Tomasz Bigaj

This book contains a concise introduction to one of the most fundamental branches of philosophy, which deals with reality and its nature. Among the topics discussed are such metaphysical questions as "Are we fundamentally free?", "Does time really pass?", "Are there any abstract objects?", "What is causation?", "What are necessary and possible truths?". The book is aimed at absolute beginners, so it does not presuppose any previous knowledge of philosophy from the reader. For those who would like to pursue the subject a bit deeper, the book comes equipped with an extended list of further reading.

This is a response to a critical review of my book Non-locality and Possible Worlds (Ontos Verlag, Frankfurt 2006) by Witold Strawiński. I present arguments why counterfactual conditionals are needed in the description of quantum-mechanical phenomena, and in particular in the analysis of the condition of locality. I rebut arguments against my choice of the relation of similarity between possible worlds offered by W. Strawiński and M. Dickson. In the later part of the article I address some other issues raised by my critics, such as the problem of the truth value of the antecedents of true counterfactuals, the problem of the semantic relation between sentences and events, and various shortcomings of my formulations of the condition of locality. In my response to the specific criticism of the definition of free-choice events, I agree with the charge that my definition is satisfied by all actual events. I present a correction to the definition which uses the distinction between actual events and merely possible events.

This paper addresses the issue of the multiplicity of various grades of discernibility that can be defined in model theory. Building upon earlier works on the subject, I first expand the known logical categorizations of discernibility by introducing several symmetry-based concepts of discernibility, including one I call “witness symmetry-discernibility”. Then I argue that only grades of discernibility stronger than this one possess certain intuitive features necessary to individuate objects. Further downsizing of the set of non-equivalent grades of discernibility can be achieved by stipulating that any relation of discernibility should be applied only to those pairs of objects which have been previously distinguished from the rest of the universe. Restricting discernibility to pairs of objects satisfying this condition gives an additional bonus in the form of restoring the transitivity of some types of indiscernibility which have been known to be non-transitive.

In the article I discuss possible amendments and corrections to Lewis’s semantics for counterfactuals that are necessary in order to account for the indeterministic and non-local character of the quantum world. I argue that Lewis’s criteria of similarity between possible worlds produce incorrect valuations for alternate-outcome counterfactuals in the EPR case. Later I discuss an alternative semantics which rejects the notion of miraculous events and relies entirely on the comparison of the agreement with respect to individual facts. However, a controversy exists whether to include future indeterministic events in the criteria of similarity. J. Bennett has suggested that an indeterministic event count toward similarity only if it is a result of the same causal chain as in the actual world. I claim that a much better agreement with the demands of the quantum-mechanical indeterminism can be achieved when we stipulate that possible worlds which differ only with respect to indeterministic facts that take place after the antecedent-event should always be treated as equally similar to the actual world. In the article I analyze and dismiss some common-sense counterexamples to this claim. Finally, I critically evaluate Bennett’s proposal regarding the truth-conditions for true-antecedent counterfactuals.

General metaphysical arguments have been proposed in favour of the thesis that all dispositions have categorical bases (Armstrong; Prior, Pargetter, Jackson). These arguments have been countered by equally general arguments in support of ungrounded dispositions (Molnar, Mumford). I believe that this controversy cannot be settled purely on the level of abstract metaphysical considerations. Instead, I propose to look for ungrounded dispositions in specific physical theories, such as quantum mechanics. I explain why non-classical properties such as spin are best interpreted as irreducible dispositional properties, and I give reasons why even seemingly classical properties, for instance position or momentum, should receive a similar treatment when interpreted in the quantum realm. Contrary to the conventional wisdom, I argue that quantum dispositions should not be limited to probabilistic dispositions (propensities) by showing reasons why even possession of well-defined values of parameters should qualify as a dispositional property. I finally discuss the issue of the actuality of quantum dispositions, arguing that it may be justified to treat them as potentialities whose being has a lesser degree of reality than that of classical categorical properties, due to the incompatibility relations between non-commuting observables.

In the first section of the chapter, I scrutinize Howard Stein’s 1991 definition of a transitive becoming relation that is Lorentz invariant. I argue first that Stein’s analysis gives few clues regarding the required characteristics of the relation complementary to his becoming—i.e. the relation of indefiniteness. It turns out that this relation cannot satisfy the condition of transitivity, and this fact can force us to reconsider the transitivity requirement as applied to the relation of becoming. I argue that the relation of becoming need not be transitive, as long as it satisfies the weaker condition of “cumulativity”: for a given observer the area of the events that have become real should not diminish as time progresses. I show that there are actually two relations of becoming that meet this weakened condition: Stein’s (transitive) relation of causal past connectibility and the (non-transitive) relation that is the logical complement of the future causal connectibility. In the second part of the chapter I defend Stein’s notion of temporal becoming against the attack that appeals to quantum-mechanical non-locality. I critically evaluate the argument given by Mauro Dorato (1996) that purports to show that space-like separated measurements done on the EPR system have to be mutually determinate. Finally, in order to account for the truth of counterfactual statements that link the space-like separated outcomes, I propose a dynamic conception of becoming, according to which the sphere of determinate events as of a given point may depend on the physical phenomena transpiring at this point.

