Determinate Truth Values Research Articles

‘Conspiracy Theory’ as a Tonkish Term: Some Runabout Inference-Tickets from Truth to Falsehood

ABSTRACT I argue that ‘conspiracy theory’ and ‘conspiracy theorist’ as commonly employed are ‘tonkish’ terms (as defined by Arthur Prior and Michael Dummett), licensing inferences from truths to falsehoods; indeed, that they are mega-tonkish terms, since their use is governed by different and competing sets of introduction and elimination rules, delivering different and inconsistent results. Thus ‘conspiracy theory’ and ‘conspiracy theorist’ do not have determinate extensions, which means that generalizations about conspiracy theories or conspiracy theorists do not have determinate truth-values. Hence conspiracy theory theory – psychological or social scientific research into conspiracy theorists and what is wrong with them – is often about as intellectually respectable as an enquiry into bastards and what makes them so mean.

A Truly Robust Signal Temporal Logic: Monitoring Safety Properties of Interacting Cyber-Physical Systems under Uncertain Observation

Signal Temporal Logic is a linear-time temporal logic designed for classifying the time-dependent signals originating from continuous-state or hybrid-state dynamical systems according to formal specifications. It has been conceived as a tool for systematizing the monitoring of cyber-physical systems, supporting the automatic translation of complex safety specifications into monitoring algorithms, faithfully representing their semantics. Almost all algorithms hitherto suggested do, however, assume perfect identity between the sensor readings, informing the monitor about the system state and the actual ground truth. Only recently have Visconti et al. addressed the issue of inexact measurements, taking up the simple model of interval-bounded per-sample error that is unrelated, in the sense of chosen afresh, across samples. We expand their analysis by decomposing the error into an unknown yet fixed offset and an independent per-sample error and show that in this setting, monitoring of temporal properties no longer coincides with collecting Boolean combinations of state predicates evaluated in each time instant over best-possible per-sample state estimates, but can be genuinely more informative in that it infers determinate truth values for monitoring conditions that interval-based evaluation remains inconclusive about. For the model-free as well as for the linear model-based case, we provide optimal evaluation algorithms based on affine arithmetic and SAT modulo theory, solving over linear arithmetic. The resulting algorithms provide conclusive monitoring verdicts in many cases where state estimations inherently remain inconclusive. In their model-based variants, they can simultaneously address the issues of uncertain sensing and partial observation.

Indeterminacy and Triviality

ABSTRACT Suppose you’re certain that a claim—say, ‘Frida is tall’—does not have a determinate truth value. What attitude should you take towards it? This is the question of the cognitive role of indeterminacy. This paper presents a puzzle for theories of cognitive role. Many of these theories vindicate a seemingly plausible principle: if you are fully certain that A, you are rationally required to be fully certain that A is determinate. Call this principle ‘Certainty’. We show that Certainty, in combination with some minimal side premises, entails a very implausible claim: whenever you’re certain that it’s indeterminate whether A, it is rationally required that you reject A. This is a surprising result, which requires abandoning at least some intuitive views about indeterminacy and cognitive role.

Art and fiction are signals with indeterminate truth values.

Menninghaus et al. distinguish art from fiction, but no current arguments or data suggest that the concept of art can be meaningfully circumscribed. This is a problem for aesthetic psychology. I sketch a solution by rejecting the distinction: Unlike most animal communication, in which signals are either true or false, art and fiction consist of signals without determinate truth values.

Eternity, knowledge, and freedom

AbstractThis article addresses the problem of divine foreknowledge and human freedom by developing a modified version of Boethius' solution to the problem – one that is meant to cohere with a dynamic theory of time and a conception of God as temporal. I begin the article by discussing the traditional Boethian solution, and a defence of it due to Kretzmann and Stump. After canvassing a few of the objections to this view, I then go on to offer my own modified Boethian solution, according to which temporal reality is fundamentally dynamic, but truth is not. My claim is that there are eternally existing, tenseless propositions, with determinate truth values, but that these are made true by events that come into existence, and are not themselves eternal.

