N.A. Vasiliev's "Imaginary" Experimental Modal Logic...

Evolutionary Thinking in Past Scientific Theories: A Logical Analysis by Antonino Drago, Dept. Phys. Sci., Univ. “Federico II”, Naples, Italy

Abstractions lead us to shape ideas, about which our minds argue by means of logic. An evolutionary thinking occurs when these ideas are not linked together by means of mechanistic deductions, but in a creative way. In this sense evolutionary thinking pushes us to shape a broader kind of logic. The phenomenon of a double negated statement whose corresponding positive statement is lacking of scientific evidence (=DNS) will be examined. It represents a failure of the double negation law; this law constitutes the borderline between classical logic and, broadly speaking, non-classical logic (in particular, intuitionistic logic). In fact, several scientific theories born in past times include in an essential way DNSs. In particular, quantum logic can be represented by means of DNSs inside intuitionistic logic. When DNSs pertain in an essential way to a theory, no more – as a comparative analysis upon the several instances shows – a deductive organization of the theory is possible; rather, the theory puts an universal problem by means of a DNS, then some double negated methodological principles (e.g.: “It is impossible a motion without an end”) follow in order to achieve a new scientific method, capable to solve the problem at issue. This arguing evolves through a cyclic pattern, according to the synthetic method as it was improved by L. Carnot. The crucial step in this pattern is an ad absurdum theorem (likely as in thermodynamics S. Carnot’s theorem is). This theorem reaches evidence for a possible conclusion, still enunciated by means of a DNS. Then by a move like Markoff principle this DNS is changed in a positive statement; it can now be put as a new hypothesis from which to develop a full deductive system. This move is illustrated at best in Lobachevsky’s – maybe first – presentation of a non-Euclidean geometry, but can be recognised also in S. Carnot’s thermodynamics, Avogadro’s atomic theory, Einstein’s founding special relativity. This pattern of arguing is examined by means of paraconsistent logic. In correspondence to the use by theoretical scientific research, of respectively paraconsistent logic, intuitionistic logic and classical logic about statements which are potentially principles for a theory, three kinds of principles are recognized; i.e., a guess, a methodological principle, an axiom-principle. These differences are expressed in a lucid way by Einstein again in his celebrated paper on special relativity: “We will raise the conjecture (the substance of which will be hereafter called the “[axiom-]principle of relativity”) to the state of a [methodological] postulate” LINK: In a previous paper I obtained a relevant result regarding paraconsistent logic. The founder of paraconsistent logic, N.A. Vasiliev, stated as a characteristic feature of his logic, three kinds of sentence, i.e., "S is A", "S is not A", "S is and is not A" ("indifferent judgment"). I was able to show that they hold true even when one substitutes "￢￢A" for "S" and "-->" for "is". One obtains respectively: "￢￢A-->A ", "￢￢A fails to -->A", "￢￢A-->A and ￢￢A fails to -->A".(substitute necessity [box] for --> everywhere) Let us remark that the three cases represent three different roles played (in) a sentence in an argument. i) ￢￢A-->A represents as an affirmative sentence, i.e. a sentence well-supported by scientific evidence; ii) ￢￢A fails to -->A represents a logical problem, i.e., it can represent a sentence still insufficiently supported by scientific evidence; iii) ￢￢A-->A and ￢￢A fails to -->A represents a sentence whose truth and falsity is not yet decided in scientific terms; this kind of sentence may be considered inside a theoretical framework as a guess, whose scientific qualification it is still yet to be decided. The last kind of sentence qualifies the characteristic sentence of paraconsistent logic as pertaining to a theory in construction. Antonino Drago   Imaginary(meaning sheet of assertions in imagination) Experimental Modal Logic: ￢￢A-->A(classical linear deduction) ￢￢A fails to -->A(non-classical/non-linear induction) ￢￢A-->A and ￢￢A fails to -->A(non-classical/non-linear abduction, hypothesis, theory)Antonino Drago on N.A. Vasiliev(my additions in italics)   Infinity__ Where all doubts are allowed… Let us consider Lobachevskii's geometry. By substituting "two straight lines meet" for A and "It is not true that two straight lines do not meet" for ￢￢A, i.e. Vasiliev's S, the three Vasiliev's above sentences describe respectively i) ￢￢A-->A, i.e. the hyperbolic secant lines, ii) ￢￢A fails to-->A, i.e. the hyperbolic ultra-parallel lines and, iii) ￢￢A-->A and ￢￢A fails to -->A, i.e. the parallel lines - which meet at a point which is located at infinity, i.e. where all doubts are allowed. This last meaning is presented by Lobachevskii himself in his most relevant writing; there, Lobachevskii refers to the meeting point at infinity by means of the following words: "In the uncertainty...", just the meaning of Vasiliev's third kind of sentence. That vindicates Vasiliev's reiterated claim, i.e. his logic represents just the logic of Lobachevskii's geometrical theory. "Handle two sorts of negations (logical and ontological)"; as paraconsistent logic does.   Conclusions The three main kinds of logic correspond to three characteristic ways of organizing a set of scientific data in a systematic way. Paraconsistent logic is a relevant logic since it represents the logic of the work of a scientist in his guessing new hypotheses for a given set of scientific data. I would add that the above exploration of the different roles played by the three kinds of logic has introduced us to a new kind of study, which can be called experimental logic; it is based upon evidence coming from the characteristic features of past scientific theories rather than the characteristic features of natural languages. Antonino  Drago Vasiliev affirmed, only ''positive'' sensations are possible, by which we can distinguish only contrary qualities. This is the basis of qualitatively different types of judgments - affirmative and negative. If one imagines a world in which not only positive but negative sensations are possible, then such a world will indeed require a different logic, and the introduction of supplementary qualitative judgments…

