Logic in the twenties: the nature of the quantifier

1979 ◽  
Vol 44 (3) ◽  
pp. 351-368 ◽  
Author(s):  
Warren D. Goldfarb
Keyword(s):  
Predicate Logic ◽  
Formal System ◽  
Recent Book ◽  
Modern Logic ◽  
Nineteen Twenties ◽  
Quantification Theory ◽  
Michael Dummett ◽  
The Road ◽  
Semantic Treatment ◽  
Do So

We are often told, correctly, that modern logic originated with Frege. For Frege clearly depicted polyadic predication, negation, the conditional, and the quantifier as the bases of logic; moreover, he introduced the idea of a formal system, and argued that mathematical demonstrations, to be fully precise, must be carried out within a formal language by means of explicitly formulated syntactic rules.Consequently Frege has often been read as providing all the central notions that constitute our current understanding of quantification. For example, in his recent book on Frege [1973], Michael Dummett speaks of ”the semantics which [Frege] introduced for formulas of the language of predicate logic.” That is, “An interpretation of such a formula … is obtained by assigning entities of suitable kinds to the primitive nonlogical constants occurring in the formula … [T]his procedure is exactly the same as the modern semantic treatment of predicate logic” (pp. 89–90). Indeed, “Frege would therefore have had within his grasp the concepts necessary to frame the notion of the completeness of a formalization of logic as well as its soundness … but he did not do so” (p. 82).This common appraisal of Frege's work is, I think, quite misleading. Even given Frege's tremendous achievements, the road to an understanding of quantification theory was an arduous one. Obtaining such understanding and formulating those notions which are now common coin in the discussion of logical systems were the tasks of much of the work in logic during the nineteen-twenties.

scholarly journals The Road to Modern Logic—An Interpretation

2001 ◽  
Vol 7 (4) ◽  
pp. 441-484 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Order Logic ◽  
Point Of View ◽  
First Order Logic ◽  
Theoretical Point ◽  
Primary Role ◽  
Philosophical Discussion ◽  
Modern Logic ◽  
Core System ◽  
First Order ◽  
The Road

AbstractThis paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplars (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which First-Order Logic matured will be seen to have played a primary role in its shaping.Mathematical logic is what logic, through twenty-five centuries and a few transformations, has become today. (Jean van Heijenoort)

AN INTENSIONAL LEIBNIZ SEMANTICS FOR ARISTOTELIAN LOGIC

2010 ◽  
Vol 3 (2) ◽  
pp. 262-272 ◽  
Author(s):  
KLAUS GLASHOFF
Keyword(s):  
Natural Deduction ◽  
Predicate Logic ◽  
Formal System ◽  
Aristotelian Logic ◽  
Categorical Logic ◽  
Characteristic Numbers ◽  
Formal Syntax ◽  
Philosophical Standpoint ◽  
Classical Philosophy

Since Frege’s predicate logical transcription of Aristotelian categorical logic, the standard semantics of Aristotelian logic considers terms as standing for sets of individuals. From a philosophical standpoint, this extensional model poses problems: There exist serious doubts that Aristotle’s terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this question—they looked at terms as standing for concepts (“Begriffe”). In 1972, Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction, while, with respect to semantics, he still made use of an extensional interpretation. In this paper we deal with a simple intensional semantics for Corcoran’s syntax—intensional in the sense that no individuals are needed for the construction of a complete Tarski model of Aristotelian syntax. Instead, we view concepts as containing or excluding other, “higher” concepts—corresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcoran’s work, furnishing his formal syntax with an adequate semantics which is free from presuppositions which have entered into modern interpretations of Aristotle’s theory via predicate logic.

Syllogism and quantification

1962 ◽  
Vol 27 (1) ◽  
pp. 58-72 ◽  
Author(s):  
Timothy Smiley
Keyword(s):  
Predicate Logic ◽  
Modern Logic ◽  
First Order ◽  
First Order Predicate Logic

Anyone who reads Aristotle, knowing something about modern logic and nothing about its history, must ask himself why the syllogistic cannot be translated as it stands into the logic of quantification. It is now more than twenty years since the invention of the requisite framework, the logic of many-sorted quantification.In the familiar first-order predicate logic generality is expressed by means of variables and quantifiers, and each interpretation of the system is based upon the choice of some class over which the variables may range, the only restriction placed on this ‘domain of individuals’ being that it should not be empty.

The “Rules of the Road”: Ethics, Firearms, and the Physician's “Lane”

2020 ◽  
Vol 48 (S4) ◽  
pp. 142-145
Author(s):  
Blake N. Shultz ◽  
Benjamin Tolchin ◽  
Katherine L. Kraschel
Keyword(s):  
Medical Ethics ◽  
Patient Autonomy ◽  
Critical Role ◽  
Firearm Injury ◽  
Agency Relationship ◽  
The Road ◽  
Firearm Violence ◽  
Patient Agency ◽  
Foundational Principle ◽  
Do So

Physicians play a critical role in preventing and treating firearm injury, although the scope of that role remains contentious and lacks systematic definition. This piece aims to utilize the fundamental principles of medical ethics to present a framework for physician involvement in firearm violence. Physicians' agency relationship with their patients creates ethical obligations grounded on three principles of medical ethics — patient autonomy, beneficence, and nonmaleficence. Taken together, they suggest that physicians ought to engage in clinical screening and treatment related to firearm violence. The principle of beneficence also applies more generally, but more weakly, to relations between physicians and society, creating nonobligatory moral ideals. Balanced against physicians' primary obligations to patient agency relationships, general beneficence suggests that physicians may engage in public advocacy to address gun violence, although they are not ethically obligated to do so. A fourth foundational principle — justice — requires that clinicians attempt to ensure that the benefits and burdens of healthcare are distributed fairly.

On formulas with valid cases

1956 ◽  
Vol 21 (2) ◽  
pp. 148-148
Author(s):  
W. V. Quine
Keyword(s):  
Atomic Formula ◽  
Quantification Theory ◽  
Truth Value ◽  
The One ◽  
Do So

Commonly, when we succeed in showing a formula of quantification theory to be consistent, we do so by producing a true interpretation. Sometimes we achieve the same effect without even exceeding the resources of quantification theory itself: we show a formula consistent by producing a valid formula from it by substitution. Example: ‘(∃x)Fx ⊃ (x)(∃y) (Gxy ▪ Fy)’ is seen consistent by noting its valid substitution case ‘(∃x)Fx ⊃ (x)(∃y)(Fx ∨ Fy ▪ Fy)’. How generally available is this latter method? I shall show that it is available if and only if the formula whose consistency is shown is satisfiable in a one-member universe.The “only if” part is immediate. For, if ψ is a substitution case of ϕ, then ϕ is satisfiable wherever ψ is; and ψ, if valid, is satisfiable in a one-member universe.Conversely, suppose a true interpretation of ϕ in a one-member universe. Each predicate letter of ϕ is thereby interpreted outright as true or false, in effect, since there is no varying the values of ‘x’, ‘y’, etc. Now form ψ from ϕ by substitution as follows: change each atomic formula ϕi(e.g. ‘Fx’, ‘Fy’, ‘Gxy’) to ⌜ϕi⊃ϕi⌝ if its predicate letter is one that was interpreted as true, or to ⌜ϕi ▪ − ⊃ϕi⌝ if its predicate letter is one that was interpreted as false. Clearly ψ under all interpretations even in large universes, will receive the truth value that ϕ received under the supposed interpretation in the one-member universe. So ψ is valid.

Recent Advances in Ordinal Analysis: Π12 — CA and Related Systems

1995 ◽  
Vol 1 (4) ◽  
pp. 468-485 ◽  
Author(s):  
Michael Rathjen
Keyword(s):  
Proof Theory ◽  
Formal System ◽  
Ordinal Analysis ◽  
The Road ◽  
Recent Success ◽  
Theoretic Proof ◽  
Independence Results ◽  
On The Road ◽  
Order Arithmetic ◽  
Made In

§1. Introduction. The purpose of this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of -analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to -formulae. The same techniques can be used to provide ordinal analyses for theories that are reducible to iterated -comprehension, e.g., -comprehension. The details will be laid out in [28]. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z2. Considerable progress has been made in proof theory since Gentzen's tragic death on August 4th, 1945, but an ordinal analysis of Z2 is still something to be sought. However, for reasons that cannot be explained here, -comprehension appears to be the main stumbling block on the road to understanding full comprehension, giving hope for an ordinal analysis of Z2 in the foreseeable future. Roughly speaking, ordinally informative proof theory attaches ordinals in a recursive representation system to proofs in a given formal system; transformations on proofs to certain canonical forms are then partially mirrored by operations on the associated ordinals. Among other things, ordinal analysis of a formal system serves to characterize its provably recursive ordinals, functions and functionals and can yield both conservation and combinatorial independence results.

Nonstandard theories of quantification and identity

1970 ◽  
Vol 35 (2) ◽  
pp. 267-294 ◽  
Author(s):  
A. Trew
Keyword(s):  
Predicate Logic ◽  
Type A ◽  
The Other ◽  
Standard System ◽  
Free Logic ◽  
Quantification Theory ◽  
Singular Terms ◽  
Free Variables

In this paper a number of nonstandard systems of predicate logic with or without identity, are translated with subsystems of applied standard system of predicate logic with identity. There are nonstandard theories of quantification which, following [16], are described as inclusive systems; their theorems are valid in all domains, including the empty domain. Theories of quantification which allow for the substitution of denotationless terms for free variables, are described, following [21], as systems of free logic; they are said to be free of the requirement that all singular terms must have denotations. Free logics and inclusive logics may each be of the other type. A nonstandard theory of identity, described, following [12] as a theory of nonreflexive identity, may be combined with a standard or with a nonstandard theory of quantification. Another kind of nonstandard system of predicate logic examined is a nonstandard version of a system of monadic predicate logic in which a distinction is made between sentence and predicate negation, and which is nonstandard in the sense that the laws relating sentence and predicate negation diverge from the standard ones. In the systems examined, this is combined with an inclusive quantification theory.

scholarly journals MUSIC MANAGEMENT- NEW OPPORTUNITIES NEW CHALLENGES

2015 ◽  
Vol 3 (1SE) ◽  
pp. 1-3
Author(s):  
Anjana Saxena
Keyword(s):  
Music Industry ◽  
File Sharing ◽  
New Music ◽  
Music Business ◽  
The Road ◽  
New Challenges ◽  
The Relationship ◽  
Do So ◽  
Financial Aspect

The transition to digital is changing the music industry. As technology has advanced over recent years, the music industry has consequently undergone a drastic change in the way it operates. This industry-wide shift has its pros and its cons: On one hand, the internet serves as an incredible platform on which anyone can exhibit their talent and potentially build a fan base. On the other hand, the presence of millions of people attempting to do so make it more and more difficult for any one person to stand out, and the reality of file sharing and illegal downloading makes the financial aspect of music much more complex. Regardless of one`s opinion about the road that the music industry has traveled down, a music manager must be flexible enough to keep up with the changes that the industry undergoes. The meaning and role of a “manager” has changed drastically over the last decade as the traditional business model has given way to the “new” music business Traditionally a manager managed an artist’s efforts to get signed to a label and once signed, he/she managed the relationship between the artist and the label. But given the state of labels today the unsigned artist must assume that he/she will never be signed and build a career accordingly. A traditional manager is often unable and ill – equipped to successfully manage and develop an artist’s career in the new environment.

scholarly journals How Do eHMIs Affect Pedestrians’ Crossing Behavior? A Study Using a Head-Mounted Display Combined with a Motion Suit

Information ◽  
2019 ◽  
Vol 10 (12) ◽  
pp. 386 ◽  
Author(s):  
Lars Kooijman ◽  
Riender Happee ◽  
Joost de Winter
Keyword(s):  
Urban Environment ◽  
Motion Tracking ◽  
Automated Vehicles ◽  
Forward Motion ◽  
The Road ◽  
Head Mounted Display ◽  
Human Machine Interfaces ◽  
Do So

In future trac, automated vehicles may be equipped with external human-machine interfaces (eHMIs) that can communicate with pedestrians. Previous research suggests that, during first encounters, pedestrians regard text-based eHMIs as clearer than light-based eHMIs. However, in much of the previous research, pedestrians were asked to imagine crossing the road, and unable or not allowed to do so. We investigated the effects of eHMIs on participants’ crossing behavior. Twenty-four participants were immersed in a virtual urban environment using a head-mounted display coupled to a motion-tracking suit. We manipulated the approaching vehicles’ behavior (yielding, nonyielding) and eHMI type (None, Text, Front Brake Lights). Participants could cross the road whenever they felt safe enough to do so. The results showed that forward walking velocities, as recorded at the pelvis, were, on average, higher when an eHMI was present compared to no eHMI if the vehicle yielded. In nonyielding conditions, participants showed a slight forward motion and refrained from crossing. An analysis of participants’ thorax angle indicated rotation towards the approaching vehicles and subsequent rotation towards the crossing path. It is concluded that results obtained via a setup in which participants can cross the road are similar to results from survey studies, with eHMIs yielding a higher crossing intention compared to no eHMI. The motion suit allows investigating pedestrian behaviors related to bodily attention and hesitation.

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