An ontological investigation of unimaginable events
aa r X i v : . [ q -f i n . R M ] J un An ontological investigation of unimaginable events
Thomas Santoli , Christoph Siebenbrunner Abstract
We show that, under mild assumptions, some unimaginable events - which we refer to as Black Swanevents - must necessarily occur. It follows as a corollary of our theorem that any computational modelof decision-making under uncertainty is incomplete in the sense that not all events that occur can betaken into account. In the context of decision theory we argue that this constitutes a stronger sense ofuncertainty than Knightian uncertainty.
Keywords: ontology; risk theory; decision theory.
1. Introduction
Ontology, loosely defined as “ the study of what there is ” (Hofweber, 2014), studies questions of theexistence of entities, their properties, and the relation between the two (Hofweber, 2014). So-called ontological arguments traditionally are proofs of the existence of god, deducing this conclusion from aset of properties attributed to the entity ‘god’ . In this article we apply the same technique to show theexistence of so-called Black Swan events.The notion of Black Swan events, originally introduced in Taleb (2005), has been popularized by aseries of popular science books (Taleb, 2016). Its formalization is work in progress by Taleb (2017).Their naming makes reference to the so-called problem of induction , defined by the Oxford EnglishDictionary as “ the process of inferring a general law or principle from the observation of particularinstances ” (OED, 2017). It is attributed to David Hume, who stated that such arguments cannot bemade rigorous by deductive reasoning alone, given the lack for a justification of the assumption that yetunobserved entities will share the same properties as those already observed (Hume and Selby-Bigge,1896). The solution to the problem that this causes to scientific reasoning proposed by Popper (1935)is to replace induction with falsification, the process of continuously trying to find empirical evidenceagainst the theses of a scientific theory (Vickers, 2016). This method was later rejected by critics as
Email address: [email protected] (Christoph Siebenbrunner) University of Oxford, Mathematical Institute. All views expressed herein are those of the authors and do not necessarilyreflect the views of any a ffi liating organization. See St. St. Anselm’s proof (Anselm and Deane, 1998) for the most well-known example, orBenzm¨uller and Woltzenlogel Paleo (2014) for a recent presentation of such a proof originally devised by Kurt G¨odel. elying on a “ whi ff ” (some extent) of inductivism itself (Newton-Smith, 1981; Salmon, 1981).The need for induction as part of the process as scientific discovery was already discussed by Aristotle,who argued that scientists should infer explanatory principles from phenomena in order to deduce furtherstatements about them. Aristotle’s method was generally accepted by medieval thinkers and many ver-sions of such methods were presented by philosophers including Roger Bacon, Duns Scotus and Williamof Ockham, amongst others. The di ffi culty of arriving at general truths instead of accidental general-izations was generally understood by these authors (Losee, 2001). A common way of presenting thisproblem is to point out that the fact that all swans observed hitherto (in Europe) were white could lead anobserver to induce that all swans are white – a case of accidental predication that Duns Scotus seeked toavoid by stating that the most that could be inferred from observations was their “ aptitudinal union ”, inthis case that swans could be white (Losee, 2001). In this sense, the discovery of a species of black swansin Australia – cygnus atratus – during the voyages of the 17th century is a good exemplification of thisproblem. First accounts of such sightings by the Dutch skipper Antony Caen in 1636 were met with skep-ticism back home, until Willem de Vlamingh brought back real specimen to the continent over 60 yearslater (Olsen, 2001). Taleb (2005) states this historical context as a reason for choosing cygnus atratus asthe namesake for events with large impact, incomputable probabilities, and surprise e ff ect properties.The theory of Black Swan events, under development by Taleb (2017), has already made significantimpact in popular language and general media. In this article we undertake to demonstrate that theoccurrence of such events is in fact implied by their definition. Several authors have written about BlackSwan events from a statistical and risk (management) theory perspective (apart from the already citedworks, further examples include e.g. Nafday (2009, 2011); Hilal et al. (2011); Taleb et al. (2012); Aven(2013)). Taleb (2005); Yudkowsky (2008) discuss Black Swans in the context of human cognition and itslimitations. However, there are – to the best of our knowledge – no works looking at Black Swans froman analytical perspective.