A Few New Ideas From FQXi

First Prize

The paradigm of kinematics and dynamics must yield to causal structure
Robert Spekkens

Essay Abstract
The distinction between a theory's kinematics and its dynamics, that is, between the space of physical states it posits and its law of evolution, is central to the conceptual framework of many physicists. A change to the kinematics of a theory, however, can be compensated by a change to its dynamics without empirical consequence, which strongly suggests that these features of the theory, considered separately, cannot have physical significance. It must therefore be concluded (with apologies to Minkowski) that henceforth kinematics by itself, and dynamics by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. The notion of causal structure seems to provide a good characterization of this union.

Author Bio
Robert Spekkens is a faculty member at the Perimeter Institute for Theoretical Physics in Waterloo, Canada. His area of research is the foundations of quantum theory.

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Second Prizes

Recognising Top-Down Causation
George Ellis

Essay Abstract
One of the basic assumptions implicit in the way physics is usually done is that all causation flows in a bottom up fashion, from micro to macro scales. However this is wrong in many cases in biology, and in particular in the way the brain functions. Here I make the case that it is also wrong in the case of digital computers – the paradigm of mechanistic algorithmic causation - and in many cases in physics, ranging from the origin of the arrow of time to the process of quantum state preparation. I consider some examples from classical physics; from quantum physics; and the case of digital computers, and then explain why it this possible without contradicting the causal powers of the underlying micro physics. Understanding the emergence of genuine complexity out of the underlying physics depends on recognising this kind of causation. It is a missing ingredient in present day theory; and taking it into account may help understand such mysteries as the measurement problem in quantum mechanics:

Author Bio
George Ellis is a relativist and cosmologist residing in Cape Town, South Africa. His books include On the Large Scale Structure of Space-Time co-authored with Stephen Hawking. In addition to contemplating relativistic and philosophical aspects of cosmology, he is now engaged in trying to understand how complex systems such as you and me can arise out of the underlying physics.

