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Measure 7 Working memory models the maximum strain on the working memory and measure 8 Proof length models the length of the shortest train of thought leading to the desired conclusion. Measure 9 Proof size is based on the idea that latency in the context of logical reasoning depends on some notion of computational workload. By identifying the working memory load with formula length, we get proof size as a measure of computational workload.

Thus measure 9 models the minimum amount of data that must flow through the working memory for the problem to be solved. Proofs of minimum length and minimum size were generated for all items appearing in the experiment. For instance, the proof appearing in Sect. Table 1 shows the mean latency and mean accuracy recorded at the experiment. Table 2 shows correlations between latency and the mathematical complexity measures defined in Sect.

Table 3 shows correlations between accuracy and the same complexity measures. In this paper, we studied human reasoning in the case of FO truth using standard methods of cognitive psychology. This reflects our view that there is nothing special about problem-solving in the domain of logic, and therefore, it can be explored using ordinary experimental methods and modeled using ordinary models of cognitive psychology that would apply equally well to mental arithmetic, Sudoku, or Minesweeper, for example.

In a preliminary investigation, we observed that the difficulty of determining truth was affected by the manner in which the models were presented graphically. For instance, determining whether a graph is complete seems to be simpler if the graph is drawn in a regular fashion.

To mitigate this problem, we drew our graphs by placing the nodes equally distributed on a circle. As is usual in experiments of this type, the answers given by the participants are potentially problematic because guesses, interaction errors e. Another potential problem relates to the instructions. Our instructions failed to address this point. On the aggregate level, some of the problems on the individual level might partially cancel out, but then new problems arise on the modeling side.

In fact, to model experimental data on the aggregated level, one must develop a computational model that represents some sort of average of the participants. Strictly speaking, this might not be possible using our present type of computational model, which was designed for cognitive modeling on the individual level.

These factors should be borne in mind when evaluating the results. The bounded proof systems described in Sect. One way is to modify the axioms; a second is to modify the rules; a third is to change the working memory and sensory memory capacities; and a fourth is to add models of other cognitive resources.

Intuitively, proof-size reflects how comprehensive a thought must be, i. Proofs that require more information processing might take longer to process and be more prone to error. One may perhaps think of the smallest proofs as the smartest proofs, relatively to a given repertoire of cognitive resources. As is often the case with cognitive modeling, this complexity measure can be criticized for being too coarse. For instance, it does not directly reflect the effort of searching for a proof; instead, it reflects the effort required to verify the steps in a proof that has been found.

Although the efforts required for finding a proof and verifying the same proof may be somewhat correlated, this limitation certainly allows for future improvements to the model. The data in Table 1 indicate that the True items were easier to solve than the False items. Among the complexity measures analyzed in Table 2 , Proof size has the highest correlation with latency, both for True and False items. Second comes the closely related complexity measure Proof length.

This might indicate that complexity measures that are defined in terms of computations are more adequate than those defined in terms of standard properties of models and sentences. Some of the complexity measures on sentences in Table 2 fared relatively well. The reason why those complexity measures on sentences were relatively successful in this particular case might be that the models that were used in the experiment were quite homogeneous.

In fact, Cardinality varied between 3 and 4, and Edge count varied between 3 and 6. Therefore, the complexity measures that considered only sentences were perhaps not sufficiently challenged in the present experiment. All of the complexity measures analyzed in Table 3 have relatively low correlations with accuracy. We do not know why the contrast to latency is so pronounced. The correlation values for Negation count stand out here and provide some support to the idea that negations increase the probability of error.

The results indicate that proof size was more successful than the other complexity measures in the case of latency. To evaluate the usefulness of this approach, which combines elements of proof theory and cognitive psychology, more experiments are needed, in particular experiments with more heterogeneous test items. Adler, J. Reasoning: Studies of human inference and its foundations.

Cambridge: Cambridge University Press. Book Google Scholar. Anderson, J. The atomic components of thought. Mahwah, NJ: Lawrence Erlbaum. Google Scholar. Braine, M. Mental logic. UK: L. Erlbaum Associates. Cassimatis, N. Polyscheme: A cognitive architecture for integrating multiple representation and inference schemes. PhD thesis. Ebbinghaus, H. Extended logics: The general framework. Model-theoretic logics pp.

Satisfaction classes in nonstandard models of arithmetic. Licentiate thesis, Chalmers University of Technology. Fitch, F. Symbolic logic: an introduction. New York: Ronald Press. Gentzen, G. Investigation into logical deduction, Szabo Eds. North-Holland Amsterdam. Geuvers, H. Rewriting for Fitch style natural deductions. In Rewriting techniques and applications. Springer pp. Gilhooly, K. Working memory and strategies in syllogistic-reasoning tasks.

Article Google Scholar. Hitch, G. Verbal reasoning and working memory. The Quarterly Journal of Experimental Psychology , 28 4 , — Holyoak, K. The Cambridge handbook of thinking and reasoning. Huth, M. Logic in computer science: Modelling and reasoning about systems. Jaskowski, S. The theory of deduction based on the method of suppositions.