2. Definitions
A Black Swan is defined by Taleb (2017) as “ a) a highly unexpected event [f] or a given observer [that] b) carries large consequences, and c) is subjected to ex-post rationalization ”.In order to formalize this definition, we define a set X of all events. We define the predicate χ ( x ) todenote that the event x ∈ X can be imagined (by a given observer). We will discuss further below whatit means in the context of our model to be able to imagine an event. In order to discuss the occurrenceor non-occurrence of events we further define ϕ ( x ) to denote an event that occurs . We make very fewassumptions about X , which are summarized by the axioms presented in section 3. Note that we do notinclude a notion of time in our theory, i.e. the chronological ordering of the occurrence of an event andits imagination by an observer carries no importance. We associate the non-imaginability of an eventto the property of being highly unexpected (denoted (a) in the definition by Taleb (2017)). We will seelater that the property of large consequences (b) follows from this definition. We do not include ex-postrationalization (c) in our definition, as we do not consider it essential to the ontological nature of Black2wans. If we denote by B ( x ) the property that event x is a Black Swan, then its definition in our theoryreads: B ( x ) ⇔ ¬ χ ( x )In order to be able to discuss the ‘size’ of consequences we introduce the partial order < which satisfiesthe following axioms for all elements a , b , c ∈ X : • Irreflexivity: ¬ ( a < a ) • Transitivity: ( a < b ) ∧ ( b < c ) ⇒ a < c Other than assuming a strict rather than a weak partial order, which we do for technical convenience here,this relation is consistent with von-Neumann-Morgenstern utility theory (von Neumann and Morgenstern,1944). It shares the important transitivity property, a consistent equivalent for continuity under a strictpartial order can be formulated, completeness and independence are not required but fully consistent withour theory. In this spirit, we will sometimes treat the notion of event y having greater consequences thanevent x , i.e. x < y , as semantically equivalent y being ‘worse’ (i.e. yielding lower utility for a givenobserver) than x . This semantic interpretation does not a ff ect the generality of the argument, and westress that our argument does not require that the size of an event is in any way related to its utility for anobserver. This interpretation does, however, help to emphasize the particular importance of unimaginableevents when they are associated with negative outcomes (for a given observer).Our definition of Black Swan events may be seen as stricter than that of Taleb (2017) in one sense, asnon-imaginability can be seen as a stronger requirement than being highly unexpected. It may also beseen as wider in the sense that it does not require ex-post rationalization. In any case, we consider it ahighly important class of events, as becomes clear when it is viewed in the context of decision-makingunder risk and uncertainty.So-called Knightian uncertainty refers to the non-quantifiability of phenomena under conditions ofuncertainty (Knight, 1921). Uncertainty in decision theory is often interpreted in the sense that theprobability distribution over future events is unknown (i.e. not allowing for probabilistic calculations thatwould be possible under conditions of risk), while the set of possible future states and their respectivepayo ff s are still known. This allows for the application of non-probabilistic computation models such asWald’s maximin-criterion or similar techniques (Wald, 1939, 1945). Black Swan events, as we considerthem here, require a stronger sense of uncertainty, whereunder not even the full set of potential eventsor their payo ff s are available to the decision maker. This notion may be seen as closer to the originaldefinition of uncertainty by Knight (1921), which states that “ We [...] restrict the term ‘uncertainty’to cases of the non-quantitive type. ”. It should be noted, however, that the emphasis here lies on adi ff erent, arguably even stronger point: the crux of Black Swan events, as defined herein, is not theirnon-quantifiability, but the fact that they cannot be considered in the decision-making process, regardless3f whether quantitative or any other methods are used. What we show in this paper is that there existevents which fundamentally cannot be taken into account when making decisions and which occurnonetheless .In order to formalize this idea, we define a standard computational model of decision-making comprisingthe following elements: • A set A of actions which are available to the agent. • A set P of information associated to the events (such as probabilities, or the property of occurrence). • A utility map Γ which maps every pair of events and actions an outcome for a given agent: Γ : ( A , X ) → O . Note that if we were to take the axioms of von Neumann and Morgenstern (1944), Γ could be made consistent with a non-strict version of the size-relation < introduced before, asshown in their proof. This is, however, not required for the point that we wish to make. • A decision map Φ which maps a vector of outcomes and a vector of associated information tothe set of actions: Φ : ( O n , P n ) → A , where