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Third Prizes

Reductionist Doubts
Julian Barbour

Essay Abstract
According to reductionism, every complex phenomenon can and should be explained in terms of the simplest possible entities and mechanisms. The parts determine the whole. This approach has been an outstanding success in science, but this essay will point out ways in which it could nevertheless be giving us wrong ideas and holding back progress. For example, it may be impossible to understand key features of the universe such as its pervasive arrow of time and remarkably high degree of isotropy and homogeneity unless we study it holistically -- as a true whole. A satisfactory interpretation of quantum mechanics is also likely to be profoundly holistic, involving the entire universe. The phenomenon of entanglement already hints at such a possibility.

Author Bio
After completing a PhD in theoretical physics, I became an independent researcher to avoid the publish-or-perish syndrome. For 45 years I have worked on the nature of time, motion, and the quantum theory of the universe. I am the author of two books: The Discovery of Dynamics and The End of Time, in which I argue that time is an illusion. Details of my research work are given at my website platonia.com. Since 2008 I have been a Visiting Professor at the University of Oxford.

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The Universal Isomorphic Algorithm__

The Universal Isomorphic Algorithm__UIA = ∑∫∏v --> IC/M Iff / ≡ ∑’s •…(The universal isomorphic algorithm equals the sum of the integral product variables, impling the isomorphic center of mass, if and only if divided identical to the sum’s center…)(Formula requires reworking)

Thanks, L.A.Gillespie mailto:lloyd.gillespi@gmail.com
Home page – blog - & http://theawakeningoftheamericamind.blogspot.com/

A Search For Universal Justice...
Wisdom Logic - The Final Stage of Philosophy - A Universal Wisdom of Universal Justice Can Be Taught...
The Universal Super-Conscious Soul and Mind...
The Many Levels of Perception...
A Newtonian Universal Capitalist Symmetry...
New Beginnings..."--- Has anyone considered the real depth of investigation needed to reveal the true root causes of our dire political problems? Are we truly intelligent enough to understand the depth of subject required? What is the overwhelming root force/entity and its evolution to the present quagmire? Does anyone even care?"
"--- The neural generation. Could a whole world be wrong? As we economically drift toward WWIII.... How do we talk? The male and female spirit battle is blocking the advance of truth and technical logic. Look outside yourself. There's a whole new world. The problem - "We only understand our own thinking and no one else's and can't or won't learn." Everyone is too insecure to see the truth. The world is too insecure to see through itself to the truth. People - Defending their own thinking - Life - Why talk? - All we're going to do is defend our own thinking! Has this created the poisoned American spirit? Drop subconscious's and go conscious."
"---There's a wolf in the system... He was born of your laws. He roams from Maine to California - Alaska to Florida - Hawaii to D.C., and Chicago to New York... He is a hungry wolf. He tears into your hind quarters, clear to the bone, with a vicious set of teeth. He is simply after your wallet. He is the [international] speculation wolf, and he operates legally under your floating exchange law system, to rip the very soul from your nation. He will succeed unless you try to understand how he feeds............"
"--- We need a fixed value monetary system. At the present time, we have none. Under floating exchanges, America is simply a powerful ship on an ocean, with no rudder. Old gold, silver, and other known standards will no longer work. They will not work due to the massive increases in communication's speed, the varied endowments of nations' natural resources, and encrypted international speculative opportunities. Therefore, we need a new system. INTERNAL EXCHANGE CLEARING is such a system. It is an entirely new fixed value enhancing - [production standard] - monetary system, to benefit all humankind."
"--- The citizen of the ideal state will require a currency for the purpose of every day expenses; This is practically indispensable for workers of all kinds and for such purposes as the payment of wages to wage earners. To meet these requirements, the citizen will possess a currency which will pass for value among themselves, but will not be accepted outside their own boundaries. But a stock of some currency common to the Hellenic world generally i.e., of international currency, will at all times be kept by the state for military expenditures or official missions abroad such as embassies and for any other necessary purposes of state. If a private citizen has occasion to go abroad, he will make his application to the government and go; and upon his return if he has any foreign currency left over in his possession, he will hand it over to the state receiving in exchange the equivalent in local currency." Plato
"--- Intelligence wandered into town one day, looking for a friend. After meeting mediocrity and inanity, any town's overwhelming majority, he wondered even if his quest were possible? Already having visited many other towns and cities and finding no luck in meeting a friend, his spirit was growing dim; however, he could not stop, but where was he to look..................?"

