# Is restatement true in syllogism?

Statement : All A is B

Conclusion : All A is B.

Does this conclusion follow?

• The argument is valid but a Syllogism is "inferences with two premises, each of which is a categorical sentence, having exactly one term in common, and having as conclusion a categorical sentence the terms of which are just those two terms not shared by the premises". Mar 10 '17 at 15:19
• @MauroALLEGRANZA has it got anything to do with valid and invalid syllogism or is it just not a syllogism anyway? Mar 10 '17 at 15:39
• A syllogism is an argument defined above; some arguments are valid ans some are not. A valid syllogism is a syllogism that is valid. Mar 10 '17 at 19:22
• The argument "A, therefore A" is valid; but it is not a syllogism. Mar 10 '17 at 19:47

Yes, the conclusion follows. Here, the form of the argument is "P, therefore P". When the premise is identical to the conclusion, the complete if-then statement is a tautology. Tautologies are necessarily true.

• Isn't reading conditional or hypothetical syllogism into the OPs statement and conclusion with the same proposition non sequitur? Mar 11 '17 at 23:14
• @Mr.Kennedy. The statement "P, therefore P" is not a syllogism. Mauro ALLEGRANZA had already made this point and I did not see it necessary to make it again. Mar 12 '17 at 2:18
• According to Gensler "P, therefore P" is a single premise syllogism where P stands for some combination of "All/some/no X are Y/notY". Trivial yes, but also not a tautology or a conditional (if..., then...). Even if "P, therefore P" were "All A is A, therefore all A is A" only the "A is A" part of "P" is tautological, not the syllogistic argument. Mar 12 '17 at 23:52
• This exchange seems to turn on a matter of definition. An argument is a syllogism if and only if, among other things, it contains three categorical sentences. Barker (1965), p. 61. "P, therefore P", having less than three sentences, is a tautology. Barker, p. 125. Mar 13 '17 at 1:46
• True, and technically I suppose "P, therefore P" is technically a tautology. Mar 13 '17 at 2:14

UPDATE per katipra's comment and find in Gensler's book:

In the book you mentioned, it says ' all A is A ' is a premiseless syllogism. It also states that a premiseless syllogism is valid iff it's impossible for its conclusion to be false. First of all I don't understand how it can be a syllogism since it not only it doesn't have two premises but none premises. Secondly, considering what it says is true, can you give an example when a premiseless syllogism is false. Also before that it gives an example of a single premise syllogism

"True" isn't so much the operative term as is "valid" - i.e. "Is repetition valid in syllogism?" As for your question in the comments, in section 2.2 ("The Star Test") of the chapter "Syllogistic Logic" Gensler discusses a "premise-less syllogism"

A premise-less syllogism is valid if and only if it’s impossible for its conclusion to be false.

...and he gives the example "∴ All A is A"

This may be true in the case of "All A is A" as it is self evident that A literally and actually is A. In your case, however, it is possible that the statement "All A is B" is false. Hence, "∴ All A is B" is not a syllogism.

Furthermore, he states that a syllogism is a "...sequence of one or more [well formed formulas] in which each letter occurs twice and the letters “form a chain”" so, yes, per Gensler's definition of syllogism "All A is A" & "All A is B, therefore All A is B" technically have terms which occur twice, but it is a bit of a stretch to say the reasoning forms a chain in the way "No P is B; Some C is B; ∴ some C is ¬P" does.

Good find tho - glad to see you read the book! I agree that it is a stretch to call a conclusion a syllogism or to call a single premise argument a syllogism, but, consider that Gensler may be doing so in order to make a point in his effort to teach how to think logically. Sometimes these kinds of technical discrepancies result from a pedagogical strategy. Also, logicians do not always agree. As pointed out in the comments to your question, there are other authors who do not consider Gensler's definition of syllogism to be correct. I like his writing style - and his software is good for practice, and most of all his book is freely available on line, but don't let that stop you from looking to other author's :)

Lastly, for a premise-less syllogism which is false, I suppose "∴ All A is ¬A" fits the bill, yes?

Statement : All A is B

Conclusion : All A is B.

Gensler may disagree, but this is also not a syllogism as there is no second premise or third term. You can, of course claim it as a syllogism per Gensler's definition but such a construction is hardly even an argument. It's just a redundancy or, argumentatively, an insistence.

Stated otherwise, "All A is B" can be represented by "X" and you are simply repeating "X", e.g.

• Statement: X
• Conclusion: X

...and in this sense if the statement of X is imagined as a justification for concluding that X is true, then the argument is not demonstrating the truth of statement "X" it is instead begging the question.

Is restatement true in syllogism?

Restating a false claim does not make it true, nor does restating a true claim make it false. Furthermore, the truth or falsity of the claims used as premises is not due to the form of the argument. Truth is a condition of statements which is satisfied when what is claimed corresponds to what is the case, irrespective of the statements use in an argument.