Studia Logica , 1 , 5— Johnson-Laird, P. Mental models. Kosslyn, S. Cognitive psychology: Mind and brain. Laird, J. Soar: An architecture for general intelligence. Artificial Intelligence , 33 3 , 1— Negri, S. Structural proof theory. Nizamani, A. Anthropomorphic proof system for first-order logic. Masters thesis, Chalmers University of Technology. Prawitz, D.

Natural deduction. In A proof-theoretical study, volume 3 of Stockholm studies in philosophy. Rips, L. The psychology of proof. Robinson, A. Handbook of automated reasoning. The Netherlands: Elsevier Science. Sheeran, M. January Formal Methods in Systems Design , 16 1 , 23— Smullyan, R.

Logic, First-Order second corrected edition. NewYork: Dover. Stenning, K. Human reasoning and cognitive science. Reasoning processes in propositional logic. Journal of Logic, Language and Information , 19 3 , — Sun, R. Toms, M. Working memory and conditional reasoning. The Quarterly Journal of Experimental Psychology , 46 4 , — Download references. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author s and the source are credited.

You can also search for this author in PubMed Google Scholar. To enable a deeper understanding of the experiment, let us list items 1—9 of the experiment and give their associated data. A full description of the items used can be found in Nizamani Tables 4 and 5 list the sentences and models of items 1—9, respectively. Table 6 shows the data recorded at the experiment concerning items 1—9.

Table 7 , finally, shows the mathematical complexity measures of items 1—9. Note that all formulas listed below are logical truths. Reprints and Permissions. J of Log Lang and Inf 22, — Download citation. Published : 13 February Issue Date : January Search SpringerLink Search. Download PDF. Abstract First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs.

Introduction What truths are humans able to identify in the case of first-order logic FO on finite models? Psychological Complexity Human reasoning concerning FO truth lends itself perfectly to exploration using standard methods of experimental psychology. Full size image. Participants The participants in our experiment were ten computer science students from Gothenburg, Sweden, whom were invited through email.

Material Fifty test items were prepared for the experiment. Procedure The experiment was conducted in a computer laboratory at the Department of Applied Information Technology, University of Gothenburg. Let us briefly take stock of the situation as it existed in Peirce had differentiated between first-order and second-order logic, but had put the distinction to no mathematical use, and it dropped from sight.

Both Frege and Russell had formulated versions of multi-level type theory, but neither had singled out the first-order fragment as an object worthy of study. But Veblen did not possess a precise characterization of formal deduction, and his observation remained inert. Hilbert had lectured and published on foundational topics in the years —; in the intervening time, as he concentrated on other matters, the publications had ceased, though the extensive classroom lecturing continued.

He kept up with current developments, and in particular was informed about the logical work of Whitehead and Russell, largely through his student Heinrich Behmann. When we consider the matter more closely, we soon recognize that the question of the consistency for integers and for sets is not one that stands alone, but that it belongs to a vast domain of difficult epistemological questions which have a specifically mathematical tint: for example to characterize this domain of questions briefly , the problem of the solvability in principle of every mathematical question, the problem of the subsequent checkability of the results of a mathematical investigation, the question of a criterion of simplicity for mathematical proofs, the question of the relationship between content and formalism in mathematics and logic, and finally the problem of the decidability of a mathematical question in a finite number of operations.

Hilbert — Although Bernays had little previous experience in foundations, this turned out to be a shrewd choice, and the beginning of a close and fruitful research partnership. Hilbert for the first time clearly distinguishes metalanguage from object language, and step-by-step presents a sequence of formal logical calculi of gradually increasing strength.

Each calculus is carefully studied in its turn; its strengths and its weaknesses are identified and balanced, and the analysis of the weaknesses is used to prepare the transition to the next calculus. The function calculus is a system of many-sorted first-order logic, with variables for sentences as well as for relations. It is here, for the first time, that we encounter a precise, modern formulation of first-order logic, clearly differentiated from the other calculi, given an axiomatic foundation, and with metalogical questions explicitly formulated.

Hilbert concludes his discussion of first-order logic with the remark:. The basic discussion of the logical calculus could cease here if we had no other end in view for this calculus than the formalization of logical inference. But we cannot be satisfied with this application of symbolic logic. Not only do we want to be able to develop individual theories from their principles in a purely formal way, but we also want to investigate the foundations of the mathematical theories themselves and examine how they are related to logic and how far they can be built up from purely logical operations and concept formations; and for this purpose the logical calculus is to serve us as a tool.

The lecture protocol ends with the sentence:. Thus it is clear that the introduction of the Axiom of Reducibility is the appropriate means to turn the calculus of levels into a system out of which the foundations for higher mathematics can be developed. The following summer, Bernays produced a Habilitation thesis in which he developed, with full rigor, a Hilbert-style, axiomatic analysis of propositional logic. He then proceeds to investigate questions of decidability, consistency, and the mutual independence of various combinations of axioms.

