The novel concept of a simulating halt decider enables halt decider H to to correctly determine the halt status of the conventional “impossible” input D that does the opposite of whatever H decides. This works equally well for Turing machines and “C” functions. The algorithm is demonstrated using “C” functions because all of the details can be shown at this high level of abstraction. ---------------------------------------------------------------------------------------------------- ---- Simulating halt decider H correctly determines that D correctly simulated by H would remain stuck in (...) recursive simulation never reaching its own final state. D cannot do the opposite of the return value from H because this return value is unreachable by every correctly simulated D. This same result is shown to be derived in the Peter Linz Turing machine based proof. (shrink)

The notion of a simulating termination analyzer is examined at the concrete level of pairs of C functions. This is similar to AProVE: Non-Termination Witnesses for C Programs. The termination status decision is made on the basis of the dynamic behavior of the input. This paper explores what happens when a simulating termination analyzer is applied to an input that calls itself.

By extending the notion of a Well Formed Formula to include syntactically formalized rules for rejecting semantically incorrect expressions we recognize and reject expressions that have the semantic error of Pathological self-reference(Olcott 2004). The foundation of this system requires the notion of a BaseFact that anchors the semantic notions of True and False. When-so-ever a formal proof from BaseFacts of language L to a closed WFF X or ~X of language L does not exist X is decided to be (...) semantically incorrect. (shrink)

That every expression of language that is {true on the basis of its meaning expressed using language} must have a connection by truth preserving operations to its {meaning expressed using language} is a tautology. The accurate model of the actual world is expressed using formal language and formalized natural language.

The novel concept of a simulating halt decider enables halt decider H to to correctly determine the halt status of the conventional “impossible” input D that does the opposite of whatever H decides. This works equally well for Turing machines and “C” functions. The algorithm is demonstrated using “C” functions because all of the details can be shown at this high level of abstraction.

A simulating halt decider correctly predicts whether or not its correctly simulated input can possibly reach its own final state and halt. It does this by correctly recognizing several non-halting behavior patterns in a finite number of steps of correct simulation. Inputs that do terminate are simply simulated until they complete.

This is an explanation of a possible new insight into the halting problem provided in the language of software engineering. Technical computer science terms are explained using software engineering terms. No knowledge of the halting problem is required. -/- It is based on fully operational software executed in the x86utm operating system. The x86utm operating system (based on an excellent open source x86 emulator) was created to study the details of the halting problem proof counter-examples at the much higher level (...) of abstraction of C/x86. (shrink)

The halting theorem counter-examples present infinitely nested simulation (non-halting) behavior to every simulating halt decider. Whenever the pure simulation of the input to simulating halt decider H(x,y) never stops running unless H aborts its simulation H correctly aborts this simulation and returns 0 for not halting.

MIT Professor Michael Sipser has agreed that the following verbatim paragraph is correct (he has not agreed to anything else in this paper) -------> -/- If simulating halt decider H correctly simulates its input D until H correctly determines that its simulated D would never stop running unless aborted then H can abort its simulation of D and correctly report that D specifies a non-halting sequence of configurations.

The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic property of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide valid the deductive inference. Conclusions of sound arguments are derived from truth preserving finite string transformations applied to true premises.

If there truly is a proof that shows that no universal halt decider exists on the basis that certain tuples: (H, Wm, W) are undecidable, then this very same proof (implemented as a Turing machine) could be used by H to reject some of its inputs. When-so-ever the hypothetical halt decider cannot derive a formal proof from its input strings and initial state to final states corresponding the mathematical logic functions of Halts(Wm, W) or Loops(Wm, W), halting undecidability has been (...) decided. (shrink)

For any natural (human) or formal (mathematical) language L we know that an expression X of language L is true if and only if there are expressions Γ of language L that connect X to known facts. -/- By extending the notion of a Well Formed Formula to include syntactically formalized rules for rejecting semantically incorrect expressions we recognize and reject expressions that evaluate to neither True nor False.

The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic value of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide the valid deductive inference. Sound deductive conclusions are the result of these finite string transformation rules.

This paper decomposes the Liar Paradox into its semantic atoms using Meaning Postulates (1952) provided by Rudolf Carnap. Formalizing truth values of propositions as Boolean properties of these propositions is a key new insight. This new insight divides the translation of a declarative sentence into its equivalent mathematical proposition into three separate steps. When each of these steps are separately examined the logical error of the Liar Paradox is unequivocally shown.

When we understand that every potential halt decider must derive a formal mathematical proof from its inputs to its final states previously undiscovered semantic details emerge. -/- When-so-ever the potential halt decider cannot derive a formal proof from its input strings to its final states of Halts or Loops, undecidability has been decided. -/- The formal proof involves tracing the sequence of state transitions of the input TMD as syntactic logical consequence inference steps in the formal language of Turing Machine (...) Descriptions. (shrink)

This is the formal YACC BNF specification for Minimal Type Theory (MTT). MTT was created by augmenting the syntax of First Order Logic (FOL) to specify Higher Order Logic (HOL) expressions using FOL syntax. Syntax is provided to enable quantifiers to specify type. FOL is a subset of MTT. The ASSIGN_ALIAS operator := enables FOL expressions to be chained together to form HOL expressions.

Within the (Haskell Curry) notion of a formal system we complete Tarski's formal correctness: ∀x True(x) ↔ ⊢ x and use this finally formalized notion of Truth to refute his own Undefinability Theorem (based on the Liar Paradox), the Liar Paradox, and the (Panu Raatikainen) essence of the conclusion of the 1931 Incompleteness Theorem.

Hypothesis: WFF(x) can be applied syntactically to the semantics of formalized declarative sentences such that: WFF(x) ↔ (x ↦ True) ∨ (x ↦ False) (see proof sketch below) For clarity we focus on simple propositions without binary logical connectives.

Minimal Type Theory (MTT) shows exactly how all of the constituent parts of an expression relate to each other (in 2D space) when this expression is formalized using a directed acyclic graph (DAG). This provides substantially greater expressiveness than the 1D space of FOPL syntax. -/- The increase in expressiveness over other formal systems of logic shows the Pathological Self-Reference Error of expressions previously considered to be sentences of formal systems. MTT shows that these expressions were never truth bearers, thus (...) never sentences of any formal logic system. (shrink)

Because formal systems of symbolic logic inherently express and represent the deductive inference model formal proofs to theorem consequences can be understood to represent sound deductive inference to deductive conclusions without any need for other representations.

We begin with the hypothetical assumption that Tarski’s 1933 formula ∀ True(x) φ(x) has been defined such that ∀x Tarski:True(x) ↔ Boolean-True. On the basis of this logical premise we formalize the Truth Teller Paradox: "This sentence is true." showing syntactically how self-reference paradox is semantically ungrounded.

If the conclusion of the Tarski Undefinability Theorem was that some artificially constrained limited notions of a formal system necessarily have undecidable sentences, then Tarski made no mistake within his assumptions. When we expand the scope of his investigation to other notions of formal systems we reach an entirely different conclusion showing that Tarski's assumptions were wrong.

Could the intersection of [formal proofs of mathematical logic] and [sound deductive inference] specify formal systems having [deductively sound formal proofs of mathematical logic]? All that we have to do to provide [deductively sound formal proofs of mathematical logic] is select the subset of conventional [formal proofs of mathematical logic] having true premises and now we have [deductively sound formal proofs of mathematical logic].

To eliminate incompleteness, undecidability and inconsistency from formal systems we only need to convert the formal proofs to theorem consequences of symbolic logic to conform to the sound deductive inference model. -/- Within the sound deductive inference model there is a (connected sequence of valid deductions from true premises to a true conclusion) thus unlike the formal proofs of symbolic logic provability cannot diverge from truth.

This sentence G ↔ ¬(F ⊢ G) and its negation G ↔ ~(F ⊢ ¬G) are shown to meet the conventional definition of incompleteness: Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)). They meet conventional definition of incompleteness because neither the sentence nor its negation is provable in F (or any other formal system). -- .

We can simply define Gödel 1931 Incompleteness away by redefining the meaning of the standard definition of Incompleteness: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ). This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system. Since self-contradictory expressions are neither provable nor disprovable (...) only because they are self-contradictory we could define them as unsound instead of defining the formal system as incomplete. (shrink)

Tarski "proved" that there cannot possibly be any correct formalization of the notion of truth entirely on the basis of an insufficiently expressive formal system that was incapable of recognizing and rejecting semantically incorrect expressions of language. -/- The only thing required to eliminate incompleteness, undecidability and inconsistency from formal systems is transforming the formal proofs of symbolic logic to use the sound deductive inference model.

Both Tarski and Gödel “prove” that provability can diverge from Truth. When we boil their claim down to its simplest possible essence it is really claiming that valid inference from true premises might not always derive a true consequence. This is obviously impossible.

An abstract machine having a tape head that can be advanced in 0 to 0x7FFFFFFF increments an unlimited number of times specifies a model of computation that has access to unlimited memory. The technical name for memory addressing based on displacement from the current memory address is relative addressing.

When we look at 800,000 year ice core data CO2 levels since 1950 have risen at a rate of 123-fold faster than the fastest rate in 800,000 years. When we see that this rise is precisely correlated with global carbon emissions the human link to climate change seems certain and any rebuttal becomes ridiculously implausible. The 800,000 year correlation between CO2 and global temperatures seems to be predicting at least 9 degrees C of more warming based on current CO2 levels.

The fail-safe makes sure the fee is high enough to meet carbon emission reduction targets. The safeguard keeps the fee from getting any higher than needed. -/- One of the ways that we could account for the unpredictability of the price elasticity of demand for carbon would be to provide a fail-safe mechanism to ensure that we definitely stay on the carbon reduction schedule. If we keep Energy Innovation Act (HR 763) essentially as it is and scale up the annual (...) carbon fee increase by Number-of-Years-Behind-Schedule * 0.15. (shrink)

The generalized conclusion of the Tarski and Gödel proofs: All formal systems of greater expressive power than arithmetic necessarily have undecidable sentences. Is not the immutable truth that Tarski made it out to be it is only based on his starting assumptions. -/- When we reexamine these starting assumptions from the perspective of the philosophy of logic we find that there are alternative ways that formal systems can be defined that make undecidability inexpressible in all of these formal systems.

One of the ways that we could account for the unpredictability of the price elasticity of demand for carbon would be to provide a fail-safe mechanism to ensure that we definitely stay on the carbon reduction schedule. If we kept Energy Innovation Act (HR 763) essentially as it is and scale up the annual carbon fee increase by Number-of-Years-Behind-Schedule * 0.15.

Not only the direct physical experiences of deployment can severely harm soldiers’ mental health. Witnessing violations of their moral principles by the enemy, or by their fellow soldiers and superiors, can also have a devastating impact. It can cause soldiers’ moral disorientation, increasing feelings of shame, guilt, or hate, and the need for general answers on questions of right and wrong. Various attempts have been made to keep soldiers mentally sane. One is to provide convincing causes for their deployment, which (...) risks an “end justifies the means” way of thinking. The good cause can provide a moral justification for horrible atrocities. Another method, introduced in the USA, Canada, and Australia, aims to strengthen military personnel’s resistance by promoting and maintaining a happy, optimistic state of mind through the use of positive psychology. Alongside making soldiers “morally fit” for all kinds of situations, the focus could also be on moral recovery and forgiveness. Such a care-based military ethics approach, aimed at mutual understanding and interdependence, could help soldiers handle the emotional impact of moral conflicts. This demands that military units reflect on their organizational culture and rethink oaths and codes of conduct that focus mainly on efficiency and readiness, as well as the soldierly self-image with its seemingly still deeply rooted warrior ethos. Today, resilience and positive psychology in the military is apparently mainly geared to assuring its soldiers’ readiness. An appropriate set of virtues and understanding of virtue ethics that are less centered on self-perfection and autonomy could point to a different form of character-building and lead to a better understanding of others. (shrink)

You don't say much about who you are teaching, or what subject you teach, but you do seem to see a need to justify what you are doing. Perhaps you're teaching underprivileged children, opening their minds to possibilities that might otherwise never have occurred to them. Or maybe you're teaching the children of affluent families and opening their eyes to the big moral issues they will face in life — like global poverty, and climate change. If you're doing something like (...) this, then stick with it. Giving money isn't the only way to make a difference. (shrink)

This is a genealogical study that traces a “broadly Cartesian” pattern of argumentation: from Augustine’s scriptural semiotic to the “narrowly Cartesian” practice of foundationalism to Charles Peirce’s pragmatic and reparative semiotic. The essay argues (1) that Augustine transformed Stoic logic into a scriptural semiotic; (2) that this semiotic breeds both Cartesian foundationalism and the pragmatic semiotic that repairs it; (3) that Peirce’s semiotic displays the latter. In sum, Augustine’s inquiry risks foundationalism but also breeds a self-corrective “reparative reasoning.” This reasoning (...) is at once scriptural and philosophic. (shrink)

This paper examines a promising probabilistic theory of singular causation developed by David Lewis. I argue that Lewis' theory must be made more sophisticated to deal with certain counterexamples involving pre-emption. These counterexamples appear to show that in the usual case singular causation requires an unbroken causal process to link cause with effect. I propose a new probabilistic account of singular causation, within the framework developed by Lewis, which captures this intuition.

How is Jewish-Christian theological dialogue possible today? Assuming that the possibility of dialogue is not something to be envisioned by any individual thinker a priori, I offer here a study of two examples of successful Jewish-Christian theological dialogue: George Lindbeck's dialogue with Jewish sources and Michael Wyschogrod's dialogue with Christian sources. To garner some general lessons from these examples, I try to reconstruct the general conditions of dialogue which they appear to share. Discovering that the two dialogues may share identifiable (...) conditions of problem-solving, I label these "pragmatic" conditions, after Charles Peirce's definition of pragmatism as the claim that the purpose of rational inquiry is to solve the problems that stimulate it. I conclude that these are the conditions that may underlie theological dialogue among the specific group of Jewish and Christian theologians to which I believe Lindbeck and Wyschogrod belong. These conditions may underlie dialogues typical of other groups, too, but there is no way to know this a priori. (shrink)

You ask if metaphysics is possible after modernity, or after Barth and Wittgenstein and Derrida and the critique of foundationalism? May I invite you, by way of response, to listen in on a conversation? It is a dialogue between what I will call a postcritical philosopher ("P") and a postcritical scriptural theologian —— I'll label the latter a "textualist" ("T"). What I mean by "postcritical" would be displayed as the pattern of inquiry traced by this dialogue. I take the term (...) "postcritical" from George Lindbeck, whose theological work is described by his commentators as more properly "textualist" rather than "philosophic." I believe my usage is, however, true to Lindbeck's discourse. This means that one of the arguments of this essay is that there is a mode of philosophic inquiry proper to the postcritical orientation in theology that we may associate with such Christian theologians as Hans Frei, etc., as well as Lindbeck. A second argument, offered indirectly, is that there is a family of Jewish thinkers whose work may also be labeled "postcritical" on the model of these Christian thinkers, and that this Jewish postcritical theology would also be displayed in dialogues between philosophic and textualist tendencies. A third argument is that, if we listen in on the philosophic side of any of these dialogues, we will hear a postcritical philosophy. By philosophy, I mean primarily logic, defined as an inquiry that examines problematic rules of reasoning and recommends testable ways of correcting them. Secondarily, philosophy includes metaphysics, defined as an inquiry that, for the sake of assuring people about the reliability of a given logic, constructs pictures of the world whose existence would warrant this logic. We need not assume either that logic and metaphysics can be conducted only in the manner of modern and scholastic foundationalism, or that non-foundationalist theology must be "textualist" in a manner that excludes any philosophic generalization. I will argue that there are, indeed, non-foundationalist ways of conceiving of both logical and metaphysical inquiry and that these ways contribute to the dialogic activity of postcritical inquiry. (shrink)

Demographic diversity might often be present in a group without group members noticing it. What are the epistemic effects if they do? Several philosophers and social scientists have recently argued that when individuals detect demographic diversity in their group, this can result in epistemic benefits even if that diversity doesn’t involve cognitive differences. Here I critically discuss research advocating this proposal, introduce a distinction between two types of detection of demographic diversity, and apply this distinction to the theorizing on diversity (...) in science. Focusing on ‘invisible’ diversity, I argue that in one common kind of group in science, if group members have full insight into their group’s diversity, this is likely to create epistemic costs. These costs can be avoided and epistemic benefits gained if group members only partly detect their group’s diversity. There is thus an epistemic reason for context-dependent limitations on scientists’ insight into the diversity of their group. (shrink)

Confirmation bias is one of the most widely discussed epistemically problematic cognitions, challenging reliable belief formation and the correction of inaccurate views. Given its problematic nature, it remains unclear why the bias evolved and is still with us today. To offer an explanation, several philosophers and scientists have argued that the bias is in fact adaptive. I critically discuss three recent proposals of this kind before developing a novel alternative, what I call the ‘reality-matching account’. According to the account, confirmation (...) bias evolved because it helps us influence people and social structures so that they come to match our beliefs about them. This can result in significant developmental and epistemic benefits for us and other people, ensuring that over time we don’t become epistemically disconnected from social reality but can navigate it more easily. While that might not be the only evolved function of confirmation bias, it is an important one that has so far been neglected in the theorizing on the bias. (shrink)

