do you want cake or ice cream” with”Yes”.

So, in propositional logic, every statement is given a truth value, which is either true or false. Other possible values are simply not discussed, which is one of the limitations of logic.

The “arithmetic” of logic is quite simple. For “or”, you have “True or true equals true”. “True or false equals true”. “False or true equals true.” False or false equals false.”

For “and”, you have “true and true equals true”. “True and false equals false.” “False and true equals false.” false and false equals false.

There is also negation “Not true equals false”, and “not false equals true”.

These are often summarized in truth tables. There is nothing mysterious about truth table: they just like addition and multiplication tables, only much simpler.

On the other hand, the algebra of logic, Boolean algebra, it is called, after Boole, is a bit more complicated. There, as in ordinary algebra, you use letters to stand for statements, or more exactly, the truth values of statements, and there are rules for the combination and simplification of variables and expressions. The goal is not so much to “solve for the unknown as it to “make the expression as simple as possible.

Since this isn’t an algebra lesson, (aren’t you thankful)? that’s as far as I will go right now. Unless you want be a computer scientist, mathematician, or logician, you probably won’t need or use it. For those few who someday might, I will assign homework. No, I won’t collect or grade it. Like writing a textbook, you’d have to pay me for that. But mathematical logic is not a spectator sport and if you don’t flog your brain through the exercises, you won’t learn it.

Alright, I’m going to bore you all to pieces and do a little boasting. I am the only living master of three valued propositional logic. You will not find what I can teach you about it you in any textbook. (Among other reasons, because I haven’t yet written the textbook, and don’t intend to until I can find some way of getting paid for it. It’s not that much fun.) That’s not because it’s a hard subject. In fact, it’s quite easy, once you know how. It’s only unknown because no one will pay any redacted attention. Like Colombus is said to have claimed about reaching the New World, it’s quite easy once you know it can be done. Being the first was the hard part. Pay attention and I’ll reveal all the secrets, including the tortuous paths of discovery, missteps and false trails. Free of charge, even, for now. It’s of the benefits of following this here blog here. It’s my thing, and like Woo Young-woo’s whales (Extraordinary Attorney Woo, currently showing on Netflix), I’m going to keep talking about it until somebody shows an interest, and then talk about it some more.

So, a little bit of background. Back in the early 1980s when I came home from my LDS mission to Bolivia, I had intended to go back to school at BYU to continue my studies in Chemical Engineering. Due to a unfortunate combination of factors, I wasn’t able to get back in. However, I was living in Provo and withing walking distance of the Harold B. Lee Library. And, since there was no one to tell me no, I spent a large portion of my days in the stacks down on the 2nd level in the mathematics section, reading up on the foundations of mathematics including Logic, set theory, and mathematical structure. gathering a smattering of understanding of the history, controversies, and unsolved problems. I’ll come back to some of those later. A lot of it was over my head and I didn’t do much with the exercises in the textbooks, but I thought about the concepts. Basic propositional logic, boolean algebra I understood, since my sophomore high school geometry had included it in in the section on logic and mathematical proofs, and so did basic computer science, which I also studied on my own without benefit of teacher.

Since I didn’t have a teacher, I developed a lot of what you might call unorthodox theories about the nature of logic. Contrary to any impression you might have gained from watching Star Trek or reading E.E. “Doc” Smith’s Skylark series (Both of which I had done, showing off my nerd credentials). I came appreciate that logic is useless in finding “absolute” truth. It is only useful for “relative” truth, that is, this statement is just as true as that one. In logical argumentation, proof, or deductive reasoning you must always start with something you assume to be true; or believe for other reasons. The reason you believe something can be quite arbitrary: It may be as simple as something you declare to be so. The fun is in examining the consequences of that declaration.

Also, it’s perfectly possible to reach true conclusions with faulty reasoning. Logically valid reasoning has another purpose, which I came to appreciate more fully somewhat later.

Also, mathematical discovery and logical deduction are two different things. Typically, mathematical discovery is more of creative, experimental process. The deductive part, where you test or prove that the ideas you have created or discovered are correct, is something else. As my High School Geometry teacher was fond of saying ” ‘It’s obvious’ is not a proof”.

I picked up a couple of other notions along the way. One of these was modal logic, which is the logic of possibility and necessity, and comes back into the story later.

Stay tuned for the next episode, in which I made an exciting original discovery, only to find that that someone else had beaten me to it long before.

This is a continuation of a series of posts on three-valued logic which began with “The learned professors”

There are about seven reasons this work is revolutionary.

1) First, it is a full-featured extension of classical logic into the realm of the uncertain. The arithmetic of the integers is an extension of the ordinary arithmetic of whole numbers into the negative, and the arithmetic of common factions is an extension of ordinary arithmetic into the realm of parts of objects. Both are fully compatible and include the whole numbers as a special case. In a similar fashion, all the laws of classical two-valued logic remain true in the two-valued case, but some of them must be modified in the three valued case.

2) It is truth functional and the same methods of truth truth tables and algebraic manipulation apply in this logic as in two valued logic. It follows the associative, commutative, distributive, identity, annihilator, idempotent, double negation, and De Morgan laws of Boolean algebra but does not follow the complementation laws x & ~x= F and x ∨ ~x = T. This is called a De Morgan algebra.

3) It is a truth functional system of modal logic. It differs from the Lewis systems S1-S5, because it uses a different version of the strict conditional, and because it does not include the law of the excluded middle. However, theorems of L3M that are parallel to the axioms of S5 can be formulated, therefore anything that can be proven in S5 can be proven with this logic. It has been established that S5 itself cannot be reduced to 3 values, but this is a narrow result and can probably be traced to the difference in conditionals.
It can also be established that that if the strict material conditional []( P =) Q) holds, then so does the strict Lukasewicz conditional [](P -> Q), but not conversely.

4) It is a truth functional version of intuitionism. In similar fashion to modal logic, if all the axioms of intuitionism are replaced using the strong negation ~<>P instead of the ordinary negation, and using the strict Lukasewicze conditional instead of the ordinary material conditional, the resulting axioms are theorems of this system. Therefore, anything that can be proven using these axioms is also true in this systems.

5) It clarifies long-standing controversies in logic, notably the meaning of the conditional and its relation with concepts of implication and entailment, and doubts about the universal validity of modus ponens. Along with fuzzy logic, it offers a resolution to the Sorites paradox. It addresses some of the same concerns as relevance logic. The Lukasiewicz conditional can be defined as (~Q \/ P \/ P==Q), if P== Q has been defined as P and Q having the same truth value.

6) It is connected to Fuzzy logic. It has some of the same features, although this logic includes all the values between T and F in the one value U, where Fuzzy logic gives each truth value a distinct number.

7) It shares features with paraconsistent logic. (P & ~P) => Q is not a tautology, so it is not explosive, and is not sufficient to prove any other proposition at all. However, this logic does include methods of indirect proof, which work by establishing contradictions. It is simply necessary to establish a contradiction and not merely a contrary. Expressions such as (p & ~<>P) “true and impossible”, ([]p & ~P) “certain and false”, ([]p & ~[]P) “certain and not certain”, (<>P & ~<>P) “possible and impossible”, and ([]P & ~<> P) “certain and impossible” all work.