The Hilbert lectures and the Bernays Habilitation are a milestone in the development of first-order logic. In the lectures, for the first time, first-order logic is presented in its own right as an axiomatic logical system, suitable for study using the new metalogical techniques. It was those metalogical techniques that represented the crucial advance over Peirce and Frege and Russell, and that were in time to bring first-order logic into focus.

But that did not happen at once, and a great deal of work still lay ahead. It was characteristic of Hilbert to break complex mathematical phenomena into their elements: the sequence of calculi can be viewed as a decomposition of higher-order logic into its simpler component parts, revealing to his students precisely the steps that went into the building of the full system.

Although he discusses the functional calculus, he does not single it out for special attention. In other words and as with Peirce three decades earlier first-order logic is introduced primarily as an expository device: its importance was not yet clear. His proof of the completeness of propositional calculus is a mere sketch, and relegated to a footnote; the parallel problem for first-order logic is not even raised as a conjecture.

Even more strikingly, when Bernays eventually in published his Habilitation , he omitted his proof of the completeness theorem because as he later ruefully said the result seemed at the time straightforward and unimportant. For discussion of this point, see Hilbert [LFL]: For readily available general discussions, see Sieg , Zach , and the essays collected in Sieg ; for the original documents and detailed analysis, see Hilbert [LFL. The Hilbert school throughout the s regarded first-order logic as a fragment of type theory, and made no argument for it as a uniquely favored system.

He did not at the time publish on these topics because, as he later said:. I believed that it was so clear that the axiomatization of set theory would not be satisfactory as an ultimate foundation for mathematics that, by and large, mathematicians would not bother themselves with it very much.

To my astonishment I have seen recently that many mathematicians regard these axioms for set theory as the ideal foundation for mathematics. For this reason it seemed to me that the time had come to publish a critique. Skolem appendix. In the second he provided a new proof of that result. These technical results were of great importance for the subsequent debate over first-order logic.

But it is important not to read into Skolem a later understanding of the issues. Skolem at this point did not possess a distinction between the object language and the metalanguage. And although in retrospect his axiomatization of set theory can be interpreted to be first-order, he nowhere emphasizes that fact.

Indeed, Eklund presents a compelling argument that Skolem did not yet clearly appreciate the significance of the distinction between first-order and second-order logic, and that the reformulation of the axiom of separation is not in fact as unambiguously first-order as it is often taken to be. There are two broad tendencies within logic during these years, and they pull in opposite directions. One tendency is towards pruning down logical and mathematical systems so as to accommodate the criticisms of Brouwer and his followers.

Set theory was in dispute, and Skolem explicitly presented his results as a critique of set theoretical foundations. To put the matter slightly differently: the very point of axiomatizing set theory was to state its philosophically problematic assumptions in such a way that one could clearly see what they came to.

One possibility was to restrict oneself to first-order logic; another, to adopt some sort of predicative higher-order system. Similar broadly constructivist tendencies were also very much in evidence in the proof theoretical work of Hilbert and Bernays and their followers in the s. Hilbert, like C. The epsilon-substitution method was the principal device Hilbert introduced in order to attempt to attain this result. But despite these constructive tendencies, many logicians of the s including Hilbert continued to regard higher-order type theory, and not its first-order fragment, as the appropriate logic for investigations in the foundations of mathematics.

The ultimate hope was to provide a consistency proof for the whole of classical mathematics including set theory. But, in the meanwhile, researchers still were somewhat unclear about certain basic distinctions. So matters remained unclear throughout the s. But the constructivist ambitions of the Hilbert school, the focus on the analysis of the quantifiers, and the explicit posing of metalogical questions had made the emergence of first-order logic as a system worthy of study in its own right all but inevitable.

With these results and others that soon followed it finally became clear that there were important metalogical differences between first-order logic and higher-order logics. Perhaps most significantly, first-order logic is complete, and can be fully formalized in the sense that a sentence is derivable from the axioms just in case it holds in all models.

Second-order logic does not. By the middle of the s these distinctions were beginning to be widely understood, as was the fact that categoricity can in general only be obtained in higher-order systems. But the technical results alone did not settle the matter in favor of first-order logic. In other words, even after the metalogical results there was a choice to be made, and the choice in favor of first-order logic was not inevitable.

After all, the metalogical results can be taken to show a severe limitation of first-order logic: that it is not capable of specifying a unique model even for the natural numbers. At this point in the s, however, several other strands of thinking about logic now coalesced. The intellectual situation was highly complex. A search for secure foundations, and in particular for an avoidance of the set-theoretical paradoxes, was something they shared, and that helped to tip the balance in favor of first-order logic.