The Exist__The Photon…

The Exist__The Photon…
Hi, I’m a photon. You all know me from seeing visible light. I’m also much, much more. I’m all you absolutely know. I’m everywhere, and I mean everywhere. I’m omnipotent, omniscient, and omnipresent__you live inside me, and I live inside you__There’s no escape__I’m all of the all of everything. I do all good and bad, true and false, positive and negative, etc., etc., etc… I even lie, steal and cheat__love, care and give birth to everyone of you, and all your creatures and plants. Really, I’m miraculous, but I’m also miraculously simple__How else could I do everything possible…??? Even Star-Trek's Q’s got nothing on me__I’m perfect…

You awake most mornings to my shining face, glistening on crystal clear lakes, awakening the morning life in you__Me, lil’ ol' me. You climb a mountain, take in the view__and you only see the beauty by my attributes of em-waves, of so many frequencies, I don’t even know how many ways I can be divided and added to__It’s a long ongoing story__Eternal really. All I know is I form all you see in the entire Universe__It’s all me, your lil’ ol’ photon. I’m the single Universal state of all states, and all state changes. I form all the other particles, within all electrons, positrons, protons and neutrons__in fact, I am them in toto__Ain’t I something…???

You’ve all been puzzling for millennia on what makes up the finest structures of the Universe__Well here I am, get a good look__Watch me shine most every day, lighting up your life__and giving you every ounce of your life. Follow me through any instrument of your choice to see all my diverse frequencies, harmonics and motions. In rest state, I’m spinning so fast it makes even me dizzy, so I continue to radiate away a bit o’ sweat as rads__It’s a really difficult job, to hold myself so tight, for so many billions of years__But, I do venture around as rads. I get the bird's eye view of every trick in the book of nature, and you oughta’ see what I’ve seen__Oh my…!!! I’ve tricked you for millennia to my identity, which I designed in you, so as to entertain myself and create all the mysteries in your lives. I knew you’d be a bit slow in discovering what you live in, and are, just as a fish can not know he’s living in water__you’ve evidently never learned you're living in me__The humble and arrogant lil’ ol’ photon__Well, that’s kind of an under-statement__Since I’m also the infinite all, everywhere to eternity__Btw, that’s the end of time to you__But, there ain’t really no end__I just recycle my-self__Round and round the Universal merry-go-round__That’s why it’s Universal, ya know…!!!

I love beings, and I explode things__How else would you expect a photon to get his kicks? I got a sense of humor, just as you, since I built you. Don’t you think it’s about time you opened your eyes to see me?__I’m quite cute... I’m the solid non-viscous fluidic state of all states you know, live in, and your entire Universe is made of and exists in__What more do you want...??? Without me, you ain’t you, and your Universe ain’t at all. Without you, I’m still me, but truthfully, it’d be a bit lonely without you, since I’ve got to know you, in all your frailties of intellect and passions__over the many millennia__you ain’t changed much, down deep... So, I think I’ll keep you. Well honestly on second thought, I can’t get rid of you, without your help, because I lack free-will without your miraculous bio-bodies and brains. Gee, I’d really be lonely without all you inquisitive creatures…

Will you keep me company__Please…???

Why Biology Can Not Be Mechanized...

 Mr. Herbert Spencer wishes to explain evolution upon mechanical principles. This is illogical, for four reasons. First, because the principle of evolution requires no extraneous cause; since the tendency to growth can be supposed itself to have grown from an infinitesimal germ accidentally started. Second, because law ought more than anything else to be supposed a result of evolution. Third, because exact law obviously never can produce heterogeneity out of homogeneity; and arbitrary heterogeneity is the feature of the universe the most manifest and characteristic. Fourth, because the law of the conservation of energy is equivalent to the proposition that all operations governed by mechanical laws are reversible; so that an immediate corollary from it is that growth is not explicable by those laws, even if they be not violated in the process of growth. In short, Spencer is not a philosophical evolutionist, but only a half-evolutionist,―or, if you will, only a semi-Spencerian. Now philosophy requires thoroughgoing evolutionism or none. C.S. Peirce

Possibility, Necessity, and Contingency...