Open-endedness, Schemas and Ontological Commitment

Second-order axiomatizations of certain important mathematical theories-such as arithmetic and real analysis—can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth-value. Second-order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth-values. The status of second-order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, Vann McGee has argued that one can get some of the technical advantages of second-order axiomatizations—categoricity, in particular—while walking free of worries about ontological commitment. In so arguing he appeals to the notion of an open-ended schema—a schema that holds no matter how the language of the relevant theory is extended. Contra McGee, we argue that second-order quantification and open-ended schemas are on a par when it comes to ontological commitment.

Objective or Subjective ‘Ought’?

The prescriptive ‘ought’ has both an objective and a subjective interpretation. In the objective sense, what one ought to do depends on what is actually true. In the subjective sense it depends on what one believes to be true. Ordinary usage seems to vacillate between these two interpretations. An example (the indecisive terrorist) is used to show that a subjective ought statement can have a determinate truth-value in situations where the corresponding objective ought statement has no truth-value, not even an unknowable one. Therefore the subjective ought is not definable in terms of the objective ought. However, definability in the other direction is not excluded by this argument.

SEA BATTLE SEMANTICS

The assumption that the future is open makes well known problems for traditional semantics. According to a commonly held intuition, today's occurrence of the sentence ‘There will be a sea battle tomorrow’, while truth-valueless today, will have a determinate truth-value by tomorrow night. Yet given traditional semantics, sentences that are truth-valueless now cannot later ‘become’ true. Relativistic semantics has been claimed to do a better job of accommodating intuitions about future contingents than non-relativistic semantics does. However, intuitions about future contingents cannot by themselves give good reasons for shifting to a new paradigm, for despite the initial appearances, standard non-relativistic semantics (plus an account of truth-value gaps) can accommodate both intuitions about future contingents.

Gödel's Program Revisited Part I: The Turn to Phenomenology

AbstractConvinced that the classically undecidable problems of mathematics possess determinate truth values, Gödel issued a programmatic call to search for new axioms for their solution. The platonism underlying his belief in the determinateness of those questions in combination with his conception of intuition as a kind of perception have struck many of his readers as highly problematic. Following Gödel's own suggestion, this article explores ideas from phenomenology to specify a meaning for his mathematical realism that allows for a defensible epistemology.

Future contingents, non-contradiction, and the law of excluded middle muddle

For whatever reason, we might think that contingent statements about the future have no determinate truth value. Aristotle, in De interpretatione ix, for instance, held that only those propositions about the future which are either necessarily true, or necessarily false, or 'predetermined' in some way have a determinate truth value. This led Lukasiewicz's in 1920 to construct a three-valued logic in an attempt to formalize Aristotle's position by giving the truth value 1/2 = indeterminate to future contingents and defining '-', '&' and 'v', where 1 = true and 0 = false, as:

Counterfactuals and Causal Explanation

This article defends the use of interventionist counterfactuals to elucidate causal and explanatory claims against criticisms advanced by James Bogen and Peter Machamer. Against Bogen, I argue that counterfactual claims concerning what would happen under interventions are meaningful and have determinate truth values, even in a deterministic world. I also argue, against both Machamer and Bogen, that we need to appeal to counterfactuals to capture the notions like causal relevance and causal mechanism. Contrary to what both authors suppose, counterfactuals are not “unscientific”—a substantial tradition within statistics and the causal modelling literature makes heavy use of them.

Contingency and Divine Knowledge in Ockham

Ockham appeared to maintain that God necessarily knows all true propositions, including future contingent propositions, despite the fact that such propositions have determinate truth values. While some commentators believe that Ockham's attempt to reconcile divine omniscience with the contingency of true future propositions amounts to little more than a simple-minded assertion of Ockham's Christian faith, I argue that Ockham's position is more sophisticated than this and rests on attributing to God a dual knowledge property: God not only knows every true proposition, but knows its modal properties as well. Future contingent propositions are determinately true when actualized, not timelessly, and God's knowledge of their truth values is knowledge of when the truth value of a proposition is actually determined.