n is the cardinality of the Cartesian product A × X .In accordance with the concept of uncertainty defined above we may assume – without loss ofgenerality – that there is no variation in the set of associated information P and write Φ ( O n ) fornotational convenience.We say that an event being non-imaginable for an agent is equivalent to her not being able to map it toan action. We consider this as being di ff erent from deciding not to react to an event, because the latterentails finishing the computation Γ of an outcome, which can then be mapped to whatever action wouldhave been chosen without knowledge of the event (one may also think of A containing another responselabeled ‘Do Nothing’). We make this distinction because it facilitates the discussion of the computationalaspects of decision-making, which we present in section 3.1, but note that it is not essential to the validityof our argument. For now we present the definition of the decision map, which states that a given set ofevents has to contain at least one imaginable event in order for it to be mapped to an action: Φ ( Γ n ( A , X )) = ↑ if ∀ x ∈ X : ¬ χ ( x ) a ∈ A otherwise , where Γ n ( A , X ) denotes the element-wise application of the Γ map to every element in the Cartesianproduct A × X , and ↑ denotes the fact that the computation has not terminated.
3. Reasoning
Our reasoning can be compactly summarized as follows:4 xiom 1 :There exists (at least) one event that occurs that is so bad that an event with greater consequences cannotbe imagined: ∃ x ( ϕ ( x ) ∧ ∀ y ( x < y → ¬ χ ( y ))) Axiom 2 :No matter how bad an event that occurs is, there exists an event with even greater consequences whichoccurs: ∀ x ( ϕ ( x ) → ∃ y ( ϕ ( y ) ∧ x < y )) Theorem :Black Swan events occur: ∃ x ( B ( x ) ∧ ϕ ( x ))In Appendix A, we present a formal proof of the above argument using a Hilbert-style system. Here wegive a proof via semantic argument. Proof.
By axiom 1, there exists an element x ∈ X such that:1. ϕ ( x )2. ∀ y ( x < y → ¬ χ ( y )).By 1. and axiom 2 we obtain that there exists y ∈ X such that ϕ ( y ) and x < y . By the latter, and by 2., weobtain ¬ χ ( y ). Therefore we have obtained y ∈ X such that ¬ χ ( y ) ∧ ϕ ( y ), that is B ( y ) ∧ ϕ ( y ). This showsthat axioms 1 and 2 imply our Theorem. (cid:3) We now move on to lay out the rationale underpinning our axioms.
Axiom 1 states that all events greater than a particular event (which occurs) are beyond our imagination,regardless of whether they occur or not. We call this argument the
Horatio-Principle , after the followingquote: 5
There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.” - Hamlet (1.5.167-8)
It has been argued that Hamlet in this quote is talking about general limitations of human thought ratherthan trying to insult Horatio’s intellect (Bradley, 1952), and some versions of the text even talk about‘our’ instead of ‘your’ philosophy . The Horatio-Principle here states the existence of events that are‘not dreamt of’ in the ‘philosophy’ of a given observer, i.e. which she cannot imagine.In order to justify the Horatio-Principle, we refer to the computation model described by the decisionmap Φ introduced in section 2. As stated in the definition, the notion of being able to imagine an eventis equivalent to being able to come up with a response to that event. The decision making process couldthus be viewed as a Turing machine (Turing, 1937) that tries to compute a response to a given set ofevents. In this case A would be the set of terminating states of the Turing machine, X would be the setof alphabet symbols, which here also serves as the set of input symbols, and Φ the transition function.As stated by the Halting problem (Turing, 1937), no such machine could be guaranteed to ever reach aterminating state, i.e. to arrive at a decision upon a given set of events. The Horatio principle states (i)that there exist events for which a response is not computable, which is motivated by the Halting problem.Furthermore (ii), it states that there exists a size threshold for the consequences of an event beyond whichthis holds for all events (i.e. that the computability of an event is to some extent proportional to the size ofits outcome). And lastly (iii), it states that there exists an event which does not exceed the size thresholdand which has the property of occurring. Hence, while the Halting problem does not fully extend to theHoratio principle, we consider it a strong motivation thereof.One may be lead to think that the possibility of deliberate Antifragility (Taleb, 2016) might induce atype of Russell paradox (Russell, 1903) in our system: we can imagine the occurrence of a Black Swanevent and thus adjust our strategies accordingly. Antifragile strategies allow us to benefit from such anoccurrence, even though we would not be able to describe the nature of the event in advance. Our set ofimaginable events includes the occurrence of events that are not imaginable. Imagination in this case doesnot specify the event itself, which would thus still be outside the set of imaginable events. Accordingto the Horatio-Principle, this set is still non-empty. In other words, even for (seemingly) antifragilestrategies, there exist events with large consequences for which the outcome cannot be known beforethey occur.