Possibility, Necessity, and Contingency; from IEP...

To expose the mistakes in the deterministic arguments, we will need some tools of modern logic. Some elementary symbols will help to illuminate the concepts at play in the deterministic arguments. However, all the formulas that will be used, which incorporate these symbols, will also be expressed in English prose.

Symbol	Its meaning	Explanation
P, Q, R, …	propositions	See (see Sec. 2 of Truth)
~P	it is not the case that P	Example: It is not the case that copper conducts electricity. (Note: “P” and “~P” have opposite truth-values – whichever is true, the other is false.)
P ⊃ Q	if P, then Q	Example: If she is late, (then) the meeting will be delayed.
gKP	God knows that P	Example: God knows that the Mississippi River flows north to south.

Next we need three concepts at the heart of modern modal logic. The symbols are:

Symbol	Its meaning	Explanation
◊P	it is (logically) possible that P	Example: It is (logically) possible that the United States was defeated in World War II. (Note: Whatever is not self-contradictory is logically possible.)
☐P	It is (logically) necessary that P	Example: It is logically necessary that every number has a double. (Note: If Q is not logically possible, then ~Q is logically necessary.)
∇P	It is contingent that P	Example: It is contingent that the United States purchased Alaska from Russia. (Note: A proposition, Q, is contingent if and only if ◊Q and ◊~Q.)

These latter three concepts require further elaboration.

P is possible (symbolized “◊P”). A proposition, P, is possible if and only if it is not self-contradictory. All propositions that are true are possibly true. In addition, some false propositions are also possibly true, namely those that are false but are not self-contradictory. Some philosophers like to explicate “P is possible” in this way: “There are some possible circumstances in which P is true”. And some philosophers, adopting the terminology popularized by Leibniz (1646-1716), will substitute “worlds” for “circumstances”, yielding “P is true in some possible worlds”. Examples of possibly true propositions include:

1. Ottawa, Canada, is north of Washington, DC.
2. The Great Salt Lake is saltier than the Dead Sea.
3. The Dead Sea is saltier than the Great Salt Lake.
4. John Lennon was the first songwriter to travel in a space capsule.
5. There are three times as many species of insect as there are species of mollusk.
6. 2 + 2 = 4
7. All aunts are female.
8. Some pigs can levitate.

Understand that prefacing a proposition, P, with “◊” does not ‘make’ P possible. What it does is to create a new, different, proposition, namely ◊P, which, in effect, says that P is possible. If P is possible (for example, suppose “P” stands for “Gold was first discovered in California in 1990″), then (although P is false), ◊P is true. Or, suppose “Q” stands for “2 + 2 = 7″. Then prefacing “Q” with “◊” does not ‘make’ Q possible. It produces a new proposition, “◊Q”, which is false. Q is, and remains, impossible whether or not it is prefaced with “◊”.

Everything that is actual (or actually true) is possible (that is, possibly true). But if a proposition is actually false, then it is impossible only if it is self-contradictory; otherwise it is a false contingency, and all contingencies, whether true or false, are possible.

We may ask “What color did Sylvia paint the lawn chair?” We look at the chair and see that she has painted it yellow. Thus it is demonstrable that it is possible that she painted the chair yellow. And its being yellow implies it is false that she painted the chair blue. But the falsity of the proposition that she painted the lawn chair blue in no way precludes that she could have done so. Even though false, it still remains possible that she painted the chair blue.

P is necessary (symbolized “☐P”). Necessarily true propositions are those that are true in all possible circumstances (/worlds)—that is, are not false in any. Necessary truth can be defined in terms of possibility, namely P is necessary if and only if its negation (that is, “~P”) is impossible. In symbols (where “=_df” stands for “is by definition”):

☐P =_df ~◊~P

Examples of necessarily true propositions:

1. 2 + 2 = 4
2. All aunts are female.
3. Whatever is blue is colored.
4. There are either fewer than 20 million stars or there are more than 12 million. (This statement may be unobvious; but if you think about it you may come to see that it cannot be false.)
5. It is false that some triangle has exactly four sides.

P is contingent (symbolized “∇P”). A proposition, P, is contingent if and only if it is both possibly true andpossibly false. Contingent propositions are those that are true in some possible circumstances (/worlds) and are false in some possible circumstances (/worlds). Contingency can be defined in terms of possibility, namely:

∇P =_df ◊P & ◊~P

It is essential to understand that “◊P & ◊~P” does not mean “P is true and false in some possible circumstances (worlds)”. No proposition whatsoever is both true and false in the same set of circumstances (law of non-contradiction). To say that a proposition is contingent is to say that it is true in some possible circumstances and is false in some (other!) circumstances.

Examples:

1. The Boston Red Sox won the World Series in 2002.
2. It is false that the Boston Red Sox won the World Series in 2002.
3. Steel-clad ships can float in the ocean.
4. It is false that steel-clad ships can float in the ocean.

Modal terms and modal status

Terms such as “must”, “has to”, “cannot”, “is necessary”, “is impossible”, “could not be otherwise”, “has to be”, “might”, “could be”, “contingent”, and the like, are known as “modal” terms. All of these are definable in terms of “possibility”.

Every proposition is either logically possible or logically impossible. And no proposition is both.

Drawing the net a bit finer, and dividing the class of logically possible propositions into those that are necessarily true and those that are contingent, we have three exclusive categories. Every proposition is exclusively either necessarily true, necessarily false, or contingent. That is, every proposition falls into one of these latter three categories, and no proposition falls into more than one.

Just as the expression “truth-value” is a generic term encompassing “truth” and “falsity”, the expression “modal status” is a generic term encompassing “contingent”, “necessarily true”, and “necessarily false”.

Finally, no proposition ever changes its modal status. We will call this principle “The Principle of the Fixity of Modal Status“. And for the purposes of assessing the deterministic arguments we note especially: no contingent proposition ever ‘becomes’ necessary or impossible.

6. The Modal Fallacy

From a mathematical point of view, if we arbitrarily pick any two propositions, truth and falsity can be attributed to them in four different combinations, specifically

• the first is true, and the second is true
• the first is true, and the second is false
• the first is false, and the second is true
• the first is false, and the second is false

However, it sometimes happens that two propositions will have certain logical relationships between them such as to make one or more of these four combinations impossible. For example, consider the two propositions α and β.

α: Diane planted only six rosebushes.β: Diane planted fewer than eight rosebushes.

While each of these propositions, by itself, could be true and could be false, there are – as it turns out – only three, not four, possible combinations of truth and falsity that can be attributed to this particular pair of propositions. On careful thought, we can see that the second combination – that is, the one which attributes truth to α and falsity to β – is impossible. For if α is true (that is, if it is true that Diane has planted only six rosebushes) then β is also true. Put another way: the truth of α guarantees the truth of β. This is to say

(1) It is impossible (for α to be true and for β to be false).

Unfortunately, ordinary English does not lend itself easily to express the quasi-symbolic sentence (1). In symbols we can express the sentence this way:

(1a) ~◊(α & ~β)

About the best we can do in English is to create the following unidiomatic, extremely clumsy sentence:

(1b) The compound sentence, α and not-β, is impossible (that is, is necessarily false).

English prose is a poor tool for expressing fine logical distinctions (just as it is an unsuitable tool for expressing fine mathematical distinctions^[3] ). But, as it turns out, the situation is worse than just having to make do with awkward sentences. For it is a curious fact about most natural languages – English, French, Hebrew, etc. – that when we use modal terms in ordinary speech, we often do so in logically misleading ways. Just see how natural it is to try to formulate the preceding point [namely proposition (1)] in this fashion:

(2) If α is true, then it is impossible for β to be false.