As a practical matter, these first-order set theories sufficed to formulate all existing mathematical practice; so for the codification of mathematical proofs, there was no need to resort to higher-order logic. This confirmed an observation that Hilbert had already made as early as , though without himself fully developing the point. Thirdly, there was an increased tendency to distinguish between logic and set theory, and to view set theory as a branch of mathematics.

By the end of the decade, a consensus had been reached that, for purposes of research in the foundations of mathematics, mathematical theories ought to be formulated in first-order terms. Let us now try to draw some lessons, and in particular ask whether the emergence of first-order logic was inevitable.

I begin with an observation. Each stage of this complex history is conditioned by two sorts of shifting background consideration. One is broadly mathematical: the theorems that had been established. The other is broadly philosophical: the assumptions that were made explicitly or tacitly about logic and about the foundations of mathematics. These two things interacted. Each thinker in the sequence starts with some more or less intuitive ideas about logic.

Those ideas prompt mathematical questions: distinctions are drawn: theorems are proved: consequences are noted, and the philosophical understanding is sharpened. Let us now consider the question: When was first-order logic discovered? That question is too general.

It needs to be broken down into three subsidiary questions:. Equipped with these distinctions, let us now ask: Why was first-order logic not discovered earlier? It is striking that Peirce, already in , had clearly differentiated between propositional logic, first-order logic, and second-order logic. He was aware that propositional logic is significantly weaker than quantificational logic, and, in particular, is inadequate to an analysis of the foundations of arithmetic.

He could then have gone on to observe that second-order logic is in certain respects philosophically problematic, and that, in general, our grasp on quantification over objects is firmer than our grasp on quantification over properties. The problem arises even if the universe of discourse is finite. We have, for example, a reasonable grasp on what it means to speak in first-order terms of all the planets , or to say that there exists a planet with a particular property.

But what does it mean to talk in second-order terms of all properties of the planets? What is the criterion of individuation for such properties? Is the property of being the outermost planet the same as the property of being the smallest planet? What are we to say about negative properties?

Is it a property of the planet Saturn that it is not equal to the integer 17? In that case, although there are only a finite number of planets, our second-order quantifiers must range over infinitely many properties. And so on. The Quinean objections are familiar. Arguments of this sort had been made in the scholastic disputes between realists and nominalists: and Peirce was steeped in the medieval literature on these topics.

That would in any case have run contrary to his logical pluralism. Why did he not make these points already in ? Any answer can only be speculative. One factor, a minor one, is that Peirce was not himself a nominalist. There are also technical considerations. Peirce, unlike Hilbert, does not present first-intentional logic as an axiomatized system, nor does he urge it as a vehicle for studying the foundations of mathematics. He does not possess the distinction between an uninterpreted, formal, axiomatic calculus and its metalanguage.

As a result, he does not ask about questions of decidability, or completeness, or categoricity; and without the metamathematical results a full understanding of the differences in expressive power between first-order and second-order logic was not available to him. He provided a flexible and suggestive notation that was to prove enormously fertile, and he was the first to distinguish clearly between first-order and second-order logic: but the tools for understanding the mathematical significance of the distinction did not yet exist.

As Henri Pirenne once remarked, the Vikings discovered America, but they forgot about it, because they did not yet need it. A related point holds for Frege and Russell. But they never considered isolating the lowest level of the hierarchy as a free-standing system.

There are both philosophical and mathematical reasons for this. And then, as a mathematical matter, second-order logic was necessary to their construction of the integers. So they had no compelling reason, either philosophical or mathematical, that would have led them to focus on the first-order fragment. There is here an instructive contrast with Peirce.

Peirce, in the spirit of the 19 th -century algebraists, was happy to explore a lush abundance of logical structures: his attitude was fundamentally pluralist. The logicists, working in the analytical tradition, were more concerned to discover what the integers actually are : their attitude was fundamentally monistic and reductionist.

But in order to single out first-order logic as was done in the s, two things were needed: an awareness that there were distinct logical systems, and an argument for preferring one to the other. Let us now turn to the question, Was the emergence of first-order logic inevitable?

It is impossible to avoid counterfactual considerations, and the answer must be more speculative. By , the metalogical results can fairly be said to have been inevitable. It would then have been an obvious next step to inquire about the completeness of higher-order systems.

That was a comparatively insignificant matter. Hilbert did not treat it as important, and appears to have viewed it primarily as an expository device, a means of simplifying the presentation of the logic of Principia Mathematica. The important step in was rather the introduction of techniques of metamathematics, and the explicit posing of questions of completeness and consistency and decidability.

To pose these questions for systems of logic was an enormous conceptual leap, and Hilbert understood it as such. And even after he had introduced his metalogical distinctions in his papers of the s, logicians of the caliber of Russell and Brouwer and Ramsey had difficulty in understanding what he was attempting to do.

This development was in anything but inevitable: and without the introduction of the metalogical techniques the history of logic and proof theory in the s and s would have looked very different. As we saw, the metalogical results of the s do not settle the primacy of first-order logic.

All these things show the continuing influence of the Grundlagenkrise of the s, which did so much to set the terms of the subsequent philosophical understanding of the foundations of mathematics. The names of Brouwer and Weyl are nowhere mentioned.

HEIDEGGER THE QUESTION CONCERNING TECHNOLOGY AND OTHER ESSAYS PDF

Among the complexity measures analyzed in Table 2 , Proof size has the highest correlation with latency, both for True and False items. Second comes the closely related complexity measure Proof length. This might indicate that complexity measures that are defined in terms of computations are more adequate than those defined in terms of standard properties of models and sentences.

Some of the complexity measures on sentences in Table 2 fared relatively well. The reason why those complexity measures on sentences were relatively successful in this particular case might be that the models that were used in the experiment were quite homogeneous.

In fact, Cardinality varied between 3 and 4, and Edge count varied between 3 and 6. Therefore, the complexity measures that considered only sentences were perhaps not sufficiently challenged in the present experiment. All of the complexity measures analyzed in Table 3 have relatively low correlations with accuracy.

We do not know why the contrast to latency is so pronounced. The correlation values for Negation count stand out here and provide some support to the idea that negations increase the probability of error. The results indicate that proof size was more successful than the other complexity measures in the case of latency. To evaluate the usefulness of this approach, which combines elements of proof theory and cognitive psychology, more experiments are needed, in particular experiments with more heterogeneous test items.

Adler, J. Reasoning: Studies of human inference and its foundations. Cambridge: Cambridge University Press. Book Google Scholar. Anderson, J. The atomic components of thought. Mahwah, NJ: Lawrence Erlbaum. Google Scholar. Braine, M. Mental logic. UK: L. Erlbaum Associates. Cassimatis, N. Polyscheme: A cognitive architecture for integrating multiple representation and inference schemes. PhD thesis. Ebbinghaus, H. Extended logics: The general framework. Model-theoretic logics pp. Satisfaction classes in nonstandard models of arithmetic.

Licentiate thesis, Chalmers University of Technology. Fitch, F. Symbolic logic: an introduction. New York: Ronald Press. Gentzen, G. Investigation into logical deduction, Szabo Eds. North-Holland Amsterdam. Geuvers, H. Rewriting for Fitch style natural deductions. In Rewriting techniques and applications. Springer pp.

Gilhooly, K. Working memory and strategies in syllogistic-reasoning tasks. Article Google Scholar. Hitch, G. Verbal reasoning and working memory. The Quarterly Journal of Experimental Psychology , 28 4 , — Holyoak, K. The Cambridge handbook of thinking and reasoning. Huth, M. Logic in computer science: Modelling and reasoning about systems.

Jaskowski, S. The theory of deduction based on the method of suppositions. Studia Logica , 1 , 5— Johnson-Laird, P. Mental models. Kosslyn, S. Cognitive psychology: Mind and brain. Laird, J. Soar: An architecture for general intelligence. Artificial Intelligence , 33 3 , 1— Negri, S. Structural proof theory. Nizamani, A. Anthropomorphic proof system for first-order logic. Masters thesis, Chalmers University of Technology.

Prawitz, D. Natural deduction. In A proof-theoretical study, volume 3 of Stockholm studies in philosophy. Rips, L. The psychology of proof. Robinson, A. Handbook of automated reasoning. The Netherlands: Elsevier Science. Sheeran, M. January Formal Methods in Systems Design , 16 1 , 23— Smullyan, R. Logic, First-Order second corrected edition. NewYork: Dover. Stenning, K.

Human reasoning and cognitive science. Reasoning processes in propositional logic. Journal of Logic, Language and Information , 19 3 , — Sun, R. Toms, M. Working memory and conditional reasoning. The Quarterly Journal of Experimental Psychology , 46 4 , — Download references.

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author s and the source are credited. You can also search for this author in PubMed Google Scholar. To enable a deeper understanding of the experiment, let us list items 1—9 of the experiment and give their associated data.

A full description of the items used can be found in Nizamani Tables 4 and 5 list the sentences and models of items 1—9, respectively. Table 6 shows the data recorded at the experiment concerning items 1—9. Table 7 , finally, shows the mathematical complexity measures of items 1—9. Note that all formulas listed below are logical truths. Reprints and Permissions. J of Log Lang and Inf 22, — Download citation. Published : 13 February Issue Date : January Search SpringerLink Search.

Download PDF. Abstract First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs. Introduction What truths are humans able to identify in the case of first-order logic FO on finite models? Psychological Complexity Human reasoning concerning FO truth lends itself perfectly to exploration using standard methods of experimental psychology.

Full size image. Participants The participants in our experiment were ten computer science students from Gothenburg, Sweden, whom were invited through email. Material Fifty test items were prepared for the experiment. Procedure The experiment was conducted in a computer laboratory at the Department of Applied Information Technology, University of Gothenburg.

Now we can define our two bounded proof systems. The maximum length of a sentence that can appear in a proof is 8. Sentence length. Range: 5—9. Range: 3—4. Edge count. Range: 3—6. Let us provide some motivation why these particular complexity measures were chosen. Results Table 1 shows the mean latency and mean accuracy recorded at the experiment. Table 1 Mean latency and mean accuracy for true and false items Full size table. Table 2 Correlations between latency and some mathematical complexity measures Full size table.

Table 3 Correlations between accuracy and some mathematical complexity measures Full size table. Discussion In this paper, we studied human reasoning in the case of FO truth using standard methods of cognitive psychology. Comments on the Experiment In a preliminary investigation, we observed that the difficulty of determining truth was affected by the manner in which the models were presented graphically.

Comments on the Proof Systems The bounded proof systems described in Sect. Comments on the Results The data in Table 1 indicate that the True items were easier to solve than the False items. Notes 1. This can be defined in a precise manner, but we omit the technical details here.

References Adler, J. Book Google Scholar Anderson, J. Google Scholar Braine, M. Google Scholar Cassimatis, N. Google Scholar Gentzen, G. Article Google Scholar Hitch, G. Article Google Scholar Holyoak, K. Google Scholar Huth, M. Book Google Scholar Jaskowski, S.

Google Scholar Johnson-Laird, P. Google Scholar Kosslyn, S. Google Scholar Laird, J. Even more strikingly, when Bernays eventually in published his Habilitation , he omitted his proof of the completeness theorem because as he later ruefully said the result seemed at the time straightforward and unimportant. For discussion of this point, see Hilbert [LFL]: For readily available general discussions, see Sieg , Zach , and the essays collected in Sieg ; for the original documents and detailed analysis, see Hilbert [LFL.

The Hilbert school throughout the s regarded first-order logic as a fragment of type theory, and made no argument for it as a uniquely favored system. He did not at the time publish on these topics because, as he later said:. I believed that it was so clear that the axiomatization of set theory would not be satisfactory as an ultimate foundation for mathematics that, by and large, mathematicians would not bother themselves with it very much.

To my astonishment I have seen recently that many mathematicians regard these axioms for set theory as the ideal foundation for mathematics. For this reason it seemed to me that the time had come to publish a critique. Skolem appendix. In the second he provided a new proof of that result.

These technical results were of great importance for the subsequent debate over first-order logic. But it is important not to read into Skolem a later understanding of the issues. Skolem at this point did not possess a distinction between the object language and the metalanguage. And although in retrospect his axiomatization of set theory can be interpreted to be first-order, he nowhere emphasizes that fact.

Indeed, Eklund presents a compelling argument that Skolem did not yet clearly appreciate the significance of the distinction between first-order and second-order logic, and that the reformulation of the axiom of separation is not in fact as unambiguously first-order as it is often taken to be.

There are two broad tendencies within logic during these years, and they pull in opposite directions. One tendency is towards pruning down logical and mathematical systems so as to accommodate the criticisms of Brouwer and his followers. Set theory was in dispute, and Skolem explicitly presented his results as a critique of set theoretical foundations. To put the matter slightly differently: the very point of axiomatizing set theory was to state its philosophically problematic assumptions in such a way that one could clearly see what they came to.

One possibility was to restrict oneself to first-order logic; another, to adopt some sort of predicative higher-order system. Similar broadly constructivist tendencies were also very much in evidence in the proof theoretical work of Hilbert and Bernays and their followers in the s.

Hilbert, like C. The epsilon-substitution method was the principal device Hilbert introduced in order to attempt to attain this result. But despite these constructive tendencies, many logicians of the s including Hilbert continued to regard higher-order type theory, and not its first-order fragment, as the appropriate logic for investigations in the foundations of mathematics. The ultimate hope was to provide a consistency proof for the whole of classical mathematics including set theory.

But, in the meanwhile, researchers still were somewhat unclear about certain basic distinctions. So matters remained unclear throughout the s. But the constructivist ambitions of the Hilbert school, the focus on the analysis of the quantifiers, and the explicit posing of metalogical questions had made the emergence of first-order logic as a system worthy of study in its own right all but inevitable. With these results and others that soon followed it finally became clear that there were important metalogical differences between first-order logic and higher-order logics.

Perhaps most significantly, first-order logic is complete, and can be fully formalized in the sense that a sentence is derivable from the axioms just in case it holds in all models. Second-order logic does not. By the middle of the s these distinctions were beginning to be widely understood, as was the fact that categoricity can in general only be obtained in higher-order systems.

But the technical results alone did not settle the matter in favor of first-order logic. In other words, even after the metalogical results there was a choice to be made, and the choice in favor of first-order logic was not inevitable.

After all, the metalogical results can be taken to show a severe limitation of first-order logic: that it is not capable of specifying a unique model even for the natural numbers. At this point in the s, however, several other strands of thinking about logic now coalesced. The intellectual situation was highly complex. A search for secure foundations, and in particular for an avoidance of the set-theoretical paradoxes, was something they shared, and that helped to tip the balance in favor of first-order logic.