Axiom 2 states that for every event that occurs, an event that is greater occurs as well. We argue thatthe set of occurring events is an open set and call this assumption
Murphy’s law , the often humorouslystated aphorism that “ anything that can go wrong will go wrong ”. Formally, Axiom 2 can be derivedfrom Murphy’s law, as expressed in our system, by using an additional assumption that set of events X is The Folio version of Hamlet uses ‘our’, while the first and second Quarto versions use ‘your’. While it is unclear whetherthis di ff erence stems from an editor’s mistake, it is commonly understood that the usually adopted ‘your’ is meant as a generaladdress and not and not as a direct attack on Horatio (Thompson and Taylor, 2014).
6n open set (the formal proof is left to the reader): • ∀ x , y ( φ ( x ) ∧ ( x < y ) ⊃ φ ( y ))Murphy’s law • ∀ x ∃ y ( x < y )Open universe • ( ∀ x ) (cid:16) ϕ ( x ) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) (cid:17) Axiom 2
4. Implications
In this section we show that it follows as a corollary of our Theorem that every computational model ofdecision-making is incomplete in the sense that not all events can be taken into account, and that this alsoconcerns events that do occur.A computational model is said to be complete if the decision map Φ has the property that for every twosets of events which di ff er by at least one element Y , Z : ( ∃ y ∈ Y : y < Z ) ∨ ( ∃ z ∈ Z : z < Y ) there exist aset of actions A and a map Γ such that Φ ( Γ n ( A , Y )) , Φ ( Γ n ( A , Z )). This means that a complete decision-model always allows an agent to act di ff erently under di ff erent circumstances if this is indicated by herpreferences, as expressed by the utility map Γ . In other words, a complete decision map Φ ( Γ n ( A , X ))takes into account all events in X . A more refined concept of completeness with respect to occurringevents only requires this for events that have the property of occurring, which means that for Y , Z suchthat ∀ y ∈ Y : ϕ ( y ) and ∀ z ∈ Z : ϕ ( z ), we have Φ ( Γ n ( A , Y )) , Φ ( Γ n ( A , Z )) if and only if Y , Z .Assume for the sake of contradiction that there exists a computational model with a complete decisionmap Φ : ( O n , P n ) → A , as defined above. Let S ⊆ X be a set of events which occur and which arenot imaginable (i.e. Black Swan events) for a given agent: S = { s : s ∈ X ∧ ϕ ( s ) ∧ ¬ χ ( s ) } . It followsfrom our theorem that the set S is non-empty. It further follows from Axiom 2 that for every eventin S there exists an event with greater consequences that occurs as well. The cardinality of S is thusat least ℵ . Thus there exist two vectors of unimaginable events which di ff er by at least one element Y , Z ∈ S j : ( ∃ y ∈ Y : y < Z ) ∨ ( ∃ z ∈ Z : z < Y ). By the definition of Φ , both vectors are not mapped to anyresponse, because the computation will not terminate. Therefore it is impossible to have Φ ( Γ n ( A , Y )) , Φ ( Γ n ( A , Z )), contradicting the assumption of the existence of a complete decision-making model withrespect to occurring events. The contradiction of the existence of a complete decision-making model canbe obtained by setting S = { s : s ∈ X ∧ ¬ χ ( s ) } .
