Let T be a complete first-order theory over a finite relational language which is axiomatized by universal and existential sentences. It is shown that T is almost trivial in the sense that the universe of any model of T can be written $F \overset{\cdot}{\cup} I_1 \overset{\cdot}{\cup} I_2 \overset{\cdot}{\cup} \cdots \overset{\cdot}{\cup} I_n$ , where F is finite and I 1 , I 2 ,...,I n are mutually indiscernible over F. Some results about complete theories with ∃∀-axioms over a (...) finite relational language are also mentioned. (shrink)

This paper offers a new semantic theory of existentials (sentences of the form There be NP pivot XP coda ) in which pivots are (second order) predicates and codas are modifiers. The theory retains the analysis of pivots as denoting generalized quantifiers (Barwise and Cooper 1981; Keenan 1987), but departs from previous analyses in analyzing codas as contextual modifiers on a par with temporal/locative frame adverbials. Existing analyses universally assume that pivots are arguments of some predicate, and that codas (...) are main or secondary predicates. It is shown that these analyses cannot account for the behavior of codas with quantifiers and for the interaction of multiple codas, both of which receive a simple treatment on the proposed theory. The assimilation of codas to frame adverbials explains several semantic properties which have not been analyzed in the semantic literature, and that distinguish existentials from their copular counterparts. Furthermore, it highlights important properties of the semantics of modification and its relation to predication. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:EXISTENTIAL PROPOSITIONS IN THE THOUGHT OF ST. THOMAS AQUINAS A REVALENT VIEW of St. Thomas Aquinas's position on the logic of propositions has been that according to him propositions of the :form, x is, hold a privileged place, that they are in a special sense " existential," and that such propositions straight.forwardly attribute the act of exi,stence to an individual or to a class of individuals.1 Some (...) texts seem to support such an interpretation. For example, in his commentary on Aristotle's Perihermeneias, Aquinas says: " To clarify this we must note that the verb 'is ' itself is sometimes predicated in an enunciation, as in ' Socrates is.' By this we intend to signify that Socrates really is." 2 This passage seems to treat the predicate " exists " or " is " as unproblematical, as though it had a straightforward meaning needing little clarification, at least in the context of a logical treatise. On the basis of such a text, one may be tempted to infer that for Aquinas " Socrates exists " has the same logical form as " Socrates sits," and that " Men exist " is of the same form as " Men are mortal.'' Peter Geach has shown, however, that Aquinas's doctrine here is not as simple as it may first appear. For Aquin,as some of the propositions which have the grammatical form of x is have 'a different logical form.3 Geach points out that for Aquinas the sentence, "Blindness exists," for example, actual1 E.g., Etienne Gilson, Being and Some Philosophers, 2nd ed. (Toronto: Pontifical Institute of Mediaeval Studies, 1952), 190-215. 2 Super Peri, Bk. II, lect. 2, # 212. Cf. Summa Contra Gentiles, Bk. II, ch. 37 at "FJ11J hoc..•." a Peter Geach, in P. T. Geach and G. E. M. Anscombe, Three Philosophers (New York: Cornell University, 1961), 88 ff; Peter Geach, "Form and Existence," in Aquinas: A Collection of Critical Essays, ed. A. Kenny (New York: Notre Dame U., 1969), pp. 41-47. 605 606 PATRICK LEE ly means, Something or other is blind. That is, blind is the real predicate rather than exists. Mo11eover, Geach argues, the logical peculiarity of this sentence is equally found in any sentence of the form, "an F exists." Such a proposition, " does not,attribute actuality (existence) to an F, but F-ness to something or other..." 4 And yet Geach will not apply rthis interpretation to all sentences of the form, " x is." He al'lgues that sentences in which " is " or " exists " is. joined to a logically pl'oper name are in a different category. Unlike in general propositions such as "an F exists," in such singular propositions, those using logically proper names, Geach,argues, actual existence is attributed to or predicated of an individual. "We have here a sense of 'is ' or 'exists ' that seems to me to be certainly a genuine predicate of individuals: the sense of ' exists ' in which one says that an individual came to exist, still exists, no longer exists, etc...."5 I believe that, while Geach's position is more accurate than the first interpretation, mentioned above, it is nevertheless still inaccurate on certain key points. While it is misleading to say that for Aquinas exists is never a genuine predicate; nevertheless, I will argue that it is his position that exists or existence is not a genuine predicate in any proposition which does not refer back to a previous proposition (or to what is known in or by previous propositional knowledge). Let us crull a proposition which refers to something known in a previous proposition a " second-order p11oposition," and one that does not thus refer a " first-order proposition." Then, I shall argue that for Aquinas exists is not a genuine predicate in any first-order proposition. Moreover, I believe that these points are crucial to understanding Aquinas's position on existence as well as his position on how knowledge is related,to reality, i.e., his realism. In this article I would like to examine Aquinas's thought on the logic of existential propositions and discuss its significance for epistemology. 4 Geach, " Form and Existence," Zoo. int., p. 45. 5 Ibid., p. 46. EXISTENTIAL PROPOSITIONS IN AQUINAS 607 In sections... (shrink)

Some theories assume that sentences like (i) with a presupposition trigger in the scope of a quantifier carry an existential presupposition, as in (ii); others assume that they carry a universal presupposition, as in (iii). No student knows that he is lucky. Existential presupposition: At least one student is lucky.Universal presupposition: Every student is lucky. This work is an experimental investigation of this issue in French. Native speakers were recruited to evaluate the robustness of the (...) inference from (i) to (iii). The main result is that presuppositions triggered from the scope of the quantifier aucun‘no’ are in fact universal. But the present results also suggest that the presuppositions triggered from the scope of other quantifiers depend on the quantifier. This calls for important changes in the main theories of presupposition projection. (shrink)

Contemporary logicians continue to address problems associated with the existential import of categorical propositions. One notable problem concerns invalid instances of subalternation in the case of a universal proposition with an empty subject term. To remedy problems, logicians restrict first-order predicate logics to exclude such terms. Examining the historical origins of contemporary discussions reveals that logicians continue to make various category mistakes. We now believe that no proposition per se has existential import as commonly understood and thus (...) it is unnecessary to restrict first-order predicate logics to non-empty classes. After introducing the problem, we trace some nineteenth century treatments of the issue to locate a source of misconstruing propositional import in misconceptions of ‘implies’ and ‘affirms’ and name the process/product fallacy, along with the translation of categorical sentences using quantifiers and accommodating an empty class. Next we treat some metalogical matters to orient our discussion by which we provide a more precise nomenclature about ‘sentence’ and ‘proposition’ to correct previous misconceptions; here we uncover a common category mistake in respect of a proposition’s efficacy. The semantic distinction between agent and force is helpful in this connection. We conclude by showing that logicians have reinserted existence as a predicate, a position previously excised by Kant, and that the Frege-Russell ambiguity thesis applies only to relationships within a categorical sentence between grammatical predicate and subject. (shrink)