A Theory of Mathematical Correctness and Mathematical Truth

A theory of objective mathematical correctness is developed. The theory is consistent with both mathematical realism and mathematical anti-realism, and versions of realism and anti-realism are developed that dovetail with the theory of correctness. It is argued that these are the best versions of realism and anti-realism and that the theory of correctness behind them is true. Along the way, it is shown that, contrary to the traditional wisdom, the question of whether undecidable sentences like the continuum hypothesis have objectively determinate truth values is independent of the question of whether mathematical realism is true.

Do appellate courts regularly cheat?

It probably will not surprise you to be told that practice of law is almost entirely remote from its philosophy. Lawyers and judges are not much interested in jurisprudence--as indeed, not many mathematicians or scientists are much interested in philosophies of their respective disciplines. (For that matter, philosophers are not much interested in philosophy of what they do either.) But I have had one jurisprudential question set for me by my appellate practice, which has been almost entirely as a criminal defender. I have found in a substantial number of my appellate cases that I was left at its end with distinct impression that court had cheated in reaching its decision: that in considering case it had early on made up its collective mind what this would be and that it had thereafter ignored contrary facts or legal principles, however weighty those might have been. And I believed in some of those cases (albeit not all) that court had as a result come to a clearly incorrect legal conclusion--with unfortunate consequences to my client. In talking shop with other appellate attorneys I frequently ask whether they too have often been left with same impression, and their invariable response is: Sure, let me tell you about. . .. So I found that I and other practitioners believed that appellate courts frequently fail to follow law-at times, as it seemed to us, deliberately so. (Because we had clearly brought those inconvenient facts and arguments to court's attention!) This raised two significant questions: one seemingly empirical (Is it true that courts regularly `cheat' as so described?) and another straightforwardly jurisprudential (Is it proper--that is, consistent with true principles of political morality--for courts so to cheat?) Somewhat surprisingly, first question--do courts often cheat?--is least tractable. It assumes a controversial jurisprudential hypothesis, namely, that at least some propositions of law have a determinate truth value. (If none do, then there is nothing for courts to follow, no way that they could cheat.) Champions of indeterminacy--by whom I understand philosophers who deny that there is determinate truth in domains wherein we commonly suppose we have knowledge: that is, physical sciences, social sciences, morality--have also been very influential in jurisprudence. Holmes, one of our greatest judges, famously remarked in his essay Path of Law that law is only the prediction of incidence of public force through instrumentality of courts.(1) Legal realists in 1930s, and Critical Legal Theorists today, agree that decisions in concrete cases are never deductions from facts and pre-existing law, but rather are mere expressions of judges' political ideologies. point is sometimes made by saying: The law is only what judge had for breakfast. So one might quite reasonably, block my first question (Do courts often cheat?) by saying, Courts can't cheat because they are not bound by anything at all. Despite attractive simplicity of skeptics' answer, I suggest that we not make such short shrift of an interesting question. (Indeed, best argument against legal skepticism, and against skepticism generally, is that it makes every interesting philosophical question seem boring.) And legal skepticism, while now much in fashion in law schools and law reviews, is not dominant view among scholars who specialize in philosophy of law. Certainly it is not dominant view among lawyers--to extent that they count as having jurisprudential views. Legal scholars often say that legal skepticism is lawyer's view of law,(2) but it is more plausibly styled law professors'. Legal education is still dominated by case method, which focuses primarily upon appellate cases published in official reporters. Compilers of casebooks tend to look for those signs of doctrinal embarrassment that prefigure changes in law: where there is no convincing theoretical justification for an established rule or where such a rule frequently works an injustice. …