5. Conclusion
We developed a first-order deductive system to show that the occurrence of Black Swan events is impliedby their definition. We make two assumptions, namely that our imagination is bounded and that the7niverse of occurring events is an open set, which we call the Horatio Principle and Murphy’s law,respectively. We motivate the Horatio principle by showing that under a computational model of humandecision-making, the question of whether all events are imaginable can be reduced to the Halting problem.We present a formal proof of our argument using a Hilbert System. We show that it follows as a corollaryof this Theorem that every computational model of decision-making is incomplete in the sense that notall events that occur can be taken into account in the decision-making process. When viewed throughthe lense of decision-making under uncertainty – as in von-Neumann-Morgenstern utility theory – weargue that Black Swans entail a stronger sense of uncertainty than Knightian uncertainty because theirexistence means that even under perfect information no decision criterion – regardless of whether it is ofquantitative nature or not – can make use of all the information available.
Acknowledgments
The idea for this paper developed out of a conversation with Davoud Taghawi-Nejad at the Institutefor New Economic Thinking at the Oxford Martin School. We thank Prof. Timothy Williamson forhis comments on the paper. We further thank Matthew Deakin and G¨unther Siebenbrunner for usefulremarks that have been incorporated into the paper.
ReferencesReferences
Anselm and Deane, S. N. (1998).
St. Anselm Basic Writings: Proslogium, Mologium, Gaunilo’s In Behalfof the Fool, Cur Deus Homo . Open Court.Aven, T. (2013). On the meaning of a black swan in a risk context.
Safety Science , 57:44–51.Benzm ¨uller, C. and Woltzenlogel Paleo, B. (2014). Automating g¨odel’s ontological proof of god’s ex-istence with higher-order automated theorem provers. In
Proceedings of the Twenty-first EuropeanConference on Artificial Intelligence , pages 93–98. IOS Press.Bradley, A. (1952).
Shakespearean Tragedy. Lectures on Hamlet, Othello, King Lear, Macbeth . MacMil-lian, London.Hilal, S., Poon, S.-H., and Tawn, J. (2011). Hedging the black swan: Conditional heteroskedasticity andtail dependence in s&p500 and vix.
Journal of Banking and Finance , 35(9):2374 – 2387.Hofweber, T. (2014). Logic and ontology. In Zalta, E. N., editor,
The Stanford Encyclopedia of Philoso-phy . Metaphysics Research Lab, Stanford University, fall 2014 edition.Hume, D. and Selby-Bigge, L. (1896).
A Treatise of Human Nature by David Hume, reprinted from theOriginal Edition in three volumes and edited, with an analytical index . Oxford: Clarendon Press.8night, F. (1921).
Risk, uncertainty and profit . Houghton Mi ffl in Company, Boston, New York.Losee, J. (2001). A historical introduction to the philosophy of science . Oxford University Press, Oxford,United Kingdom.Nafday, A. M. (2009). Strategies for managing the consequences of black swan events.
Leadership andManagement in Engineering , 9(4):191–197.Nafday, A. M. (2011). Consequence-based structural design approach for black swan events.
StructuralSafety , 33(1):108 – 114.Newton-Smith, W. (1981).
The rationality of science . Routledge, London, New York.OED (2017). induction, n. [Logic] . Oxford University Press. Accessed online October 09, 2017.Olsen, P. (2001).
Feather and Brush: Three Centuries of Australian Bird Art . Csiro Publishing.Popper, K. (1935). Logik der forschung.
Journal of Philosophy , 32(4):107–108.Russell, B. (1903).
The Principles of Mathematics (Classic Reprint) . Forgotten Books (16 Nov. 2016).Salmon, W. C. (1981). Rational Prediction.