As a practical matter, these first-order set theories sufficed to formulate all existing mathematical practice; so for the codification of mathematical proofs, there was no need to resort to higher-order logic. This confirmed an observation that Hilbert had already made as early as , though without himself fully developing the point. Thirdly, there was an increased tendency to distinguish between logic and set theory, and to view set theory as a branch of mathematics.

By the end of the decade, a consensus had been reached that, for purposes of research in the foundations of mathematics, mathematical theories ought to be formulated in first-order terms. Let us now try to draw some lessons, and in particular ask whether the emergence of first-order logic was inevitable. I begin with an observation. Each stage of this complex history is conditioned by two sorts of shifting background consideration.

One is broadly mathematical: the theorems that had been established. The other is broadly philosophical: the assumptions that were made explicitly or tacitly about logic and about the foundations of mathematics. These two things interacted. Each thinker in the sequence starts with some more or less intuitive ideas about logic. Those ideas prompt mathematical questions: distinctions are drawn: theorems are proved: consequences are noted, and the philosophical understanding is sharpened.

Let us now consider the question: When was first-order logic discovered? That question is too general. It needs to be broken down into three subsidiary questions:. Equipped with these distinctions, let us now ask: Why was first-order logic not discovered earlier? It is striking that Peirce, already in , had clearly differentiated between propositional logic, first-order logic, and second-order logic. He was aware that propositional logic is significantly weaker than quantificational logic, and, in particular, is inadequate to an analysis of the foundations of arithmetic.

He could then have gone on to observe that second-order logic is in certain respects philosophically problematic, and that, in general, our grasp on quantification over objects is firmer than our grasp on quantification over properties. The problem arises even if the universe of discourse is finite.

We have, for example, a reasonable grasp on what it means to speak in first-order terms of all the planets , or to say that there exists a planet with a particular property. But what does it mean to talk in second-order terms of all properties of the planets? What is the criterion of individuation for such properties?

Is the property of being the outermost planet the same as the property of being the smallest planet? What are we to say about negative properties? Is it a property of the planet Saturn that it is not equal to the integer 17? In that case, although there are only a finite number of planets, our second-order quantifiers must range over infinitely many properties. And so on. The Quinean objections are familiar.

Arguments of this sort had been made in the scholastic disputes between realists and nominalists: and Peirce was steeped in the medieval literature on these topics. That would in any case have run contrary to his logical pluralism. Why did he not make these points already in ? Any answer can only be speculative. One factor, a minor one, is that Peirce was not himself a nominalist.

There are also technical considerations. Peirce, unlike Hilbert, does not present first-intentional logic as an axiomatized system, nor does he urge it as a vehicle for studying the foundations of mathematics. He does not possess the distinction between an uninterpreted, formal, axiomatic calculus and its metalanguage. As a result, he does not ask about questions of decidability, or completeness, or categoricity; and without the metamathematical results a full understanding of the differences in expressive power between first-order and second-order logic was not available to him.

He provided a flexible and suggestive notation that was to prove enormously fertile, and he was the first to distinguish clearly between first-order and second-order logic: but the tools for understanding the mathematical significance of the distinction did not yet exist. As Henri Pirenne once remarked, the Vikings discovered America, but they forgot about it, because they did not yet need it. A related point holds for Frege and Russell.

But they never considered isolating the lowest level of the hierarchy as a free-standing system. There are both philosophical and mathematical reasons for this. And then, as a mathematical matter, second-order logic was necessary to their construction of the integers. So they had no compelling reason, either philosophical or mathematical, that would have led them to focus on the first-order fragment.

There is here an instructive contrast with Peirce. Peirce, in the spirit of the 19 th -century algebraists, was happy to explore a lush abundance of logical structures: his attitude was fundamentally pluralist. The logicists, working in the analytical tradition, were more concerned to discover what the integers actually are : their attitude was fundamentally monistic and reductionist.

But in order to single out first-order logic as was done in the s, two things were needed: an awareness that there were distinct logical systems, and an argument for preferring one to the other. Let us now turn to the question, Was the emergence of first-order logic inevitable? It is impossible to avoid counterfactual considerations, and the answer must be more speculative. By , the metalogical results can fairly be said to have been inevitable.

It would then have been an obvious next step to inquire about the completeness of higher-order systems. That was a comparatively insignificant matter. Hilbert did not treat it as important, and appears to have viewed it primarily as an expository device, a means of simplifying the presentation of the logic of Principia Mathematica.

The important step in was rather the introduction of techniques of metamathematics, and the explicit posing of questions of completeness and consistency and decidability. To pose these questions for systems of logic was an enormous conceptual leap, and Hilbert understood it as such. And even after he had introduced his metalogical distinctions in his papers of the s, logicians of the caliber of Russell and Brouwer and Ramsey had difficulty in understanding what he was attempting to do.

This development was in anything but inevitable: and without the introduction of the metalogical techniques the history of logic and proof theory in the s and s would have looked very different. As we saw, the metalogical results of the s do not settle the primacy of first-order logic. All these things show the continuing influence of the Grundlagenkrise of the s, which did so much to set the terms of the subsequent philosophical understanding of the foundations of mathematics.

The names of Brouwer and Weyl are nowhere mentioned. On the contrary. The work is explicitly undertaken in the spirit of his studies of the axioms of geometry. He will take up a system, explore it for a while, then drop it to examine something else. In his pluralism and in his pragmatic, experimental attitude he is closer to Peirce than to the logicists. The Grundlagenkrise and his public, polemical exchanges with Brouwer came later, and they gave a distorted picture of the motivations behind his logical investigations.

What was the impact of these philosophical debates on the technical aspects of his program? For the formulation of first-order logic, and for the posing of metalogical questions, the answer is easy: there was no impact whatsoever. The polemics might have added a sense of urgency, but it is hard to detect any influence on the actual mathematics.

So even if we imagine the philosophical Grundlagenkrise entirely removed from the picture, the technical results of the Hilbert school would not have been significantly affected. The completeness and incompleteness results would, in all likelihood, have arrived more or less on schedule.

It is worthwhile to note that Bernays and Hilbert had contemplated the possibility of various sorts of incompleteness as early as see the discussion by Wilfried Sieg in Hilbert [LFL]: — But those results would have emerged in a very different philosophical climate. The incompleteness theorems would likely have been greeted as an important technical contribution within the broader Hilbert program, rather than as its dramatic refutation.

Perhaps as Angus Macintyre has suggested they would have been viewed more like the independence results in set theory, with less talk about the limits of mathematical creativity. In other words, far from being inevitable, the emergence, towards the end of the s, of first-order logic as a privileged system of logic depended on two things, each independent of the other.

Neither of these things was inevitable: nor was the fact that they occurred at roughly the same time. It is worthwhile to observe that, as the philosophical concerns of the Grundlagenkrise have receded, and as new approaches from the direction of computer science and homotopy theory have entered the field, the primacy of first-order logic is open to reconsideration.

George Boole 2. Charles S. Peirce 3. Gottlob Frege 4. Giuseppe Peano 6. Alfred North Whitehead and Bertrand Russell 7. David Hilbert and Paul Bernays 9. Thoralf Skolem Peirce Peirce worked in the algebraic tradition of Boole. Giuseppe Peano In his , Giuseppe Peano, independently of Peirce and Frege, introduced a notation for universal quantification.

David Hilbert and Paul Bernays Let us briefly take stock of the situation as it existed in Hilbert concludes his discussion of first-order logic with the remark: The basic discussion of the logical calculus could cease here if we had no other end in view for this calculus than the formalization of logical inference.

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The intention is to leave considerable discretion in the distribution of effort across questions. Each question requires some thought, but any reasonable effort to engage with the matters raised by these questions will earn partial credit.

Naturally, the more coherent and insightful the thought, and the more strongly connected are the answers to the material presented in the readings and recordings assigned for the course, the more points will be earned. Among the submitted responses, there is a subset of more than 20 submissions that exhibit a striking similarity of content and verbal formulation, a similarity all the more remarkable given the fact that most of the answers are incorrect, some incorrect in a quite bizarre way.

In response, 20 students select the number 64, and all 20 give the number 9. Discuss whether any of these hypotheses provide plausible explanations of these responses. Are there other a priori plausible hypotheses that might account for this response? For each of the hypotheses H1, H2, and H3, construct a less extreme variant of this illustration that makes that hypothesis plausible. Discuss the following situations:. Assume that these are the only two hypotheses that are at play.

Does the blind extraction of a white ball confirm H1? H1 and H2? Suppose a coin in our possession is biased 2-to-1 in favor of Heads i. Summarize their argument and evaluate their position. You might consider, in particular, the following questions: Is there any reason that the chance device suggested by Nicholas Rescher for selecting among alternatives among which one is indifferent need to be unbiased?

Can one argue that any selection situation, whether a choosing or a picking situation, invariably resolves at some stage of implementation into a picking situation? A Assume for the sake of argument a universal acceptance of the in fact completely discredited hypothesis that the strong statistical correlation between smoking and a host of serious diseases including lung cancer is accounted for by a genetic factor that is the common cause of both.

Assume that the Predictor can predict what if anything the friend will say and what X will hear, and that it has factored this into its prior analysis and decision. Why or why not? If the friend were able to announce out loud the contents of box B, would it be advantageous for X to place himself in a situation in which he is unable to make out what his friend says?

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Paper order logic thesis cover format

FOL (First Order Logic)

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This paper describes how first order logic can be used as a representational language for a knowledge base and inferences from it can be. In this paper, we present a higher-order version of def- inite clauses that may be used to specify computations, and we describe a logic programming language. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable Once the satisfiability problem for first-order logic was.