Merleau-Ponty's Phenomenology of Perception - a canonical text of twentieth-century philosophy - concludes with an appeal to 'heroism' by citing a series of enigmatic sentences drawn from Saint-Exupe;ry's Pilote de guerre. Surprisingly, however, these lines are antithetical to the philosophical thrust of Merleau-Ponty's project. This book aims to explain this situation. Foregrounding liminal themes in Merleau-Ponty's thought that have been largely overlooked - e.g., sacrifice, death, myth, faith - and showing how these themes support Merleau-Ponty's reinterpretation of Husserlian phenomenology, (...) Smyth shows that Merleau-Ponty's appeal to 'heroism' represents an extra-philosophical appeal to a historical purposiveness as a universal feature of human nature, and that Merleau-Ponty makes this appeal in virtue of his recognition of the intrinsic methodological limitations of philosophy as a theoretical endeavor. The book thus recovers the 'militant' dimension of Merleau-Ponty's thought. This sheds considerable new light on his work. It does so in a way that challenges some of the basic parameters of existing Merleau-Ponty scholarship by illuminating the intrinsic normativity of his existential phenomenology, and its epistemic reliance on forms of non-reason such as faith and myth. (shrink)

We present various results regarding the decidability of certain sets of sentences by Simple Type Theory. First, we introduce the notion of decreasing sentence, and prove that the set of decreasing sentences is undecidable by Simple Type Theory with infinitely many zero-type elements ; a result that follows directly from the fact that every sentence is equivalent to a decreasing sentence. We then establish two different positive decidability results for a weak subtheory of math formula. Namely, the decidability (...) of math formula and math formula. Finally, we present some consequences for the set of existential-universal sentences. All the above results have direct implications for Quine's theory of “New Foundations” and its weak subtheory math formula. (shrink)

This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal-existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–Suszko preservation theorems (...) follow. (shrink)

We study the relations between abelian groups B and C that every universal (resp. universal-existential) sentence true in B is also true in C, and give algebraic criteria for these relations to hold. As a consequence we characterize the inductive complete theories of abelian groups and prove that they are exactly the model-complete theories.

Sylvia Wynter ends her monumental essay “‘No Humans Involved:’ An Open Letter to My Colleagues” (1994) with an urgent call to address the dire condition of the jobless and poor: “We must now undo their narratively condemned status.” Who are “we”? The sentence separates the university and its “narratively condemned” other. In fact, what the pronoun “we” in the open letter refers to is situated and far from universal, for it is “we in academia” that institute the Western imperial (...) constructs of race and class through “rigorous elaboration” of the present order of knowledge, the one that fails to register its misfits as humans, thus demoting them to the category of N.H.I. (“no humans involved”). I take Wynter’s letter as a point of departure by attending to its mode of address, its conscious use of the pronoun “we,” in light of her genealogy of the Eurocentric conceptions of humanity fundamentally entrenched in university vocation. In so doing, I re-raise the ethical and existential questions around the university, that is, its accountability for the enduring abjection of the N.H.I.-categorized Other, as well as the purpose and very reason for its being as an institutional structure. (shrink)

The problem concerns quantifiers that seem to hover between universal and existential readings. I argue that they are neither, but a different quantifier that has features of each. NOTE the published paper has a mistake. I have corrected this in the version on this site. A correction note will appear in Analysis.

The current study aims to examine the hope levels of theology students in the context of their education level. The correlational (relational) screening method was used in this study. The sample of the study consists of a total of 429 students (328 girls, 101 boys) studying at the Faculty of Theology at Erzincan Binali Yildirim University. Hope levels of the students were determined by Karaca-Kandemir Hope Scale developed by Karaca and Kandemir. The scale consists of three sub-dimensions: goal-oriented, hope and (...) significance, and stability. One-way analysis of variance (ANO-VA) was used for data analysis. According to the findings obtained as a result of the analysis, it is seen that the education levels of the Theology Faculty students cause differentiation in terms of goal-oriented and stability which are sub-dimensions of hope; however, it was found that the scale did not cause any difference in terms of hope and meaningfulness subscale and total hope levels. This finding did not corroborate the research hypothesis that the higher the education level of theology faculty students, the higher their hope level. The results, however, give an idea that theology education can have a positive effect on students' hope levels.Summary: Hope, which is a fundamental feature of human existence, has many functions, both individually and socially. Hope, which is one of the main characteristics that distinguish human beings from other beings, is one of the most powerful tools that help create and implement future designs of individuals and societies. It is possible to see this powerful effect of hope in the object of all human works and actions such as art, literature, and architecture. However, despite the expressed impact of hope, religion, which accompanies it in the existential journey of humanity and is one of its most influential sources of motivation, has the potential to produce a highly useful function on hope. Because religion demands the individual to be future-oriented rather than the past. This demand of religion does not mean that the past actions of the individual and society have no importance. While building the future of the individual and society on lessons learned from the past, religion orders that he should not be a prisoner of his past and that he hopes for the energy he has produced by taking shelter in the attributes of Allah, Most Gracious and Merciful. For this reason, the Qur'an equates despair of Allah's mercy with coming out of religion (Yusuf 12 / / 87).Despite the strong relations between religion and hope, empirical studies on the issue are not yet adequate. However, in recent times, especially with the effect of Positive Psychology, it is a fact that empirical studies have been carried out to determine the relationship of these emotions with religiosity about positive emotions such as patience, optimism, empathy, and self-regulation. One of the positive emotions expressed is hope. However, it is seen that some empirical studies have been conducted on the relationship between hope and religiosity. However, apart from these studies, empirical studies on higher education levels are needed in order to reveal the relationship between hope and religiosity in a healthier way. Because of the strong relationship between religion and hope, it is expected that there will be an increase in parallel with the religious education taken in higher education. From this point of view, this study primarily aims to reveal the strong connection between religion and hope. The main problem of this study, which is based on the primary hypothesis that learning religion at a higher education level, will increase the level of hope of students, is the relationship between theology education and hope. In this study, which was conducted to determine the hope levels of the students of the Faculty of Theology, the method of collecting information related to the questionnaire technique used in the screening model was used.The population of the study consists of 744 students enrolled in Erzincan Binali Yıldırım University Theology Faculty in the fall semester of 2018-2019 academic year. The sample group of the research is 429 students in the faculty during the application period. Accordingly, 144 (33.6%) of the total students in the sample group were in the “preparatory class,” 79 (18.4%) were in the “first grade”, and 78 (18.2%) were in the “second grade,” while 60 (14.0%) were in the “third grade,” 68 (15.9%) were in the “fourth grade.”“Personal Data Questionnaire” and “Karaca-Kandemir’s Hope Scale” were used as data collection instruments. In the “Personal Data Questionnaire,” there are questions about the learning levels of the working group.Karaca-Kandemir’s Hope Scale, which is used in the study, is a scale developed by Karaca and Kandemir based on their cultural codes to determine the hope levels of individuals.The scale is a 5-point Likert type and consists of three sub-dimensions: “goal-oriented,” “hope and meaningfulness,” and “stability.” Of the 13 items in the scale, 8 (1, 2, 4, 5, 7, 9, 10 and 12) had a positive sentence structure such as “Dreaming about the future excites me,” 5 of them (3, 6, 8, 11 and 13) it has a negative sentence structure such as “I cannot bring the end of a job easily after starting a job.” In the scoring phase of the hope scale, negative sentences were reversed. While the lowest score that can be obtained from the “goal-oriented” dimension of the scale is 5, the highest score is 25; the lowest score that can be obtained from the dimension of “hope and meaningfulness” is 4, the highest score is 20; the lowest score that can be obtained from the “stability” sub-dimension is 4, while the highest score is 20. As a result, the lowest score that can be obtained from the total of the scale is 13, and the highest score is 65. Low scores from the scale indicate that the individual's level of hope is low, and high scores indicate high levels of hope.In order to determine the differentiation between class levels, Single Factor Analysis of Variance-ANOVA was applied to the data set. As a result of the analysis, multiple comparisons were made between the means with statistically significant differences. For this purpose, Post Hoc LSD and Tukey HSD tests were used to determine the source of the difference. The significance level of “.05 “ was taken as a criterion for all these operations. In the research, it was seen that the education they received from the Faculty of Theology students caused a differentiation in the “goal-oriented and stability” sub-dimension of hope; however, “hope and meaningfulness” subscale and “total hope” did not cause any differentiation. This result did not confirm the research hypothesis that “the level of hope differs according to class (education) level and the level of hope increases as the level of education (education) increases hope. However, in the “goal-oriented” dimension of hope, the second and third-grade students of fourth-grade students; In terms of “stability” dimension, the fact that the first-grade students scored significantly higher than the preparatory class students gives an important idea that the education received in the Faculty of Theology can have an effect on hope. However, it is thought that similar studies that will be conducted with a large sample group and centering on all dimensions and hopes of religiosity will produce remarkable results. (shrink)