The British Journal for the Philosophy of Science , 32(2):115–125.Taleb, N.-N. (2005).
The Black Swan: Why Don’t We Learn that We Don’t Learn?
Random House, NewYork.Taleb, N.-N. (2016).
Incerto: Fooled by Randomness, The Black Swan, The Bed of Procrustes, Antifrag-ile . Random House.Taleb, N.-N. (2017).
Silent Risk . available at . Accessed online October09, 2017.Taleb, N. N., Canetti, E., Kinda, T., Loukoianova, E., and Schmieder, C. (2012). A New HeuristicMeasure of Fragility and Tail Risks: Application to Stress Testing.
IMF Working Paper 12 / .Thompson, A. and Taylor, N. (2014). Hamlet . Bloomsbury Arden Shakespeare, London.Turing, A. M. (1937). On Computable Numbers, with an Application to the Entscheidungsproblem.
Proceedings of the London Mathematical Society , s2-42(1):230–265.Vickers, J. (2016). The problem of induction. In Zalta, E. N., editor,
The Stanford Encyclopedia ofPhilosophy . Metaphysics Research Lab, Stanford University, spring 2016 edition.von Neumann, J. and Morgenstern, O. (1944).
Theory of Games and Economic Behavior . PrincetonUniversity Press, Princeton, NJ.Wald, A. (1939). Contributions to the Theory of Statistical Estimation and Testing Hypotheses.
TheAnnals of Mathematical Statistics , 10(4):299–326.9ald, A. (1945). Statistical Decision Functions Which Minimize the Maximum Risk.
Annals of Mathe-matics , 46(2):265–280.Yudkowsky, E. (2008). Cognitive biases potentially a ff ecting judgment of global risks. In Bostrom, N.and Cirkovic, M. M., editors, Global catastrophic risks . Oxford University Press.
Appendix A. Formal proof
In this section we present the formal system which in which we will establish the theorem of the occur-rence of Black Swan events and its proof.
Appendix A.1. Hilbert System
We use a deductive system for first-order logic. This system consists of the following axioms and rules.
Appendix A.1.1. Axioms (FO1) A ⊃ ( B ⊃ A )(FO2) ( A ⊃ B ) ⊃ ( A ⊃ ( B ⊃ C )) ⊃ ( A ⊃ C )(FO3) A ⊃ ( A ∨ B )(FO4) B ⊃ ( A ∨ B )(FO5) ( A ⊃ C ) ⊃ ( B ⊃ C ) ⊃ ( A ∨ B ⊃ C )(FO6) ( A ⊃ B ) ⊃ ( A ⊃ ¬ B ) ⊃ ¬ A (FO7) ¬¬ A ⊃ A (FO8) A ∧ B ⊃ A (FO9) A ∧ B ⊃ B (FO10) A ⊃ B ⊃ ( A ∧ B )(FO11) A ( t ) ⊃ ( ∃ x ) A ( x ) where t can be any term(FO12) ( ∀ x ) A ( a ) ⊃ A ( t ) where t can be any term 10 ppendix A.1.2. Rules (MP) A A ⊃ BB (R1) C ⊃ A ( x ) C ⊃ ( ∀ x ) A ( x ) where the variable x must not occur free in C (R2) A ( x ) ⊃ C ( ∃ x ) A ( x ) ⊃ C where the variable x must not occur free in C (R3) ( A ∧ B ) ⊃ CB ⊃ A ⊃ C Appendix A.2. Proof of the theorem
Using the deductive system described above, we show a proof of our theoremThm : = ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) . Our proof will be a list of formulas A , . . . , A n such that: • A n = Thm • for every i = . . . n , A i is either one of the axioms (FO1)–(FO12), or it is one of our two axiomsAx1 : = ( ∃ x ) (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) Ax2 : = ( ∀ x ) (cid:16) ϕ ( x ) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) (cid:17) , or it is deduced by applying one of the rules (MP),(R1)–(R3) to some formula(s) A , . . . , A i − . Proof.