How We Learn Mathematical Language

Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning (so that 'e' stands for the elementhood relation and the quantifiers range over all the sets there are), each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is En2 or whether there are measurable cardinals, whether or not those facts are knowable by us. There are numerous objections to realism, some of them quite penetrating. Some objections come from philosophers, who frequently find talk about abstract entities either unintelligible or incredible. Other objections come from mathematicians, many of whom are affably willing to go along with realist talk about natural numbers or real numbers, but who find that when we get to set theory their credulity has been exhausted, so that they think of set theory the way we think of group theory, as a formal calculus whose models we examine, without any supposition that any of the models are preferred. Here I do not intend to defend realism but rather to presume it-though this be a very large presumption-and to discuss a problem that arises internally within the realist conception of mathematics. The internal problem is this: the realist conception supposes that the meaning of mathematical terms is fixed with sufficient preci-

On the structure of quantal proposition systems

I define sublaltices of quantum propositions that can be taken as having determinate (but perhaps unknown) truth values for a given quantum state, in the sense that sufficiently many two-valued maps satisfying a Boolean homomorphism condition exist on each determinate sublattice to generate a Kolmogorov probability space for the probabilities defined by the slate. I show that these sublattices are maximal, subject to certain constraints, from which it follows easily that they are unique. I discuss the relevance of this result for the measurement problem, relating it to an early proposal by Jauch and Piron for defining a new notion of state for quantum systems, to a recent uniqueness proof by Clifton for the sublattice of propositions specified as determinate by modal interpretations of quantum mechanics that exploit the polar decompostion theorem, and to my own previous suggestions for interpreting quantum mechanics without the projection postulate.

On Bohr's response to EPR: A quantum logical analysis

Bohr's complementarity interpretation is represented as the relativization of the quantum mechanical description of a system to the maximal Boolean subalgebra (in the non-Boolean logical structure of the system) selected by a classically described experimental arrangement. Only propositions in this subalgebra have determinate truth values. The concept of a minimal revision of a Boolean subalgebra by a measurement is defined, and it is shown that the nonmaximal measurement of spin on one subsystem in the spin version of the Einstein—Podolsky—Rosen experiment actually selects an appropriate maximal Boolean subalgebra ℛ′ in the logical structure of the composite system, via a minimal revision of the maximal Boolean subalgebra & associated with the preparation of the singlet spin state. This provides an explanation for the determinate truth values of propositions in ℛ′ referring to the second subsystem within the framework of the complementarity interpretation.

On the conservative extensions of semantical systems: A contribution to the problem of analyticity

The attempt to distinguish within an empirical theory between a factual and a conventional component is an important task. It bears especially on the following methodological issues. 'Cognitive status' of theories: The dispute between conventionalism and its critics concerns just where this distinction should be drawn. But equally, if the conventional part of a theory determines the reference of some of its expressions, the question whether theoretical assertions of empirical science include non-conventional statements of determinate truth-value cannot be settled until an exact characterization of the con­ ventional part is given. Inter-theory relations: It is clear that logical relations between theories­ inconsistency, equivalence, reducibility, etc. - depend critically on intra­ theoretic relations. These are determined for each theory in a way which depends on the choice of conventional component. Evidential support: It is to be expected that an adequate measure of support furnished to theory by evidence should assign the highest value to statements belonging to the conventional part. It follows that the measure appropriate to the language of a given theory depends on how its factual and conventional components are distinguished. The problem of distinguishing the factual and conventional component was solved by R. Carnap for finite sets of postulates as follows [4], [5, pp. 958-66]. Let n be the conjunction of theoretical postulates and correspondence postulates and let the descriptive constants of n be distinguished into O-terms and T-terms. Let Rn be the existential closure of the formula obtained from n by proper simultaneous substitution of predicate variables for T-terms. Then the factual part of n is identified with the consequence class of Rn and its conventional part with the con­ sequence class of Rn;:) n. It is understood in addition that a partial inter­ pretation is effected for T-terms by restricting their proper interpre­ tations to those that establish the truth of Rn;:) n given prior interpretations of O-terms. Carnap made the following expository observations: