Axioms logic

Deterministic—relying upon axioms, logic, and proof... 

The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms... 

If the conclusion seems to be in error, and the chain of reasoning is valid, we have then demonstrated that one (or more) of our initial assumptions must be erroneous. For example, if one of the premises was the implication If a ketone is present then the compound is an alkane , then the conclusion that would follow from the observation a ketone is present would be obviously false. Since the PC is logically correct (in fact defines logically correct ) any erroneous conclusions must arise from an erroneous axiom, and cannot be the fault of the calculating procedure used. This allows us to focus our attention on the assumptions and frees us from having to worry about the procedure. 

Why More Systems Haven t Been Axiom it i zed. Geometry is unique in that it can be expressed in a simple logic, the results are either true or false, and that the actual experiments were capable of being done with thought alone. In chemistry there was not sufficient knowledge to enumerate the basic definitions and postulates. The recent explosion of knowledge in chemistry has made it feasible to begin the process of axiomatization of chemical theories. 

Unlike chemistry, mathematics often deals with infinite domains, and infinite axiom sets. If we allow the fact that two axioms infer the same conclusion to increase the truth value of that conclusion, we must choose some increment that reflects the importance of each individual axiom. If there are an infinite number of such axioms, then each axiom becomes infinitesimally important. Thus LT logic chooses to err on the side of conservatism, assuring that the conclusions will be valid, though perhaps less strong than they could actually be. 

The multi-valued predicate calculus logic as implemented in QED has been demonstrated to be suitable for cleanly representing strategic axioms of chemical synthesis. QED is a powerful tool for exploring inference in the planning of synthesis strategies. QED helped us elucidate key strategic concepts and their interdependence and... 

A set of thermodynamic laws governing the behavior of macroscopic systems lead to a large amount of equations and axioms that are exact, based entirely on logic, and attached to well-defined constraints. These laws are summarized in the following sections. 

Christian theology was presented as a logically coherent and rationally defensible system, derived from syllogistic deductions based on known axioms. In other words, theology began from first principles, and proceeded to deduce its doctrines on their basis. 

Hegel, in The Science of Logic, derived the various ontological categories from each other according to certain deductive principles which have resisted analysis to this day. The connection is neither that of cause to effect, nor that of axiom to theorem, nor finally that of given fact to its condition of possibility. The "self-determination of the concept" appears... 

Deduction By means of logical connectives, correct inferences can be derived in the sense of deductive reasoning. In general, for deduction, true axioms (postulates) are given and the conclusions drawn are again true. This is denoted a legal inference. 

What we have discussed so far is not strictly mathematics the systems L and Ki are systems of logic. The absence of restrictions on the language Xmake the coiiclusions deducible from them very general and they are interpretable in many different ways. If is interpreted in a mathematical way then the theorems of are mathematical truths by virtue of their logical structure. Earlier the symbol A was interpreted as =, and one cannot get far in mathematics without it. For example the statement ( V Xi) ( V X2) (Aj (xi, X2) Ay (X2,xi)) is interpreted as (for all natural numbers Xj, ifxj = X2 then X2 = x,). This is a consequence of the meaning of =, for the w/as it stands is not logically valid and so is not a theorem of K. To introduce this idea into a mathematical interpretation of At, the axioms are extended by axioms of equality such as, for example, (xi, Xi) which means Xi = x,. The other axioms ensure that for example / >,.. . y. .. )=/I>i... z. .. )ify = z. 

One of the fundamental ideas of mathematics based on an extension of At is Group Theory. A group consists of variables x,X2... an identity constant / function symbols predicate symbol = punctuation (,) and logical symbols V, D. Three extra axioms are... 

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