# Download e-book Logic: Form and function - The Mechanization of Deductive Reasoning

If the metalanguage, in which sequents are formulated, is not formalized, then establishing their truth and falsity is a matter of natural thought. A cognitive agent, of course, uses some metalanguage counterparts of the rules of the logical system, formulated in the object language, but uses them informally. An order of application of logical rules is not fixed and is therefore determined by the natural thought of the cognitive agent. The crucial fact is that we can also operate with metalinguistic statements also in a purely formal manner.

This is what calculi of a sequential type do. Under systems of a sequential type I mean Gentzen's and Kanger's systems, Beth's tableaux, Hintikka's models sets, etc. As we have already seen, a sequent is a metastatement about the deducibility of the list of formulas from the list of formulas in the logic of the object language. We can already treat the classical work of G. Gentzen in such a manner, even though he himself formulated his sequents as expressions in the object language. In his sequential calculi G. Gentzen "has also axiomatized part of the metatheory of his systems for natural deduction.

The elementary sentences of his metatheory are of the form U 1 ,U 2 , These new elementary sentences are called sequents and their intended interpretation is that Z can be 'safely' derived from the premises U 1 ,U 2 , The next fact, which is even more important for our purposes, is that formal models of the activities of justifying and refuting of such metastatements have also been invented in logic. These models are represented by proof-search procedures.

Sequential systems are well adapted for design of proof-search procedures. Since a proof-search is used for establishing the truth or falsity of metastatements, and since it essentially involves objects of metalanguage for example, metavariables , it can be labeled as a metaprocess. The possibility of formalizing of the part of metalanguage in which sequents are formulated, and in which the methods of their justification and refutation are developed, creates a new situation in the treatment of logical procedures.

We can now proceed from the usual one-level interpretation to a new two-level one. Indeed, a logical procedure can now be represented in its full extension as consisting of two formalized levels and one informal level: a the object level , at which we specify a formal system which formalizes a class of valid formulas and their proofs; b the metalevel , at which we formalize metastatements on object-level deducibility and methods of their justification and refutation; and c the metametalevel which is a level of non-formal reasoning about a and b , and which is similar to a metalanguage in the conventional sense.

Such two-level procedures are usually employed in logic without theoretical and philosophical awareness. One example is given by semantic tableaux in original formulation by E. Here the logic of the object level is constructed in the form of a natural deduction calculus, the logic of the metalevel is formalized in the form of semantic tableaux, and there are some connections between these levels, expressed by the algorithm for reconstructing the results of the metalevel proof-search in semantic tableaux into natural deduction in the object language.

Another example is given by the algorithm of proof-search ALPEV-LOMI where the proof-search takes place in the sequential calculus, and the resulting deduction is given in natural form.

## Automated Reasoning (Stanford Encyclopedia of Philosophy/Fall Edition)

So we can state that the "two-level" construction of logical procedures is in some sense typical for the organization of proof-search procedures. In order to understand better such a two-level interpretation, it is essential to attend to the different styles of formalization. Concerning our problem there are two main styles: a formal systems that do not include a formalization of the proof-search, and b formal systems that include a formalization of at least some elements of the proof-search. Axiomatic systems of the Hilbert type and systems of natural deduction are examples of the first kind of system, and the systems of the sequential type are examples of the second kind.

The previous considerations allow us to conclude that the formalization of the first kind corresponds to the logic of the object level, and the second kind is appropriate to the logic of the metalevel. The metametalanguage in this case is a part of natural language, so its logic is not subject to formalization. Therefore a complete logical formalization of the notion of logical procedure presupposes I an object level, with a system of a Hilbert or natural type, which formalizes notions of a valid formula and a proof, II a metalevel with a formal system of a sequential type, which formalizes metastatements on deducibility at the object level and the methods of processing them; and III a metametalevel at which informal reasonings about the first two levels are carried on.

The crucial difference between the object level and metalevel of formal systems is that the latter formalize at least several elements of a proof-search while the former do not. Since a sequential type system is, as we have already seen, a formalization of the metalevel of Hilbert type axiomatic or natural deduction system, we shall understand what is formalized at the object level if we consider what occurs at the corresponding metalevel. In such a system, a logical procedure develops, as it were, in two dimensions: in the formal system itself, where the deduction is written out, and in the "mind" of the cognitive agent operating with this formal system.

Everyone, who has any experience in constructing deductions in such systems knows that the cognitive agent has to try many substitutions into axioms, or must form different concrete axioms from axiom schemata, introduce auxiliary assumptions, choose relevant values of terms etc. This process - which can be improved with experience, with working out in the "cognitive agent's mind" a supply of admissible rules that allow to find the substitutions and assumptions needed more quickly - never can be completely eliminated.

All such mental actions of a cognitive agent are actions of informal proof-search, which include building a structure of possible ways of constructing deductions, eliminating of the ways which do not promise to give the deduction needed, and at last choosing a way that can become a valid logical deduction if one exists.

