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Therefore Q must also be true. Modus ponens is closely related to another valid form of argument, modus tollens. Both have apparently similar but invalid forms such as affirming the consequent , denying the antecedent , and evidence of absence. Constructive dilemma is the disjunctive version of modus ponens. Hypothetical syllogism is closely related to modus ponens and sometimes thought of as "double modus ponens. The history of modus ponens goes back to antiquity. The form of a modus ponens argument resembles a syllogism , with two premises and a conclusion:.
The first premise is a conditional "if—then" claim, namely that P implies Q. The second premise is an assertion that P , the antecedent of the conditional claim, is the case. From these two premises it can be logically concluded that Q , the consequent of the conditional claim, must be the case as well. This argument is valid , but this has no bearing on whether any of the statements in the argument are actually true ; for modus ponens to be a sound argument, the premises must be true for any true instances of the conclusion. An argument can be valid but nonetheless unsound if one or more premises are false; if an argument is valid and all the premises are true, then the argument is sound.
For example, John might be going to work on Wednesday. In this case, the reasoning for John's going to work because it is Wednesday is unsound. The argument is only sound on Tuesdays when John goes to work , but valid on every day of the week. A propositional argument using modus ponens is said to be deductive. In single-conclusion sequent calculi , modus ponens is the Cut rule. The cut-elimination theorem for a calculus says that every proof involving Cut can be transformed generally, by a constructive method into a proof without Cut, and hence that Cut is admissible.
In artificial intelligence , modus ponens is often called forward chaining. The modus ponens rule may be written in sequent notation as. The validity of modus ponens in classical two-valued logic can be clearly demonstrated by use of a truth table. However, this hoard pre-dates both the stones and the Vikings. After the tip off from the detectorist, archaeologists finished excavating at Vindelev, in the shadow of Jelling and unearthed almost a kilogram of gold jewelry. Researcher examines part of the golden treasure hoard. Ravn told TV2 News that the discovery was made last December by Ole Ginnerup Schytzusing, a local metal detectorist, and that the find was kept secret until now. The total weight of the treasure is just under a kilogram grams and it dates to around years before the reigns of Old Gorm and Harald Bluetooth, having been buried around 1, years ago.
Saucer sized medallions are a stunning part of the remarkable golden treasure hoard. The objects are widely different in their nature and design. Some of the gold pieces are marked with ancient motifs and runic inscriptions. It is thought these symbols refer to regional rulers of the time. One thin gold bracteate bears a male head with a horse and a bird below it.
A Science. Maybe to save it from enemies, or maybe to appease the gods. Therefore, these 1, year old golden treasures serve as metallic Sagas, offering us a story of a climatological Armageddon from a bygone age. Top image: A few of the many items found in the golden treasure hoard recently discovered in Denmark. Source: Vejle Museum. Ashley is a Scottish historian, author, and documentary filmmaker presenting original perspectives on historical problems in accessible and exciting ways.
He was raised in Wick, a small fishing village in the county of Caithness on the north east coast of Read More. Ancient Origins has been quoted by:. At Ancient Origins, we believe that one of the most important fields of knowledge we can pursue as human beings is our beginnings. Thus, good and bad are qualities which belong to objects independently of our opinions, just as much as round and square do; and when two people differ as to whether a thing is good, only one of them can be right, though it may be very hard to know which is right. One very important consequence of the indefinability of good must be emphasized, namely, the fact that knowledge as to what things exist, have existed, or will exist, can throw absolutely no light upon the question as to what things are good.
There might, as far as mere logic goes, be some general proposition to the effect whatever exists, is good , or whatever exists, is bad , or what will exist is better or worse than what does exist. But no such general proposition can be proved by considering the meaning of good , and no such general proposition can be arrived at empirically from experience, since we do not know the whole of what does exist, nor yet of what has existed or will exist. We cannot therefore arrive at such a general proposition, unless it is itself self-evident, or follows from some self-evident proposition, which must to warrant the consequence be of the same general kind.
But as a matter of fact, there is, so far as I can discover, no self-evident proposition as to the goodness or badness of all that exists or has existed or will exist. It follows that, from the fact that the existent world is of such and such a nature, nothing can be inferred as to what things are good or bad. The belief that the world is wholly good has, nevertheless, been widely held.
It has been held either because, as a part of revealed religion, the world has been supposed created by a good and omnipotent God, or because, on metaphysical grounds, it was thought possible to prove that the sum-total of existent things must be good. With the former line of argument we are not here concerned; the latter must be briefly dealt with. The belief that, without assuming any ethical premiss, we can prove that the world is good, or indeed any other result containing the notion of good, logically involves the belief that the notion of good is complex and capable of definition. If when we say that a thing is good we mean for example that it has three other simpler properties, then by proving that a thing has those three properties we prove that it is good, and thus we get a conclusion involving the notion of good, although our premisses did not involve it.
But if good is a simple notion, no such inference will be possible; unless our premisses contain the notion of good, our conclusion cannot contain it. The case is analogous to the case of elements and compounds in chemistry. By combining elements or compounds we can get a new compound, but no chemical operation will give an element which was not present in the beginning. So, if good is simple, no propositions not containing this notion can have consequences which do contain it.
As a matter of fact, those who have endeavoured to prove that the world as a whole is good have usually adopted the view that all evil consists wholly in the absence of something and that nothing positive is evil. This they have usually supported by defining good as meaning the same as real. Spinoza says : By reality and perfection I mean the same thing ; and hence it follows, with much less trouble than metaphysicians have usually taken in the proof, that the real is perfect.
This is the view in Abt Vogler : The evil is null, is nought, is silence implying sound. Whenever it is said that all evil is limitation, the same doctrine is involved; what is meant is that evil never consists in the existence of something which can be called bad, but only in the non-existence of something. Hence everything that does exist must be good, and the sum-total of existence, since it exists most, must be the best of all. And this view is set forth as resulting from the meaning of evil. The notion that non-existence is what is meant by evil is refuted exactly as the previous definitions of good were refuted.
And the belief that, as a matter of fact, nothing that exists is evil, is one which no one would advocate except a metaphysician defending a theory. Pain and hatred and envy and cruelty are surely things that exist, and are not merely the absence of their opposites; but the theory should hold that they are indistinguishable from the blank unconsciousness of an oyster. Indeed, it would seem that this whole theory has been advanced solely because of the unconscious bias in favour of optimism, and that its opposite is logically just as tenable.
We might urge that evil consists in existence, and good in non-existence; that therefore the sum-total of existence is the worst thing there is, and that only non-existence is good. Indeed, Buddhism does seem to maintain some such view. It is plain that this view is false; but logically it is no more absurd than its opposite. We cannot, then, infer any results as to what is good or bad from a study of the things that exist. This conclusion needs chiefly, at the present time, to be applied against evolutionary ethics. The phrase survival of the fittest seems to have given rise to the belief that those who survive are the fittest in some ethical sense, and that the course of evolution gives evidence that the later type is better than the earlier.
On this basis, a worship of force is easily set up, and the mitigation of struggle by civilization comes to be deprecated. It is thought that what fights most successfully is most admirable, and that what does not help in fighting is worthless. Such a view is wholly destitute of logical foundation. The course of nature, as we have seen, is irrelevant to deciding as to what is good or bad.