This chapter provides an introduction to higher-order metaphysics as well as to the contributions to this volume. We discuss five topics, corresponding to the five parts of this volume, and summarize the contributions to each part. First, we motivate the usefulness of higher-order quantification in metaphysics using a number of examples, and discuss the question of how such quantifiers should be interpreted. We provide a brief introduction to the most common forms of higher-order logics used in metaphysics, and indicate a (...) number of questions which can be raised in such systems using logical vocabulary alone. Using a further example, we return to applications of higher-order logics in metaphysics. We also mention key developments in the history of higher-order logic as it pertains to metaphysics. Finally, we mention certain arguments which have been raised against the use of higher-order logic, and some ways of responding to them. (shrink)

It has recently been suggested that politically motivated cognition leads progressive individuals to form beliefs that underestimate real differences between social groups and to process information selectively to support these beliefs and an egalitarian outlook. I contend that this tendency, which I shall call ‘egalitarian confirmation bias’, is often ‘Mandevillian’ in nature. That is, while it is epistemically problematic in one’s own cognition, it often has effects that significantly improve other people’s truth tracking, especially that of stigmatized individuals in academia. (...) Due to its Mandevillian character, egalitarian confirmation bias isn’t only epistemically but also ethically beneficial, as it helps decrease social injustice. Moreover, since egalitarian confirmation bias has Mandevillian effects especially in academia, and since progressives are particularly likely to display the bias, there is an epistemic reason for maintaining the often-noted political majority of progressives in academia. That is, while many researchers hold that diversity in academia is epistemically beneficial because it helps reduce bias, I argue that precisely because political diversity would help reduce egalitarian confirmation bias, it would in fact in one important sense be epistemically costly. (shrink)

When do two sentences say the same thing, that is, express the same content? We defend two-component (2C) semantics: the view that propositional contents comprise (at least) two irreducibly distinct constituents: (1) truth-conditions and (2) subject-matter. We contrast 2C with one-component (1C) semantics, focusing on the view that subject-matter is reducible to truth-conditions. We identify exponents of this view and argue in favor of 2C. An appendix proposes a general formal template for propositional 2C semantics.

We propose a solution to the problem of logical omniscience in what we take to be its fundamental version: as concerning arbitrary agents and the knowledge attitude per se. Our logic of knowledge is a spin-off from a general theory of thick content, whereby the content of a sentence has two components: an intension, taking care of truth conditions; and a topic, taking care of subject matter. We present a list of plausible logical validities and invalidities for the logic of (...) knowledge per se for arbitrary agents, and isolate three explanatory factors for them: the topic-sensitivity of content; the fragmentation of knowledge states; the defeasibility of knowledge acquisition. We then present a novel dynamic epistemic logic that yields precisely the desired validities and invalidities, for which we provide expressivity and completeness results. We contrast this with related systems and address possible objections. (shrink)

Phenomenal intentionality theories have recently enjoyed significant attention. According to these theories, the intentionality of a mental representation (what it is about) crucially depends on its phenomenal features. We present a new puzzle for these theories, involving a phenomenon called ‘intentional identity’, or ‘co-intentionality’. Co-intentionality is a ubiquitous intentional phenomenon that involves tracking things even when there is no concrete thing being tracked. We suggest that phenomenal intentionality theories need to either develop new uniquely phenomenal resources for handling the puzzle, (...) or restrict their explanatory ambitions. (shrink)

Peter Ludlow shows how word meanings are much more dynamic than we might have supposed, and explores how they are modulated even during everyday conversation. The resulting view is radical, and has far-reaching consequences for our political and legal discourse, and for enduring puzzles in the foundations of semantics, epistemology, and logic.

Scientists may sometimes generalize from their samples to broader populations when they have not yet sufficiently supported this generalization. Do such hasty generalizations also occur in experimental philosophy? To check, we analyzed 171 experimental philosophy studies published between 2017 and 2023. We found that most studies tested only Western populations but generalized beyond them without justification. There was also no evidence that studies with broader conclusions had larger, more diverse samples, but they nonetheless had higher citation impact. Our analyses reveal (...) important methodological limitations of many experimental philosophy studies and suggest that philosophical training may not protect against hasty generalizations. (shrink)