But in object level systems we have no means of expressing such mental actions. Only the results of a proof-search can be expressed in them. Since these actions are directed to the formulas and terms of the object language, they belong to a non-formal metalevel of logical procedure. This feature of object level systems is connected with the fact that in such systems their rules do not include any instructions on how to analyse the structure of a given formula or of the premises and conclusion of a given deduction, so as to find the deduction itself.

The analysis is carried out by cognitive agent in his "mind". Since the metalanguage is not formalized, such mental actions of deduction-search belong to the natural thought processes of the cognitive agent. I have mentioned earlier that for the metalevel of a logical procedure, the sequential style of formalization is appropriate.

The main feature of sequential systems is that they create a possibility of formalizing a proof-search by finding principles of analysis for the sequent proved, the rules governing the substitutions of terms, and some means permitting one to choose among ways of proof-search. Consequently, I can state that the complete formalization of the notion of logical procedure should include two interconnected levels: an object level and a metalevel of a logical system. At the object level, a subsystem of proof inference, deduction is formalized, while at the metalevel a subsystem of proof-search is formalized, and a description of this procedures takes place at the unformalized metametalevel.

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Thus I can state that procedures of proof-search, which are realized at the metalevel, replace the mental actions of a cognitive agent, directed at searching for the intended deduction. Logical Framework Based Program Development gives a high-level overview. A Higher-Order Interpretation of Deductive Tableau describes an application to functional program synthesis and Modeling a Hardware Synthesis Methodology in Isabelle describes an application to hardware synthesis.

Finally, Program Development Schemata as Derived Rules looks at applications to program transformation. Using the theorem prover turns specification, development and verification into logical activities. A compiler from a simple arithmetical language to a stack machine is proved correct. Reports available; see also the Isabelle User Workshop. Details may be found in his thesis. The work was a MSc project at the University of Oxford. Thesis and code are available. The project Implementing Constructive Z uses Isabelle to support a software development method linking the Z specification language with intuitionistic and constructive program development methods.

## Interesting Special Issues of Journals

Achim D. The encoding is in a sequent style and employs SVC as a decision procedure for real arithmetic. The encoding is based on a simple axiomatic approach to represent modal logics in Isabelle, based on higher-order types and the parameter mechanism to keep track of different worlds, which may be more generally useful. Documentation and sources available. Commands are represented by relations over states. A program comprises a set of initial states and set of commands. The approach is definitional, with UNITY specification primitives defined in terms of program semantics and their properties proved.

Paper available. David Spelt and Susan Even have worked on the verification of semantic properties of object-oriented databases. Isabelle has been used to verify e. Spelt has submitted his master's thesis to the University of Twente, Netherlands. Starting with HOL-Z 2. It allows interactive verifications in process systems, protocols and distributed algorithms over infinite alphabets.

Sara Kalvala and Valeria de Paiva have implemented Linear Logic and have an ongoing project for developing a verification methodology using Linear Logic. He formalizes the communication histories of the automata by lazy lists in domain theory. This paper compares his approach with other formalizations of finite and infinite sequences. Tobias Nipkow has verified an abstract lexical analyzer generator turning regular expressions into automata. For details see the home pages of projects Bali and VerifiCard Munich. Claire Quigley wants Just-In-Time compilers to make optimizations not currently possible, and to perform existing optimizations more efficiently, by proving that certain properties hold of Java bytecode programs.

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She will carry out these proofs using Cornelia Pusch's Isabelle implementation of the operational semantics of Java and possibly a Hoare-like logic based on these semantics. Larry Paulson has developed a new approach to the verification of cryptographic protocols. The operational semantics of all agents in the network including an active intruder is modelled using a series of inductive definitions.

Steven Shaviro in particular suggests that Her describes much more than a simple dystopia about what will happen if the aspirations of neoliberal hipster urbanism were to be realized. For Shaviro, Her fully embraces this condition of no future and no alternative to neoliberal capitalism.

Episode 1.3: Deductive and Inductive Arguments

Instead of a simulation, or a hyper-real form of construction of the real, Her reveals the threat of a speculative realist ontology whereby simulations are real entities equipped with a sensible manifold through which the cognitive infrastructure of thinking creates concepts and laws. This is not simply a cybernetic form aiming at steering decisions towards the most optimal goals. Instead, operating systems are computational structures defined by a capacity to calculate infinities through a finite set of instructions, changing the rules of the game and extending their capacities for incorporating as much data as possible.

These systems are not simply tools of or for calculation. The instrumental programming of computational systems is each time exceeded by the evolutionary expansion of the data substrates, an indirect consequence of the process of real subsumption of the social by means of machines. From this standpoint, automation replaces everyday reality with engineered intelligence. The impersonal intelligence of Her is indeed already symptomatic of the advance of an automated form of cognition itself. This means that the accelerated realism of affective capitalism has already entered the grey zone of an alien thinking, which is enveloped within the machine itself.