This is an attempt to defend Field's nominalistic program from the criticism raised by K. Wójtowicz in his article. The author argues for the following theses: (a) that Wójtowicz uses the notion of „mathematical theory” broader than Field does it; (b) that he misinterprets the conception of the „abstract counterparts” of nominalistic statements; (c) and that his general evaluation of Field's program is based on too high methodological standards which he applies to the possible nominalistic versions of empirical theories. The second part of this paper contains an attempt to generalize the results of Field's analysis. The following fact is proved: every open sentence expressible in the language of an empirical theory and being empirically contentful is implicitly definable by the set of certain qualitative predicates. In the case of first-order language this result can be strengthened via Beth's definability theorem, to the theorem stating that every open sentence fulfilling conditions formulated above is definable explicitly with the help of certain nominalistic formula. The philosophical significance of this result is that each mathematized empirical theory for which representation theorem is true, can be translated into purely qualitative version.

In the article the possibility of breaking the eigenvalue-eigenstate link in quantum mechanics is considered. An argument is presented to the effect that there are some non-maximal observables for which the implication from eigenstates to eigenvalues is not valid, i.e. such that although the probability of revealing certain value upon measurement is one, they don't possess this value before the measurement. It is shown that the existence of such observables leads to contextuality, i.e. the thesis that one Hermitean operator can represent more than one physical observable. Finally, contextuality brought about by these considerations is compared with contextuality suggested by the Kochen-Specker paradox.

Bell’s theorem in its standard version demonstrates that the joint assumptions of the hidden-variable hypothesis and the principle of local causation lead to a conflict with quantum-mechanical predictions. In his latest counterfactual strengthening of Bell’s theorem, Stapp attempts to prove that the locality assumption itself contradicts the quantum-mechanical predictions in the Hardy case. His method relies on constructing a complex, non-truth functional formula which consists of statements about measurements and outcomes in some region R, and whose truth value depends on the selection of a measurement setting in a space-like separated location L. Stapp argues that this fact shows that the information about the measurement selection made in L has to be present in R. I give detailed reasons why this conclusion can and should be resisted. Next I correct and formalize an informal argument by Shimony and Stein showing that the locality condition coupled with Einstein’s criterion of reality is inconsistent with quantum-mechanical predictions. I discuss the possibility of avoiding the inconsistency by rejecting Einstein’s criterion rather than the locality assumption.

he paper addresses the issue of the applicability of David Lewis’s possible world semantics of counterfactual conditionals to the explication of some quantum-mechanical phenomena. Three main reasons why counterfactual semantics may be useful for this task are given. It is further argued that two possible semantic approaches to counterfactuals involving spatiotemporal events which satisfy requirements of special relativity should be taken into account. The main problem considered in the article is how to expand both approaches into full semantic systems. The first of the approaches is known to be amenable to a generalization within the Lewis-style semantics. The second one, however, poses a greater challenge, as it has been proven (Bigaj 2004) that it cannot be incorporated into a similarity-based counterfactual semantics. In this article, an alternative method of generalization for the second counterfactual semantics is developed, which goes beyond Lewis’s framework based on the rigid similarity relation between possible worlds. The proposed method of evaluating counterfactuals is then put to the test using an example from the quantum theory. As a result of this test, a small correction of the method turns out to be necessary.

As it is well known, Jan Lukasiewicz invented his three-valued logic as a result of philosophical considerations concerning the problem of determinism and the status of future contingent sentences. In the article I critically analyse the thesis that the sentential calculus introduced by Lukasiewicz himself actually fulfills his philosophical assumptions. I point out that there are some counterintuitive features of Lukasiewicz three-valued logic. Firstly, there is no clear explanation for adopting specific truth-tables for logical connectives, such as conjunction, disjunction and first of all implication. Secondly, it is by no means clear, why certain classical logical principles should be invalid for future contingents. And thirly, I show that within Lukasiewicz logic it is possible to construct a „paradoxical” sentence, namely a conditional which changes in time its logical value from truth to falsity. This fact obviously contradicts Lukasiewicz's philosophical reading of his three truth values, according to which true sentences are already positively determined, false sentences are negatively determined, and possible sentences are neither positively, nor negatively determined. Above-mentioned facts justify in my opinion the thesis that Lukasiewicz's three-valued logic does not satisfy his philosophical intuitions. For this purpose more appropriate seems to be sentential calculus based on the so-called supervaluation. It is three-valued, non-extentional calculus, which nevertheless preserves all tautologies of the classical logic. At the end of the article I consider the possibility of introducing to this calculus modal operators.

In the paper, the proof of the non-locality of quantum mechanics, given by Bedford and Stapp (1995), and appealing to the GHZ example, is analyzed. The proof does not contain any explicit assumption of realism, but instead it uses formal methods and techniques of the Lewis calculus of counterfactuals. To ascertain the validity of the proof, a formal semantic model for counterfactuals is constructed. With the help of this model it can be shown that the proof is faulty, because it appeals to the unwarranted principle of “elimination of eliminated conditions” (EEC). As an additional way of showing unreasonableness of the assumption (EEC), it is argued that yet another alleged and highly controversial proof of non-locality of QM, using the Hardy example, can be made almost trivial with the help of (EEC). Finally, a general argument is produced to the effect that the locality condition in the form accepted by Stapp and Bedford is consistent with the quantum-mechanical predictions for the GHZ case under the assumption of indeterminism. This result undermines any future attempts of proving the incompatibility between the predictions of quantum theory and the idea of no faster-than-light influence in the GHZ case, quite independently of the negative assessment of the particular derivation proposed by Stapp and Bedford.

In recent years, the so-calledindispensability argument has been given a lotof attention by philosophers of mathematics.This argument for the existence of mathematicalobjects makes use of the fact, neglected inclassical schools of philosophy of mathematics,that mathematics is part of our best scientifictheories, and therefore should receive similarsupport to these theories. However, thisobservation raises the question about the exactnature of the alleged connection betweenexperience and mathematics (for example: is itpossible to falsify empirically anymathematical theorems?). In my paper I wouldlike to address this question by consideringthe explicit assumptions of different versionsof the indispensability argument. My primaryclaim is that there are at least three distinctversions of the indispensability argument (andit can be even suggested that a fourth,separate version should be formulated). I willmainly concentrate my discussion on thisvariant of the argument, which suggests thepossibility of empirical confirmation ofmathematical theories. A large portion of mypaper will focus on the recent discussion ofthis topic, starting from the paper by E.Sober, who in my opinion put reasonablerequirements on what is to be counted as anempirical confirmation of a mathematicaltheory. I will develop his model into threeseparate scenarios of possible empiricalconfirmation of mathematics. Using an exampleof Hilbert space in quantum mechanicaldescription I will show that the most promisingscenario of empirical verification ofmathematical theories has neverthelessuntenable consequences. It will be hypothesizedthat the source of this untenability lies in aspecific role which mathematical theories playin empirical science, and that what is subjectto empirical verification is not themathematics used, but the representabilityassumptions. Further I will undertake theproblem of how to reconcile the allegedempirical verification of mathematics withscientific practice. I will refer to thepolemics between P. Maddy and M. Resnik,pointing out certain ambiguities of theirarguments whose source is partly the failure todistinguish carefully between different sensesof the indispensability argument. For thatreason typical arguments used in the discussionare not decisive, yet if we take into accountsome metalogical properties of appliedmathematics, then the thesis that mathematicshas strong links with experience seems to behighly improbable.

The existence of non-local correlations between outcomes of measurements in quantum entangled systems strongly suggests that we are dealing with some form of causation here. An assessment of this conjecture in the context of the collapse interpretation of quantum mechanics is the primary goal of this paper. Following the counterfactual approach to causation, I argue that the details of the underlying causal mechanism which could explain the non-local correlations in entangled states strongly depend on the adopted semantics for counterfactuals. Several relativistically-invariant interpretations of spatiotemporal counterfactual conditionals are discussed, and the corresponding causal stories describing interactions between parts of an entangled system are evaluated. It is observed that the most controversial feature of the postulated causal connections is not so much their non-local character as a peculiar type of circularity that affects them.

David Lewis’s latest theory of causation defines the causal link in terms of the relation of influence between events. It turns out, however, that one event’s influencing another is neither a necessary nor sufficient condition for its being a cause of that event. In the article one particular case of causality without influence is presented and developed. This case not only serves as a counterexample to Lewis’s influence theory, but also threatens earlier counterfactual analyses of causation by admitting a particularly troublesome type of preemption. The conclusion of the article is that Lewis’s influence method of solving the preemption problem fails, and that we need a new and fresh approach to the cases of redundant causation if we want to hold on to the counterfactual analysis of causation.

The author presents and critically analyses different accounts of causal relation given by the main representants of Lvov-Warsaw School in philosophy. Although there are considerable differences between particular approaches to this problem, it is possible at least to distinguish the key questions, analysed and anwsered by these philosophers. Among them are such questions as: how to define „causal relation”, what are its formal features, what is the space-time localization of the effect and the cause, what are causal laws.

This book uses the formal semantics of counterfactual conditionals to analyze the problem of non-locality in quantum mechanics. Counterfactual conditionals enter the analysis of quantum entangled systems in that they enable us to precisely formulate the locality condition that purports to exclude the existence of causal interactions between spatially separated parts of a system. They also make it possible to speak consistently about alternative measuring settings, and to explicate what is meant by quantum property attributions. The book develops the possible-world semantics of quantum counterfactuals using David Lewis's famous approach as a starting point but modifying it significantly in order to achieve compatibility with the demands of the special theory of relativity as well as quantum mechanics. There have been several attempts to use counterfactuals semantics to strengthen Bell's theorem and its cognates such as the GHZ and Hardy theorems. These are critically evaluated in the book. Finally, a counterfactual reconstruction of the EPR argument and Bell's theorem is proposed that sheds a new light on their philosophical consequences regarding the relations between realism and local causation.