1. Ax2 ⊃ (cid:16) ϕ ( x ) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) (cid:17) axiom (FO12)2. Ax2axiom (Ax2)3. ϕ ( x ) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y ))by (MP) from 2. and 1.4. (cid:16) ϕ ( x ) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) (cid:17) ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (cid:16) ϕ ( x ) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) (cid:17)i axiom (FO1) 11. (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (cid:16) ϕ ( x ) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) (cid:17) by (MP) from 3. and 4.6. (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ϕ ( x )axiom (FO8)7. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ϕ ( x ) i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (cid:16) ϕ ( x ) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) (cid:17)i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) i axiom (FO2)8. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (cid:16) ϕ ( x ) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) (cid:17)i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) i by (MP) from 6. and 7.9. (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y ))by (MP) from 5. and 8.10. ( ϕ ( y ) ∧ ( x < y )) ⊃ ( x < y )axiom (FO9)11. (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y ))axiom (FO9)12. ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) ⊃ (( x < y ) ⊃ ¬ χ ( y ))axiom (FO12)13. h ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) ⊃ (( x < y ) ⊃ ¬ χ ( y )) i ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ h ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) ⊃ (( x < y ) ⊃ ¬ χ ( y )) i axiom (FO1)14. (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ h ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) ⊃ (( x < y ) ⊃ ¬ χ ( y )) i by (MP) from 12. and 13.15. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ h ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) ⊃ (( x < y ) ⊃ ¬ χ ( y )) ii ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (( x < y ) ⊃ ¬ χ ( y )) i axiom (FO2)16. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ h ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) ⊃ (( x < y ) ⊃ ¬ χ ( y )) ii ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (( x < y ) ⊃ ¬ χ ( y )) i by (MP) from 11. and 15.17. (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (( x < y ) ⊃ ¬ χ ( y ))by (MP) from 14. and 16. 128. (cid:16) ( ϕ ( y ) ∧ ( x < y )) ⊃ ( x < y ) (cid:17) ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ( x < y )) ⊃ ( x < y ) (cid:17) axiom (FO1)19. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ( x < y )) ⊃ ( x < y ) (cid:17) by (MP) from 10. and 18.20. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( ϕ ( y ) ∧ ( x < y ))axiom (FO9)21. hh(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( ϕ ( y ) ∧ ( x < y )) i ⊃ hh(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ( x < y )) ⊃ ( x < y ) (cid:17)i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( x < y )axiom (FO2)22. hh(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ( x < y )) ⊃ ( x < y ) (cid:17)i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( x < y )by (MP) from 20. and 21.23. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( x < y )by (MP) from 19. and 22.24. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (( x < y ) ⊃ ¬ χ ( y )) i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (( x < y ) ⊃ ¬ χ ( y )) i axiom (FO1)25. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (( x < y ) ⊃ ¬ χ ( y )) i by (MP) from 17. and 24.26. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) axiom (FO8)27. hh(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (( x < y ) ⊃ ¬ χ ( y )) i ⊃ hh(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (( x < y ) ⊃ ¬ χ ( y )) i axiom (FO2) 138. "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ (( x < y ) ⊃ ¬ χ ( y )) i ⊃ hh(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (( x < y ) ⊃ ¬ χ ( y )) i by (MP) from 26. and 27.29. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (( x < y ) ⊃ ¬ χ ( y ))by (MP) from 25. and 28.30. hh(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( x < y ) i ⊃ hh(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (( x < y ) ⊃ ¬ χ ( y )) i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ¬ χ ( y )axiom (FO2)31. hh(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (( x < y ) ⊃ ¬ χ ( y )) i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ¬ χ ( y )by (MP) from 23. and 30.32. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ¬ χ ( y )by (MP) from 29. and 31.33. ( ϕ ( y ) ∧ ( x < y )) ⊃ ϕ ( y )axiom (FO8)34. (cid:16) ( ϕ ( y ) ∧ ( x < y )) ⊃ ϕ ( y ) (cid:17) ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ( x < y )) ⊃ ϕ ( y ) (cid:17) axiom (FO1)35. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ( x < y )) ⊃ ϕ ( y ) (cid:17) by (MP) from 33. and 34.36. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( ϕ ( y ) ∧ ( x < y ))axiom (FO9)37. "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( ϕ ( y ) ∧ ( x < y )) ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ( x < y )) ⊃ ϕ ( y ) (cid:17) ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ϕ ( y ) axiom (FO2) 148. "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ( x < y )) ⊃ ϕ ( y ) (cid:17) ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ϕ ( y ) by (MP) from 36. and 37.39. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ϕ ( y )by (MP) from 35. and 38.40. ϕ ( y ) ⊃ ¬ χ ( y ) ⊃ ( ϕ ( y ) ∧ ¬ χ ( y ))axiom (FO10)41. (cid:16) ϕ ( y ) ⊃ ¬ χ ( y ) ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) (cid:17) ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ϕ ( y ) ⊃ ¬ χ ( y ) ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) (cid:17) axiom (FO1)42. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ϕ ( y ) ⊃ ¬ χ ( y ) ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) (cid:17) by (MP) from 40. and 41.43. "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ϕ ( y ) ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ϕ ( y ) ⊃ ¬ χ ( y ) ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) (cid:17) ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ¬ χ ( y ) ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) (cid:17) axiom (FO2)44. "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ϕ ( y ) ⊃ ¬ χ ( y ) ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) (cid:17) ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ¬ χ ( y ) ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) (cid:17) by (MP) from 39. and 43.45. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ¬ χ ( y ) ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) (cid:17) by (MP) from 42. and 44.46. "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ¬ χ ( y ) ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ¬ χ ( y ) ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) (cid:17) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) axiom (FO2) 157. "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ¬ χ ( y ) ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) (cid:17) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) by (MP) from 32. and 46.48. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( ϕ ( y ) ∧ ¬ χ ( y ))by (MP) from 45. and 47.49. ( ϕ ( y ) ∧ ¬ χ ( y )) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z ))axiom (FO11)50. (cid:16) ( ϕ ( y ) ∧ ¬ χ ( y )) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) (cid:17) ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ¬ χ ( y )) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) (cid:17) axiom (FO1)51. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ¬ χ ( y )) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) (cid:17) by (MP) from 49. and 50.52. "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( ϕ ( y ) ∧ ¬ χ ( y )) ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ¬ χ ( y )) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) (cid:17) ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) axiom (FO2)53. "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ (cid:16) ( ϕ ( y ) ∧ ¬ χ ( y )) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) (cid:17) ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) by (MP) from 48. and 52.54. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ∧ ( ϕ ( y ) ∧ ( x < y )) i ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z ))by (MP) from 51. and 53.55. ( ϕ ( y ) ∧ ( x < y )) ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z ))by (R3) from 54.56. ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z ))by (R2) from 55.57. h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y ))axiom (FO8) 168. h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) i ⊃ h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) i axiom (FO1)59. h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) i by (MP) from 56. and 58.60. "h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ⊃ "h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) i ⊃ "h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) axiom (FO2)61. "h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) i ⊃ "h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) by (MP) from 57. and 60.62. h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z ))by (MP) from 59. and 61.63. h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) axiom (FO9)64. "h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ "h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) ⊃ "h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) axiom (FO2)65. "h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) ⊃ "h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) by (MP) from 63. and 64.66. h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ∧ (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17)i ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z ))by (MP) from 62. and 65.67. (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) i by (R3) from 66.68. h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ y )( ϕ ( y ) ∧ ( x < y )) i ⊃ "(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) i ⊃ "h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) i axiom (FO2)69. "(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ h ( ∃ y )( ϕ ( y ) ∧ ( x < y )) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) i ⊃ h(cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z )) i by (MP) from 9. abd 68.70. (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z ))by (MP) from 67. and 69.71. ( ∃ x ) (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) ⊃ ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z ))by (R2) from 70.72. ∃ x (cid:16) ϕ ( x ) ∧ ( ∀ y )(( x < y ) ⊃ ¬ χ ( y )) (cid:17) axiom (Ax1)73. ( ∃ z )( ϕ ( z ) ∧ ¬ χ ( z ))by (MP) from 72. and 71. (cid:3)(cid:3)