Although the theory of definability had many important antecedents—such as the descriptive set theory initiated by the French semi-intuitionists in the early 1900s—the main ideas were first laid out in precise mathematical terms by Alfred Tarski beginning in 1929. We review here the basic notions of languages, explicit definability, and grammatical complexity, and emphasize common themes in the theories of definability for four important languages underlying, respectively, descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic. We review (...) the history of previous studies of the similarities and differences in the theories of definability of the first three of these four languages. A seminal role leading toward unification of the theories has been played by the separation principles introduced by Nikolai Luzin in 1927. Emphasizing analogies and driving toward further unification embracing finite-universe logic we concentrate on a simple example—the first and second separation principles for existential-universal first-order sentences . Using this as a test case for the fundamental problem of how to “finitize” arguments in classical pure logic to the finite-universe case, we are led to the analogous negative solution by using the theory of certain special graphs: a graph is - special for any positive integers m , n , p , q iff it is bipartite with m red points and n blue points and for every p -tuple of red points there is a blue point to which they are all connected . As an aside we introduce for further study a natural “Ramseyesque” increasing sequence A of positive integers, where A is the least positive integer n for which an -special graph exists. (shrink)

We identify complete fragments of the simple theory of types with infinity and Quine’s new foundations set theory. We show that TSTI decides every sentence ϕ in the language of type theory that is in one of the following forms: ϕ=∀x1r1⋯∀xkrk∃y1s1⋯∃ylslθ where the superscripts denote the types of the variables, s1>⋯>sl, and θ is quantifier-free, ϕ=∀x1r1⋯∀xkrk∃y1s⋯∃ylsθ where the superscripts denote the types of the variables and θ is quantifier-free. This shows that NF decides every stratified sentence ϕ in the language (...) of set theory that is in one of the following forms: ϕ=∀x1⋯∀xk∃y1⋯∃ylθ where θ is quantifier-free and ϕ admits a stratification that assigns distinct values to all of the variables y1,…,yl, ϕ=∀x1⋯∀xk∃y1⋯∃ylθ where θ is quantifier-free and ϕ admits a stratification that assigns the same value to all of the variables y1,…,yl. (shrink)

Peirce’s True Gorgias is a brief dialogue from his essay “Grounds of Validity of the Laws of Logic”, published in 1869. The True Gorgias exposes the fallacy of existential import. It has received no sustained attention in the secondary literature, perhaps because the fallacy is now familiar. Peirce’s assessment of the fallacy involved in the reasoning, however, changes between 1865 and 1869, and he only arrives at the contemporary account of existential import in 1880. Moreover, a careful examination (...) of the context in which the True Gorgias appears reveals it provides an argument in support of Peirce’s early Aristotelian realism about universals in contradistinction to non-Aristotelian varieties of realism advocated by W.T. Harris, editor of the journal in which the True Gorgias was published. (shrink)

Sentences such as Olivia can take Logic or Algebra (‘♢∨-sentences’) are typically interpreted as entailing that Olivia can take Logic and can take Algebra. Given a standard semantics for modals and disjunction, those ‘Free choice’ (FC) readings are not predicted from the surface form of ♢∨-sentences. Yet the standard semantics is appropriate for the ‘double prohibition’ reading typically assigned to ¬♢∨-sentences like Olivia can’t take Logic or Algebra. Several extant approaches to FC can account for those (...) two cases, but face challenges when ♢∨, ¬♢∨ and related sentences appear embedded in certain environments. In this paper, we present a novel account of FC that builds on a ‘grammatical’ theory of scalar implicatures — proposed by Bassi et al. (2021) and Del Pinal (2021) — according to which covert exhaustification is a presupposition trigger such that the prejacent forms the assertive content while any excludable or includable alternatives are incorporated at the non-at issue, presuppositional level. Applied to ♢∨, ¬♢∨, and similar sentences, ‘presuppositional exhaustification’ predicts that their default interpretations have an assertive component (roughly, the classical interpretation of the prejacent) and a homogeneity presupposition which projects in standard ways. Those predictions, we then show, support a uniform account of the puzzling behavior of ♢∨, ¬♢∨, and related sentences when embedded under (negative) factives (Marty & Romoli 2020), disjunctions (Romoli & Santorio 2019), and in the scope of universal, existential (Bar-Lev & Fox 2020) and non-monotonic quantifiers (Gotzner et al. 2020). (shrink)

There are several ways for defining the notion submodel for Kripke models of intuitionistic first‐order logic. In our approach a Kripke model A is a submodel of a Kripke model B if they have the same frame and for each two corresponding worlds Aα and Bα of them, Aα is a subset of Bα and forcing of atomic formulas with parameters in the smaller one, in A and B, are the same. In this case, B is called an extension of (...) A. We characterize theories that are preserved under taking submodels and also those that are preserved under taking extensions as universal and existential theories, respectively. We also study the notion elementary submodel defined in the same style and give some results concerning this notion. In particular, we prove that the relation between each two corresponding worlds of finite Kripke models A ≤ B is elementary extension (in the classical sense) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

Graham Priest's new book is about things being about things—about what it is for things which are about things, such as beliefs, hopes and fears, and the like, and sentences which express them, to be about the things they are about, and about the range of things about which things which are about are about—in a word, intentionality. It has two principal objectives—to develop a formal semantics for intentionality, and to promote and defend a philosophical thesis about what exists (...) and what does not. The semantics proposed is an extension of the familiar ‘world’1 semantics for modal logic. The philosophical thesis asserts that while some things exist, not everything does, or equivalently that while some things exist, some others do not. Priest, following Richard Routley , calls this Noneism to mark the divergence from Meinong's view, with which, nevertheless, it has a good deal in common. Meinong likewise denies that everything exists, and asserts that some things exist and some do not, but further holds that some things which, according to him, do not exist do have another ‘form of being’—they ‘subsist’ rather than exist—while other things which, according to him, do not exist do not exist simpliciter . Noneism avoids this complication, at least, and—or so Priest claims—avoids proliferating kinds of being altogether, requiring only a single kind of being and a distinction between two kinds of quantifier. In the noneist view, the basic universal and particular quantifiers are to be understood in an ontologically unloaded way, as ranging over a single all-inclusive domain of objects, some of which exist and some of which do not. The usual—existentially loaded—quantifiers are construed as a type of restricted quantifier, definable using an existence predicate: ‘GraphicxAx’ as ‘For some x, …. (shrink)

In this paper, I defend a grammatical account of scalar implicatures. In particular, I submit new evidence in favor of the contextual blindness principle, assumed in recent versions of the grammatical account. I argue that mismatching scalar implicatures can be generated even when the restrictor of the universal quantifier in a universal alternative is contextually known to be empty. The crucial evidence consists of a hitherto unnoticed oddness asymmetry between formally analogous existential sentences with reference failure (...) NPs. I conclude that the generation of mismatching scalar implicatures does not require contextual access. (shrink)

Popper's well-known demarcation criterion has often been understood to distinguish statements of empirical science according to their logical form. Implicit in this interpretation of Popper's philosophy is the belief that when the universe of discourse of the empirical scientist is infinite, empirical universal sentences are falsifiable but not verifiable, whereas the converse holds for existential sentences. A remarkable elaboration of this belief is to be found in Watkins's early work on the statements he calls “all-and-some,” such (...) as: “For every metal there is a melting point.” All-and-some statements are both universally and existentially quantified in that order. Watkins argued that AS should be regarded as both nonfalsifiable and nonverifiable, for they partake in the logical fate of both universal and existential statements. This claim is subject to the proviso that the bound variables are “uncircumscribed” ; i.e., that the universe of discourse is infinite. (shrink)

A logically impossible sentence is one which entails a contradiction, a logically necessary sentence is one whose negation entails a contradiction, and a logically possible sentence is one which does not entail a contradiction. Metaphysically impossible, necessary and possible sentences are ones which become logically impossible, necessary, or possible by substituting what I call informative rigid designators for uninformative ones. It does seem very strongly that a negative existential sentence cannot entail a contradiction, and so ”there is a (...) God’ cannot be a metaphysically necessary truth. If it were such a truth, innumerable other sentences which seem paradigm examples of logically possible sentences, such as ”no one knows everything’ would turn out to be logically impossible. The only way in which God could be a logically necessary being is if there were eternal necessary propositions independent of human language or God’s will, such that the proposition that there is no God would entail -- via propositions inaccessible to us -- a contradiction. But if there were such propositions, God would have less control over the universe than he would have otherwise. (shrink)

What good is logic? -- Seventeen ways this book is different -- The two logics -- All of logic in two pages : an overview -- The three acts of the mind -- I. The first act of the mind : understanding -- Understanding : the thing that distinguishes man from both beast and computer -- Concepts, terms and words -- The problem of universals -- The comprehension and extension of terms -- II. Terms -- Classifying terms -- Categories -- (...) Predicables -- Division -- III. Material fallacies -- Fallacies of language -- Fallacies of diversion -- Fallacies of oversimplification -- Fallacies of argumentation -- Inductive fallacies -- Procedural fallacies -- Metaphysical fallacies -- Short story : love is a fallacy -- IV. Definition -- The nature of definition -- The rules of definition -- The kinds of definition -- The limits of definition -- V. Second act of the mind : judgment -- Judgments, propositions, and sentences -- What is truth? -- The four kinds of categorical propositions -- Logical form -- Euler's circles -- Tricky propositions -- The distribution of terms -- VI. Changing propositions -- Immediate inference -- Conversion -- Obversion -- Contraposition -- VII. Contradiction -- What is contradiction? -- The square of opposition -- Existential import -- Tricky propositions on the square -- Some practical uses of the square of opposition -- VIII. The third act of the mind : reasoning -- What does reason mean? -- The ultimate foundations of the syllogism -- How to detect arguments -- Arguments vs. explanations -- Truth and validity -- IX. Different kinds of arguments -- Three meanings of because -- The four causes -- A classification of arguments -- Simple argument maps -- Deductive and inductive arguments -- Combining deduction and induction : socratic method -- X. Syllogisms -- The structure and strategy of the syllogism -- The skeptics objection to the syllogism -- The empiricist's objection to the syllogism -- Demonstrative syllogisms -- How to construct convincing syllogisms -- XI. Checking syllogisms for validity -- By Euler's circles -- By Aristotle's six rules -- Barbara Celarent : mood and figure -- Venn diagrams -- XII. More difficult syllogisms -- Enthymemes : abbreviated syllogisms -- Sorites : chain syllogisms -- Epicheiremas : multiple syllogisms -- Complex argument maps -- XIII. Compound syllogisms -- Hypothetical syllogisms -- Reductio ad absurdum arguments -- The practical syllogism : arguing about means and ends -- Disjunctive syllogisms -- Conjunctive syllogisms -- Dilemmas -- XIV. Induction -- What is induction? -- Generalization -- Causal induction : Mill's methods -- Scientific hypotheses -- Statistical probability -- Arguments from analogy -- A fortiori and a minore arguments -- XV. Some practical applications of logic -- How to write a logical essay -- How to write a socratic dialogue -- How to have a socratic debate -- How to use socratic method on difficult people -- How to read a book socratically -- XVI. Some philosophical applications of logic -- Logic and theology -- Logic and metaphysics -- Logic and cosmology -- Logic and philosophical anthropology -- Logic and epistemology -- Logic and ethics. (shrink)

Presents a set-theoretic version of the analysis of "there be" as predicating instantiation of a property, a property-theoretic version of which was developed in McNally 1992. This paper provides a solution to the criticism that McNally 1992's analysis could not account for sentences in which postverbal nominal contains a monotone decreasing or nonmonotonic determiner.

L'A. si interroga con Tommaso sul dogma trinitario che pone una difficoltà molto grande soprattutto rispetto alla questione degli universali: come è possibile che il Padre sia Dio, allo stesso tempo una persona e allo stesso tempo che la paternità sia una relazione? Questi predicati che si applicano alla Trinità sono predicati con le stesse modalità con cui si usano tutti gli altri? L'A. esamina i modelli G-S e S-I e distingue la dimensione concettuale da quella ontologica, per delineare cosa (...) sia l'essere in una specie e in un genere, per concludere sull'impossibilità di applicare il modello universale-particolare a Dio. L'Aquinate propone un'alternativa ai modelli G-S e S-I che trova insufficienti: propone un modello che può essere chiamato comune-proprio: una cosa può essere comune in due modi distinti, da una parte numericamente, dall'altra secondo la definizione. Quindi communitas secundum rem, che è quella della natura divina; communitas secundum rationem, che è quella del concetto di relazione divina e persona divina. Il concetto di relazione divina è in ultima istanza diverso da quello di animal. (shrink)

In this paper we define a suitable version of the notion of homomorphism for Kripke models of intuitionistic first-order logic and characterize theories that are preserved under images and also those that are preserved under inverse images of homomorphisms. Moreover, we define a notion of union of chain for Kripke models and define a class of formulas that is preserved in unions of chains. We also define similar classes of formulas and investigate their behavior in Kripke models. An application to (...) intuitionistic first-order arithmetic is also given. (shrink)

In this paper, Universal any and Negative Polarity Item any are uniformly analyzed as ‘counterfactual’ donkey sentences (in disguise). Their difference in meaning is reduced here to the distinction between strong and weak readings of donkey sentences. It is shown that this explains the universal and existential character of Universal- and NPI-any, respectively, and the positive and negative contexts in which they are licensed. Our uniform analysis extends to the use of any in command (...) and permission sentences. It predicts that whereas the use of any in permission sentences is licensed and gives rise to a universal reading, it is not licensed in command sentences. (shrink)

An AE-sentence is a sentence in prenex normal form with all universal quantifiers preceding all existential quantifiers, and the AE-theory of a structure is the set of all AE-sentences true in the structure. We show that the AE-theory of \, \cap, \cup, 0, 1)\) is decidable by giving a procedure which, for any AE-sentence in the language, determines the truth or falsity of the sentence in our structure.

The purpose of the book is to clarify the notion of existential dependence and cognate notions, such as supervenience and the notion of an internal relation. I defend the view that such notions are best understood in terms of the concept of metaphysical grounding, i.e. the concept of one fact obtaining in virtue of other facts, where ‘in virtue of’ has a distinctively metaphysical meaning.

Plain disjunctive sentences, such as _The mystery box contains a blue ball or a yellow ball_, typically imply that the speaker does not know which of the two disjuncts is true. This is known as an ignorance inference. We can distinguish between two aspects of this inference: the negated universal upper bound part (i.e., the speaker is uncertain about each disjunct), which we call uncertainty, and the existential lower bound part (i.e., the speaker considers each disjunct possible), (...) which we call possibility. In the traditional approach, uncertainty is derived as a primary implicature, from which possibility follows. In this paper, we report on two experiments using a sentence-picture verification task based on the mystery box paradigm that challenge the traditional implicature approach. Our findings show that possibility can arise without uncertainty, and we thus call for a reevaluation of the traditional view of disjunction and ignorance inferences. Our experimental findings are related to similar results involving disjunction in embedded contexts and pave the way for alternative theories that can account for the observed patterns of inference derivation in a unified fashion. We discuss how recent implicature and non-implicature theories can account for the derivation of existential lower bound inferences without the presence of negated universal upper bound inferences. (shrink)

We argue that the extant evidence for Stoic logic provides all the elements required for a variable-free theory of multiple generality, including a number of remarkably modern features that straddle logic and semantics, such as the understanding of one- and two-place predicates as functions, the canonical formulation of universals as quantified conditionals, a straightforward relation between elements of propositional and first-order logic, and the roles of anaphora and rigid order in the regimented sentences that express multiply general propositions. We (...) consider and reinterpret some ancient texts that have been neglected in the context of Stoic universal and existential propositions and offer new explanations of some puzzling features in Stoic logic. Our results confirm that Stoic logic surpasses Aristotle’s with regard to multiple generality, and are a reminder that focusing on multiple generality through the lens of Frege-inspired variable-binding quantifier theory may hamper our understanding and appreciation of pre-Fregean theories of multiple generality. (shrink)

Simple locative sentences show a variety of pseudo-quantificational interpretations. Some locatives give the impression of universal quantification over parts of objects, others involve existential quantification, and yet others cannot be characterized by either of these quantificational terms. This behavior is explained by virtually all semantic theories of locatives. What has not been previously observed is that similar quantificational variability is also exhibited by locative sentences containing indefinites with the ‘a’ article. This phenomenon is especially problematic for (...) traditional existential treatments of indefinites. We propose a solution where indefinites denote properties and are assigned locations similarly to other spatial descriptions. This Property-Eigenspace Hypothesis accounts for the correlation between the interpretations of locative indefinites and the pseudo-quantificational effects with simple entity-denoting NPs. Thereby, the proposal opens up a new empirical domain for property-based theories of indefinites, with implications for the analysis of collective descriptions, generics, negative polarity items and part–whole structure. (shrink)

This book presents and defends a large number of theses in ontology and meta-ontology. The meta-ontological theses are broadly Quinean: that existence or being is what is expressed by the existential quantifier of formal logic; that the variables the quantifiers bind are essentially third-person-singular pronouns; that the “ontological commitments” of a person or theory are best revealed when the sentences of the person or theory are translated into the quantifier-variable idiom. Much of the book is devoted to ontological, (...) as opposed to meta-ontological, questions: Are there things that do not exist—or even things that have no sort of being whatever? Are there universals—or, more generally, abstract objects? What sort of thing is a property (quality, attribute…)? What are the ontological commitments of modal discourse? Are there possible worlds? If there are, do some of them contain “non-actual” individuals? (The meta-ontology severely constrains the ontology. For example, a non-actual but possible individual is an individual thing that might exist but does not. Therefore, non-actual individuals do not exist. That is to say, there aren’t any.) The book closes with the presentation and defense of an ontology consistent with everything said in the earlier parts of the book. The central thesis of this ontology is that everything belongs to one of two ontological categories, substance and relation. Relations are universals, and propositions are special kinds of relation. Special consideration is given to the absence of sets from the ontology. (shrink)

Plural definite descriptions across many languages display two well-known properties. First, they can give rise to so-called non-maximal readings, in the sense that they ‘allow for exceptions’. Second, while they tend to have a quasi-universal quantificational force in affirmative sentences, they tend to be interpreted existentially in the scope of negation. Building on previous works, we offer a theory in which sentences containing plural definite expressions trigger a family of possible interpretations, and where general principles of language (...) use account for their interpretation in various contexts and syntactic environments. Our theory solves a number of problems that these previous works encounter, and has broader empirical coverage in that it offers a precise analysis for sentences that display complex interactions between plural definites, quantifiers and bound variables, as well as for cases involving non-distributive predicates. The resulting proposal is briefly compared with an alternative proposal by Križ, which has similar coverage but is based on a very different architecture and sometimes makes subtly different predictions. (shrink)

‘Quantified pure existentials’ are sentences (e.g., ‘Some things do not exist’) which meet these conditions: (i) the verb EXIST is contained in, and is, apart from quantificational BE, the only full (as against auxiliary) verb in the sentence; (ii) no (other) logical predicate features in the sentence; (iii) no name or other sub-sentential referring expression features in the sentence; (iv) the sentence contains a quantifier that is not an occurrence of EXIST. Colin McGinn and Rod Girle have alleged that (...) standard first-order logic cannot adequately deal with some such existentials. The article defends the view that it can. (shrink)

This paper will be concerned with the conjunctive interpretation of a family of disjunctive constructions. The relevant conjunctive interpretation, sometimes referred to as a “free choice effect,” (FC) is attested when a disjunctive sentence is embedded under an existential modal operator. I will provide evidence that the relevant generalization extends (with some caveats) to all constructions in which a disjunctive sentence appears under the scope of an existential quantifier, as well as to seemingly unrelated constructions in which conjunction (...) appears under the scope of negation and a universal quantifier. (shrink)

We characterize pairs of monotone generalized quantifiers Q1 and Q2 over finite domains that give rise to an entailment relation between their two relative scope construals. This relation between quantifiers, which is referred to as scope dominance, is used for identifying entailment relations between the two scopal interpretations of simple sentences of the form NP1–V–NP2. Simple numerical or set-theoretical considerations that follow from our main result are used for characterizing such relations. The variety of examples in which they hold (...) are shown to go far beyond the familiar existential-universal type. (shrink)

A quasivariety of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A. It is structurally complete if and only if the free ℵ0‐generated algebra in can serve as A. A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of all satisfy the same existential positive sentences. We prove that if is PSC then it still has the JEP, and if it has the (...) JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if is relatively semisimple then it is generated by one ‐simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics. (shrink)

This paper addresses two puzzles in the semantics of connected exceptive phrases (EP): (i) the compatibility of EPs modifying noun phrases headed by the negative polarity item (NPI) determiner any and (ii) the ability of a negative universal quantifier modified by an EP to license strong NPIs. Previous analyses of EPs are shown to fail to solve these puzzles. A new unified solution to the two puzzles is proposed. The crucial insight of the analysis is to allow von Fintel’s (...) (Natural Language Semantics 1: 123–148, 1993; Restrictions on quantifier domains, PhD dissertation, 1994) leastness condition on EPs to be imposed at scopes non-local to the surface position of the EP. A general result is derived concerning the truth conditions of sentences in which an existential modified by an EP occurs in the scope of a downward entailing operator. The distribution of EPs is argued to depend on how the leastness condition interacts with other pragmatic strengthening conditions, such as the one imposed on any by Kadmon and Landman (Linguistics and Philosophy 16: 353–422, 1993). (shrink)

The application of the maximum entropy principle to determine probabilities on finite domains is well-understood. Its application to infinite domains still lacks a well-studied comprehensive approach. There are two different strategies for applying the maximum entropy principle on first-order predicate languages: applying it to finite sublanguages and taking a limit; comparing finite entropies of probability functions defined on the language as a whole. The entropy-limit conjecture roughly says that these two strategies result in the same probabilities. While the conjecture is (...) known to hold for monadic languages as well as for premiss sentences containing only existential or only universal quantifiers, its status for premiss sentences of greater quantifier complexity is, in general, unknown. I here show that the first approach fails to provide a sensible answer for some \-premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture. (shrink)

Still, some may still want to say it. If so, my replies may gain nothing better than a stalemate against such persistence, though I can hope that earlier revelations will discourage others from persisting. But two replies are possible. Both come down, one circuitously, to an issue with us from the beginning: whether the language of the right side of (10) is suspect. For if (10) is to support instances for (6) which are about objects, that clause must itself be (...) about objects. (These would be ones assigned by variants of I it mentions to constants it mentions.) Yet Barwise and I would call it implicitly about functions. Ironically, the discussion surrounding (10), hoping to settle that issue, could only do so if the issue were already settled, revealing decisively the ontology of (10). If irritating, this is also inevitable, given the Tarskian spirit of the ensivioned semantics for QLB.A second reply begins by noting that a QLB semantics which implies (10) cannot simply be assumed. Even a QL semantics augmented to imply (11) is not trivial to frame, as I will let readers confirm. On the QLB project, Barwise comments thus:It is not possible to explain the meaning of an essential use of branching quantification ... inductively, by treating one quantifier at a time in a first-order fashion. Some use of higher-type abstract objects is essential. (BQE75)But believers in a modest ontology for QLB can claim a non sequitur here. For (10), as they read it, succeeds without mentioning abstract objects, though not in the “one quantifier at a time” way which Barwise rightly finds impossible. This is not yet to say the same about a QLB semantics implying (10). But that too can be said, as it happens, with as good a conscience as with (10). A suitable treatment can begin by somehow linearizing non QL sentences like (6). Still assuming prefixes whose rows each consist, speaking loosely, of n universal quantifiers followed by a single existential, we could simply line up these rows in any order, with unique deconcatenatability being assured. From then on, it gets both tedious and complex. A full syntax is essential, and some surprising categories arose in mine. I will spare readers all details, except to say that one can indeed “treat one quantifier at a time”, if not “in first-order fashion”: schematic rules for treating n quantifiers at once can be eschewed. Unsurprisingly, heavy use is made of the “depending only on” idiom seen in (10). Nor does it surprise me that this semantics can succeed, with that idiom available. Roughly, if open talk about functions works in a semantics for QLF, what I read as implicit talk about them should work in a semantics for QLB See [6] for a start on a semantics aspiring to handle sentences like (6) without invoking abstract objects. See also my comments on its notion of dependence, in [9]. . So even from a bare sketch, we can see that this new semantics settles nothing. It just leads back to the stalemate. Barwise and I will read crucial clauses as talking implicitly about functions, but this general charge against an idiom is equally deniable wherever the idiom occurs: in a reading for (6), say, or in the metalanguage can hardly repress a motherhood slogan: better dead than obscurely read. But that just denies the denial, unhelpfully. A better comment is the reminder that the claim stays alive only in a form which no one has ever imagined for it. The QLB quantifiers that cannot be shown not to range over objects are not the items anyone would ever have pointed to in illustrating the unkillable claim. (shrink)

. Fresh evidence from Free Choice Items (FCIs) in French question the current perception of the class. The role of some standard distinctions found in the literature is weakened or put in a new perspective. The distinction between universal and existential is no longer an intrinsic property of FCIs. Similarly, the opposition between variation-based vs intension-based analyses is relativized. We show that the regime of free choiceness can be characterized by an abstract constraint, that we call Non-Individuation (NI), (...) and which can be satisfied in different ways that match current distinctions. NI says that the information conveyed by a sentence containing a FCI should not be reducible to a referential situation, that is a situation in which particular individuals satisfy the sentence in the current world. The widely used resource of modal variation becomes a particular scenario of free-choiceness, not its essence. In fact, we show that under certain conditions, FCIs can occur in episodic, non-modal sentences, a fact that NI can accommodate. We also discuss more fine-grained aspects of the semantics of FCIs, such as their emotional colour. (. . .) the tripod fell spontaneously, because, though it stood on its feet so as to serve for a seat, it did not fall so as to serve for a seat. Aristotle, Physics II,6. (shrink)

We define notions of homomorphism, submodel, and sandwich of Kripke models, and we define two syntactic operators analogous to universal and existential closure. Then we prove an intuitionistic analogue of the generalized (dual of the) Lyndon-Łoś-Tarski Theorem, which characterizes the sentences preserved under inverse images of homomorphisms of Kripke models, an intuitionistic analogue of the generalized Łoś-Tarski Theorem, which characterizes the sentences preserved under submodels of Kripke models, and an intuitionistic analogue of the generalized Keisler Sandwich (...) Theorem, which characterizes the sentences preserved under sandwiches of Kripke models. We also define several intuitionistic formula hierarchies analogous to the classical formula hierarchies $\forall_n (= \Pi^0_n)$ and $\exists_n (=\Sigma^0_n)$ , and we show how our generalized syntactic preservation theorems specialize to these hierarchies. Each of these theorems implies the corresponding classical theorem in the case where the Kripke models force classical logic. (shrink)

The paper contains a survey of existence problems in the folowing calculi of formal logic: calculi of classes, syllogistics, classic predicate calculus, Lesniewski's ontology, quantificational modal calculi, quantificational tense logics, sentential calculi. The particular attention is given to metaphysical (ontological) presuppositions and commitments of logical calculi especially to the problems of empty set, existential axiom (axiom of the non-emptiness of universe of discourse), existential operators (quantifiers and functors), possible worlds, temporal predicates and operators, objects of sentences.

Sophists’ apologia. -/- Sophists were the first paid teachers ever. These ancient Greek enlighteners taught wisdom. Protagoras, Antiphon, Prodicus, Hippias, Lykophron are most famous ones. Sophists views and concerns made a unified encyclopedic system aimed at teaching common wisdom, virtue, management and public speaking. Of the contemporary “enlighters”, Deil Carnegy’s educational work seems to be the most similar to sophism. Sophists were the first intellectuals – their trade was to sell knowledge. They introduced a new type of teacher-student relationship – (...) the mutually beneficial communication on equal terms. They taught pupils how to think independently and how to persuade others, which was inseparably connected with the rise of democracy in the most advanced Greek polices. Sophists were the first to proclaim that all people are naturally equal. They put forward the idea of natural rights and social contract, working out the fundamentals of the present-day law. In addition, they developed the basics of philology, psychology, logic, and gave a scientific explanation to the origins of religions. They embodied definite positive ideals of their epoch. Sophistry flourished in the second half of the Vth through he first third of the IVth centuries B. C. The period coincides with the heyday of the entire Greek history – the so called Greek Miracle epoch. Sophists expressed some present-day concepts of Ancient Greek ideology and mentality, characteristic of its Golden Age times. The Ancient Athens of the times of Periclus were similar to the Florence of Renaissance as to the type of social structure. That is why (?) the art of eloquence (public speaking) was highly respected in Florence. The two cities had a lot of features of the present-day capitalist society, which was due to their role as the main centres of industry of their “worlds-economies”. During the subsequent Hellenic period, the Old Order was restored, based on the feudal rent recipients’ rule. The degradation of society resulted in the decline of sophistry. Sophistry represents the highest point in the evolution of Greek philosophy. It is also the most prominent school as to the novelty of its ideas. That was owing to the common propensity for innovation typical of Periclus’ Golden Age, which in its turn resulted from the establishment of the individualistic and rationalistic thinking similar to present-day mentality. Among other aspects, close attention was paid to ethnic issues; some humanistic ideas such as Master and Servant ethics and Enlightment as the primary condition of man’s freedom, received scientific grounding. The genre of philosophical dialogue was developed, as well as the rationalistic scientific gnoceology, aimed at consistently opposing any religion. Dialectics, the most outstanding achievement of the Antiquity, was also worked out by the sophists. Zeno of Helleas was one of the sophists. Democrites’ ideas were, too, quite close to dialectics. Socrates’ contemporaries regarded him ironically, as shown, for example, in the Aristophanes’ comedy “Clouds”. Nevertheless, this scholar can still be referred to as a sophist, although of somewhat weird personality. Socrates’ popularity, which came later, should be attributed to the fact that he was the first philosopher ever to have been sentenced and executed by law, rather than to any valuable scientific contributions. Sophistry as such, rather than Socrates, marks the line between pre-Socratic and post- Socratic schools. Plato, with the exception of his latest works, is considered to belong to the same philosophical school. Specifically, there are valid reasons to believe that he was paid fees by his disciples. In Plato’s Dialogues, all points of view will be proven and then disproved, so that an integral philosophical system is hard to observe. Plato’s “Idealism” is but one of the many possible theories. His ideal philosophy is a high-minded argument of wise men. As late as towards the end of his life, Plato betrayed the ideals of sophistry, turning into the world’s first proponent of totalitarianism. Sophistry is Greek classical literature. Later on, philosophy, as well as the entire Greek civilization, started to fall into decline. Over 500 subsequent years, Hellenistic-Roman scholars never came up with anything novel. The entire process of antique philosophical evolution can be subdivided into the following six stages: Phoenician (IX–VIIth centuries B. C.), archaic (VIth – the first half of IVth century B. C., classic (the epoch of sophistry: the second half of the Vth through the first decades of the IIIrd centuries B. C.), late antique (middle of the IIIrd – first decades of the VI th centuries B.C.) The fact that early in the VIth century B. C. sophistry was rejected in Athens, has a lot to do with the consequences of their defeat in Peloponnesian war, a social catastrophe. Ever since, the prevalent part of the criticism of sophistry is a criticism of sophists’ immoral ways of life and personalities, rather than the essentials of their philosophy. For instance, Aristotle’s criticism primarily concerns their pragmatism and tendency to make a profit out of teaching. However, Aristotle himself taught his disciples along similar lines, showing them how to “be able to prove both opposites”. It was Aristotle who summed up sophists’ century-long elaborations. The traditional modern interpretation of sophists as shown in Plato’s dialogues appears to be groundless. The truth of the matter is that Plato regards sophists as more serious philosophers than Socrates was. It becomes especially clear in the dialogue titled “Protagoras”, which shows the latter as gaining an undoubted victory over Socrates. Generally speaking, Plato’s works never regard sophists as weak or worthless opponents. A common opinion as to the essence of sophists’, in particular, Protagoras’s philosophy, has hitherto not been worked out. Many researchers renounce the very existence of any specific philosophical position in sophistry. The term “relativism” itself now has a bad name. However, it should be borne in mind that Protagoras doctrine emerged as a result of an attempt to get out of the ontological dead-end to which the entire Greek natural philosophy had come. To this purpose, he employed Anaxagoras’s idea about everything comprising everything, whereupon he built his physical-ontological theory, and Zeno of Helleas’s concept that all objects are such and different at the same time in the same respect. According to Protagoras’s philosophy, every object does not only seem, but is different from every other object, including human beings. Therefore, every person is his or her own measure of all other entities. The objective reality is the Ego’s relations with the outer world. Every object’s essence is defined through its interaction with ourselves. Nothing remains stagnant – every existence already contains non-existence, and vice-versa. However, the relativity of being is not to be regarded as absolute. That means, that relatively stable and therefore relatively cognizable entities do exist, among them the words we say. Doubt is not considered absolute by relativists either, which differentiates them from skeptics and agnostics. They dare to believe. Uncertainty is Godly – here, Protagoras becomes very close to the concept of apathetic theology. Gorgius shared Protagoras’s views and tried to prove them applying the same method as Zeno applied when upholding Parmenidis’s philosophy. He argued that if one accepts the opposite point of view and agrees that nothing can exist and not exist at the same time, one is bound to come to an the absurd conclusion that nothing is cognizable and nothing exists. To prove that there is a universal contradiction within the Existence itself, Protagoras and his followers in their turn put forward a number of special logical arguments, called rules of contraries, or sophisms. The most well-known of them is the Euathle’s sophism, which proves that an object can be such and different at the same time in the same respect. Generally speaking, all relationships of any entity are extraneous. The idea that all entities, including concepts, exist and do not exist at the same time, was also accepted by Plato. It should be stressed, that Plato’s philosophy as a whole is relativists’ greatest achievement. In his dispute against relativism, Aristotle put forward the main concepts of his ontology: the existence falls into the existence in possibility and the existence in reality, or substance and accidence – the ever changing “sublunary” world and the stable world of divine heavenly bodies. This division made his system self-contradictory to the point of absurd. Aristotle quite realized, that the direct disproof of relativism is hardly possible, which fact makes his criticism a mere tautology. It is very indicative in this respect that materialists, idealists and even skeptical agnostics applied the same arguments do disprove relativism. That demonstrates that the inner contradiction between any forms of “absolutism” and relativism is the most fundamental philosophical problem. Relativist ideas were repeatedly renewed throughout history, provided the social conditions were favourable. In variations, the concepts were advanced by Tzuan Tsi and ancient Chinese sophists, apostle Paul, Nicolas of Cusa, Galileo, Rene Descartes, G. Berkley, D. Hume, Ch. S. Pierce, E. Mach, A. Bogdanov, F. de Saussure, N. Bore, P. Feuerabend, U. Quain and others. However, no consistent relativist ontological system was developed after Protagoras. Meanwhile, a valid relativist theory could provide a clue to a fuller and more adequate general physical concept of the world, and to a deeper understanding of quantum mechanics. -/- Relativism as an ontological system of beliefs. -/- The relativist principle of relativity of all that exists is fundamental for the entire system of philosophy. Absolutely everything exists, but, at the same time, no existence is absolute. Anything is possible, but those entities only exist for us with which we interact this way or another, i. e., reality is interaction, “I interact – hence, I exist”. However, an overly close relationship between entities leads to non-existence – in other words, there is a certain existential optimum. For all of us, information, or perceptible heterogeneities, is real. Each of us is the center of a definite system of interactions with the world. If God is an endless Possibility, then the World is but the man’s dialogue with God. However, the ties that bind us with the world are of some definite kind, which predetermines our reality. The principle of relativity constrains itself, as any relativity is relative. On the one hand, that calls forth the Evil – caused by our own weakness. On the other hand, it makes our world relatively stable and cognizable. Generally speaking, the world is what we are. Everybody has an own Universe, but since people are fairly similar, our universes can be regarded as one Universe with common laws and regulations. It is possible for us to change the degree of objects’ existence in relation to ourselves and one another. It is equally important, though, to remain ourselves (keep following our “warrior’s path”). Since everything exists and no existence is absolute, the process of cognition should involve a criticism of everything but never a complete rebuttal of anything. The criterion of verity for any theory then is in its capacity to build itself upon a bigger number of diverse statements than in any other theory. Theoretically, all philosophies should ideally be complementary. The question of what in fact happened is meaningless unless we specify the “we”, the “where” and the “when”. The past can only exist as part of the present. When we change, our past undergoes corresponding qualitative objective changes, all of which are irreversible. The contemporary physics regards all objects of the Universe as existing for one another, which results in absurd conclusions. In fact, only what we interact with, exists. The so called “speed of light in vacuum” c is the maximum speed of direct interaction and direct exchange of signals. Specifically, at the oncoming speed c with a significant blue parallax the gamma-quantum is unable to interact with the matter. When this speed is reached, the gravitational mass of objects in relation to one another becomes equal to zero (imaginary), whereas their inertial mass equals infinity. Further acceleration will bring us into an entirely different Universe. That means that the Universe is an open system, which fact destroys all present-day cosmological models. Anything is possible in this world. Our reality is magic. Every bit of space contains a potentially limitless number of bits of other, however virtual, universes, as well as an infinite number of particles, which are still virtual for us (as postulated by quantum mechanics). In other words, there exists an infinity of different realities. With all the above in mind, the different levels of entities’ existence, in particular, of Life and Reason, should be taken into consideration. For example, to which extent a live system exists in relation to the inanimate forms of being is very indefinite, as what life is has to be determined by living creatures. An even greater compass of levels of existence can be brought about through interaction – that is the core idea of Godel’s theorem. As a result, the range of forms of existence tends towards infinity – the evolution of matter through super-intelligence will bring us up to God’s heights. A more generalized mathematical model of the physical world, including a universal operator of the co-existence of objects, can be developed based on the above considerations. The bonds between objects become weaker owing to 1) a high speed, 2) acceleration, 3) rotation, 4) strong gravity, 5) strong fields of other kinds, 6) the difference of levels of existence , 7) phase shifts. Yet, there always is an existential optimum. Such mathematical model is to operate certain non-transitive algebras. Relativist ontology assumes that the notion of “temperature” is relative, thus making it possible to ignore the Second Law of Thermodynamics. In its turn, that will allow for phase shifts. A perpetual motion machine of the second type will become a reality. Rotation could presumably be used to increase inertial and decrease gravitational mass, slowing down the time. An experimental research could shed light on the mentioned effects. (shrink)