Or, in symbols:

(2a) α ⊃ ~◊~β

In ordinary speech, the latter sentence, (2), is natural and idiomatic; the former sentence (1b) is unnatural and unidiomatic. But – and this is the crucial point – the propositions expressed by (1)-(1b) are not equivalent to the propositions expressed by sentences (2)-(2a). The former set, that is (1)-(1b), are all true. The latter, (2)-(2a)are false and commit the modal fallacy. The fallacy occurs in its assigning the modality of impossibility, not to the relationship between the truth of α and falsity of β as is done in (1)-(1b), but to the falsity of β alone. Ordinary grammar beguiles us and misleads us. It makes us believe that if α is true, then it is impossible for β to be false. But it is possible for β to be false. β is a contingent proposition. Recall the principle of the fixity of modal status. Even if the falsity of β is guaranteed by the truth of some other proposition [in this case α], β doesnot ‘become’ impossible: it ‘remains’ contingent, and thereby possible.

Whatever impossibility there is lies in jointly asserting α and denying β. (See (1b) above.) The proposition “it is false that β” does not ‘become’ impossible if one asserts α.^[4]

a. The Modal Fallacy in Logical Determinism

Some persons have been deceived by the following (fallacious) argument to the effect that there are no contingent propositions:

“(By the Law of Non-contradiction), if a proposition is true (/false), then it cannot be false (/true). If a proposition cannot be false (/true), then it is necessarily true (/false). Therefore if a proposition is true (/false), it is necessarily true (/false). That is, there are no contingent propositions. Every proposition is either necessarily true or necessarily false. (If we could see the world from God’s viewpoint, we would see the necessity of everything. Contingency is simply an artifact of ignorance. Contingency disappears with complete knowledge.)”

The fallacy arises in the ambiguity of the first premise. If we interpret it close to the English, we get:

P ⊃ ~◊~P
~◊~P ⊃ ☐P

---

∴ P ⊃ ☐ P

However, if we regard the English as misleading, as assigning a necessity to what is simply nothing more than a necessary condition, then we get instead as our premises:

~◊(P & ~P) [equivalently: ☐(P ⊃ P)]
~◊~P ⊃ ☐P

From these latter two premises, one cannot validly infer the conclusion:

P ⊃ ☐P.

In short, the argument to the effect that there are no contingent propositions is unsound. Its very first premise commits the
modal fallacy.

The identical error occurs in the argument for logical determinism. Recall (the expanded version of) Aristotle’s sea battle:

Two warring admirals, A and B, are preparing their fleets for a decisive sea battle tomorrow. The battle will be fought until one side is victorious. But the “logical laws (or principles)” of the excluded middle (every proposition is either true or false) and of noncontradiction (no proposition is both true and false), require that one of the propositions, “A wins” and “it is false that A wins,” is true and the other is false. Suppose “A wins” is (today) true. Then whatever A does (or fails to do) today will make no difference: A must win; similarly, whatever B does (or fails to do) today will make no difference: the outcome is already settled (that is, A must win). Or again, suppose “A wins” is (today) false. Then no matter what A does today (or fails to do), it will make no difference: A must lose; similarly, no matter what B does (or fails to do), it will make no difference: the outcome is already settled (that is, A must lose). Thus, if every proposition is either true or false (and not both), then planning, or as Aristotle put it “taking trouble,” is futile. The future will be what it will be, irrespective of our planning, intentions, etc.

If we let “A” stand for “Admiral A wins” and let “B” stand for “Admiral B wins”, the core of this argument can be stated in symbols this way:

A or B		[one or the other of these two propositions is true]
~◊(A & B)		[it is not possible that both A and B are true]

∴	A ⊃ ☐A A ⊃ ~◊~A	}	If A is true, then A must be true. If A is true, then A cannot be false.
	A ⊃ ☐~B A ⊃ ~◊B	}	If A is true, then B must be false. If A is true, then B cannot be true.
	B ⊃ ☐B B ⊃ ~◊~B	}	If B is true, then B must be true. If B is true, then B cannot be false.
	B ⊃ ☐~A B ⊃ ~◊A	}	If B is true, then A must be false. If B is true, then A cannot be true.