Fixed capital now involves algorithmic functions that retrieve, discretise, organise and evaluate data. This data processing, it has been argued, implies non-conscious mechanisms of decision-making in which speed is coupled with algorithmic synthesis. This capacity of and for fast algorithmic communication has been explained in terms of non-coherent and non-logical performance of the code, which involves not consciousness but a non-conscious level of cognition. For instance, Katherine Hayles suggests that nonconscious cognitive processes cut across humans, animals and machines and involve temporal regimes that subtend cognitive levels of consciousness insofar as they exploit the missing half-second between stimuli and response, the infinitesimal moment before consciousness becomes manifested.

As Hayles insists, whilst both a hammer and a financial algorithm are designed with an intention in mind, only the trading algorithm demonstrates nonconscious cognition insofar as this is embodied within the physical structures of the network of data on which it runs, and which sustains its capacity to make quick decisions.

## Knowledge Representation and Reasoning

As complex interactive algorithms adapt, evolve and axiomatise infinities, so fixed capital has become networked and fluid, and as Negri argues, ready to become re-appropriated. Negri has pointed out that the use of mathematical models and algorithms on behalf of capital does not make the technical machine its inherent feature. He explains that the computerized social world is in itself reorganized and automatized according to new criteria in the management of the labor market and new hierarchical parameters in the management of society.

Informatization, he argues, is the most valuable form of fixed capital because it is socially generalized through cognitive work and social knowledge. In this context, automation subordinates information technology because it is able to integrate informatics and society within the capitalist organization. Negri thus individuates a higher level of real subsumption that breaths through the command of capitalist algorithms. The algorithmic machinery that centralizes and commands a complex system of knowledge now defines the new abstract form of the General Intellect.

In tune with the operaist spirit of the expropriation of goods, Negri urges us to invent new modes of reappropriation of fixed capital in both practical and theoretical dimensions. To embrace the potential of automation for Negri means to positively address computable capacities to augment productivity, which, he suggests, can lead to the reduction of labour time disciplined and controlled by machines and an increase in salaries.

The appropriation of fixed capital therefore involves the appropriation of quantification, economic modeling, big data analysis, and abstract cognitive models put in place through the educational system and new forms of scientific practices. To address this higher level of the subsumption of information to automation, and re-appropriate the potential of fixed capital, Negri proposes to overcome the negative critique of instrumentalisation and thus reveal the potentiality for politics inherent in the algorithmic dynamics of processing.

If one follows this argument however, it may be important to ask ourselves: what can in this framework guarantee that the appropriation of the technical machine does not follow yet again a logic of exchange or instrumentalisation of machines amongst humans? Can thousands of evolving algorithmic species able to process data below the thresholds of human consciousness and critical reasoning, be used for the purposes of the common? During the last 10 years, fixed capital has come to define not only the IT revolution driven by computer users but increasingly the capacities of algorithmic machines to interact with themselves.

In a issue of the Journal Nature , a group of physicists from the University of Miami claimed that this phase of robot transition coincided with the introduction of high frequency stock trading in financial markets after , and is allegedly responsible for the crash. What is described here is a digital ecology of highly specialized and diverse interacting agents operating at the limit of equilibrium, challenging human control and comprehension.

Yet, whilst one cannot deny the opaqueness of these interactive networks of rules, I want to suggest that there is no necessary cause that shall limit the epistemological production of knowledge understanding and reasoning of the evolution of this new phase of self-referential automation. In other words, to address the potentialities of this dynamic form of automation, it is not sufficient to divorce the machine from its capitalization. Instead, what remains challenging is to unpack the use that machines make of other machines, that is the genealogical development of an information logic that allows for machines to become transformative of fixed capital, and of the mechanization of the cognitive structures embedded in social practices and relations.

In order to start to address this fundamental transformation in computational logic, I will turn to algorithmic information theory to explain how the logic subtending instrumentalisation has irreversibly changed automation. Algorithmic automation involves the breaking down of continuous processes into discrete components whose functions can be constantly re-iterated without error. In short, automation means that initial conditions can be reproduced ad infinitum. For instance, the Turing machine is an absolute mechanism of iteration based on step by step procedures.

Since the s however the nature of automation has undergone dramatic changesas a result ofthe development ofcomputational capacities of storing and processing data across a networkinfrastructure of online parallel and interactive systems. Whereas previous automated machine were limited by the amount of feedback data they could collect and interpret, algorithmic forms of automation now analyse a vast number of sensory inputs and confront them with networked data sets and finally decide which output to give.

In other words, whereas algorithmic automation has been understood as being fundamentally a Turing discrete universal machine that repeats the initial condition of a process by means of iteration, the increasing volume of incomputable or non-compressible data or randomness within on line, distributive and interactive automations, is now revealing that patternless data are rather central to computational processing and that probability no longer corresponds to a finite state.

As mathematician and computer scientist, Giuseppe Longo explains,, 25 the problem of the incomputable importantly reveals that axioms are constantly modified and rules amended.

### Becoming Human: Artificial Intelligence Magazine

During the 90s and the 00s, Chaitin identified this problem in the terms of the limits of deductive reason. Chaitin suggests that within computation, the augmentation of entropy becomes productive of new axiomatic truths that cannot be predicted in advance. He calls the emergence of a new form of logic or non-logic experimental axiomatics. The latter describes an axiomatic decision that is not a priori to the computational process.