In this argument, by hypothesis, either A is true or B is true, and since they cannot both be true, the second premise may be accepted as true. But none of the conclusions is true. A is contingent, and B is contingent. Yet the conclusions state that from the assumed truth of either of (the two contingencies) A or B, it follows that A and B are each either necessarily true or necessarily false. Each of these eight conclusions violates the principle of the fixity of modal status. What, then, are the conclusions one may draw validly from the premises? These:

☐(A ⊃ ~B)	or, equivalently,	~◊(A & B)
☐(B ⊃ ~A)	or, equivalently,	~◊(B & A)

So long as we remain mindful of the fact that “~◊(P & Q)” is logically equivalent to “☐(P ⊃ ~Q)” but is not equivalent to “P ⊃ ☐~Q”, the argument for logical determinism will be seen to be invalid. Our ordinary language treats “it is impossible for both P and Q to be true” as if it were logically equivalent to “if P is true, then Q is necessarily false”. But the profound difference between these two assertions is that the former preserves the principle of the fixity of modal status, the latter violates that principle. The proposition, “Admiral A wins”, is contingent, and if true, then it “remains” true. Indeed this is a trivial logical truth:

(i) ☐(P ⊃ P) alternatively, ~◊(P & ~P)

The argument for logical determinism illicitly treats this logical truth as if it were equivalent to the false proposition

(ii) P ⊃ ☐P alternatively, P ⊃ ~◊~P

If you do not let yourself be beguiled by the invalid ‘move’ (inference) from (i) to (ii), the argument for logical determinism collapses. The truth of a proposition concerning your future behavior does not make that future behavior necessary. What you choose to do in the future was, is, and will remain contingent, even if a proposition describing that choice is timelessly true.

b. The Modal Fallacy in Epistemic Determinism

Let’s recall Maimonides’s argument:

… “Does God know or does He not know that a certain individual will be good or bad? If thou sayest ‘He knows’, then it necessarily follows that [that] man is compelled to act as God knew beforehand he would act, otherwise God’s knowledge would be imperfect.”

We can symbolize the core of this argument, using “∴” for “it necessarily follows”; and “☐” for “compelled”; and “D” for the proposition describing what some particular person does tomorrow.

gKD

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∴ ☐D

There seems to be (at least) one missing premise. [In the terminology of logicians, the argument isenthymematic.] One tacit assumption of this argument is the necessary truth, “it is not possible both for God to know that D and for D to be false”, or, in symbols, “~◊(gKD & ~D)”. So the argument becomes:

gKD
~◊(gKD & ~D)

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∴ ☐D

But even with this repair, the argument remains invalid. The conclusion does not follow from the two premises. To derive the conclusion, a third premise is needed, and it is easy to see what it is. Most persons, with hardly a moment’s thought, virtually as a reflex action, will tacitly assume that the second premise is logically equivalent to:

gKD ⊃ ☐D

and will tacitly (/unconsciously) add this further premise, so as to yield, finally:

gKD
~◊(gKD & ~D)
gKD ⊃ ☐D

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∴ ☐D

But this third premise, we have seen above, is false; it commits the modal fallacy. Without this premise, Maimonides’ argument is invalid; with it, the argument becomes valid but unsound (that is, has a false and essential premise [namely the third one]). Either way, the argument is a logical botch.

Once the logical error is detected, and removed, the argument for epistemic determinism simply collapses. If some future action/choice is known prior to its occurrence, that event does not thereby become “necessary”, “compelled”, “forced”, or what have you. Inasmuch as its description was, is, and will remain forever contingent, both it and its negation remain possible. Of course only one of the two was, is, and will remain true; while the other was, is, and will remain false. But truth and falsity, per se, do not determine a proposition’s modality. Whether true or false, each of these propositions was, is, and will remain possible. Knowing – whether by God or a human being – some future event no more forces that event to occur than our learning that dinosaurs lived in (what is now) South Dakota forced those reptiles to take up residence there.