Imagine your "All A is B" claim were false, for example, "All vegans are carnivores". The falsehood of this statement is self-evident as "vegan" is a term for someone that does not eat meat. For the sake of argumentation, we can express this falsehood as "All A is B" such that A stands for "vegans" and B stands for "carnivore". This claim also has a truth value of (F) for "false". If an argument takes the form:

• Statement: "All A is B" (F)
• Conclusion: "All A is B" (F)

The formal construction does not alter the content or the falsehood of the claim.

Keep in mind that the claims are either true or false in addition to their status in the argument as either premise or conclusion. Also, an argument is technically neither true nor false, but how the conclusion is drawn is considered in terms of validity and soundness.

If we put your restatement in the form of a syllogism, as I will demonstrate below, it is invalid, unsound and arguably not a syllogism at all because there is no third term:

• Premise: "All A is B" (True or False) and
• Premise: "All A is B" (True or False) therefore
• Conclusion: "All A is B"(True or False)

Does this conclusion follow?

If we were to restate the claim "All A is B" as the first and second premises of a syllogism - what could be drawn from the restatement for a conclusion?

• Premise: "All A is B" (True or False) and
• Premise: "All A is B" (True or False) therefore
• Conclusion: Nothing
reiterating a premise is not an argument and there is no third term.

If you are stating, "All A is B, therefore All A is B" then sure, this is a valid argument from which it can be accurately said that the conclusion follows. Again tho, this is not a syllogism, and furthermore, it is only an argument in a mathematically trivial sense as nothing more has been said said than "All A is B" and "therefore". Technically such a trivial argument is a tautology of the form "X = X" and specifically a symmetrical, one term identity statement. This is unlike a tautology such as "2 + 2 = 4" where "4 = 2 + 2" or a self-evident tautology such as "dividends require financing" which is axiomatically true by definition of the term "dividend." Note that I use trivial in a mathematical and not a pejorative sense, though in a rhetorical sense a tautology may be considered a fault of style because it is simply saying the same thing twice. As stated above tho, Stating "All A is B = All A is B" does not at all determine or demonstrate that it is true that "All A is B".

As for syllogism, a syllogism draws (infers, deduces...) a conclusion from two premises which share a term. Syllogism is of the form:

• Premise 1: two terms, A and B (True or False)
• Premise 2: two terms, (either A or B) and C (True or False)
• Conclusion: two terms excluding the term shared by the two premises. (True or False)

That an argument is in a syllogistic form does not guarantee the truth value of the conclusion. For example, a valid syllogism with a false conclusion:

• Premise 1: All mountains are heavy (T)
• Premise 2: All cars are heavy (T)
• Conclusion: All cars are mountains (F)

This argument can also be expressed as, "All M are H. All C are H. All C are M."
Note that the two premises share a term (H) and the conclusion is composed of the terms not shared by the premises (C an M).

When both premises are true and the conclusion is soundly drawn from the premises, then the argument is valid and sound:

• Premise 1: Trump is President of the U.S. (T)
• Premise 2: All Presidents of the U.S. are also Commander-in-Chief. (T)
• Conclusion: Trump is Commander-in-Chief. (T)

This argument can also be expressed as, "T is P. All P are C. T is C."
Again: three terms, T, P and C. The premises share one of the terms (P) and the conclusion is composed of the two terms not shared by the premises (T and C).

Note as well that the order of the two premises is irrelevant, e.g. "All P are C. T is P. T is C." is the same argument as the one above. In either case, the argument is both valid and sound.

Now, that the premises may be true, share a term and the conclusion true, but, if the conclusion is not drawn from the premises, then the argument is unsound:

• Premise 1: George Washington was born February 22, 1732. (T)
• Premise 2: All people born in 1732 are now dead. (T)
• Conclusion: Venus is the second planet from the Sun. (T)

This should be obvious when the argument is expressed as: "G is B. All B are D. V is S." The premises share a term, but nothing from the two premises is drawn into the conclusion.

An example of an unsound argument which draws it's conclusion with the two terms the premise does not share:

• Premise 1: George Washington was born February 22, 1732. (T)
• Premise 2: All people born in 1732 are now dead. (T)
• Conclusion: All dead people are George Washington. (F)

I.e. "G is B. All B are D. D is G." draws a false conclusion.

Hopefully by now, it the sound argument and true conclusion from this example of two premises which share one term should be plain:

• Premise 1: George Washington was born February 22, 1732. (T)
• Premise 2: All people born in 1732 are now dead. (T)
• Conclusion: George Washington is dead. (T)

I.e. "G is B. All B are D. G is D."

You might enjoy Harry Gensler's book, "Introduction to Logic." He also has a free software application for doing drills and practicing analysis of syllogisms and such, LogiCola. And here's an article about validity and soundness.

• Good job Mr. Kennedy! However, I'd like to add just this to amplify it; were it a true syllogism 'restatement' could take it in either direction as to its truth value. Political statements made during pol. campaigns are good examples of syllogisms placed into self-serving modalities of the speaker's ideological outlook on any particular issue. Mar 11 '17 at 3:27
• Perhaps you could elaborate upon your comment and give an example? Do you mean equivocation, or just using the same premise in a different sense - such as "build a border wall" implying "we will build the wall" then being used to argue that "they should the wall" such to shift the burden of responsibility if the wall isn't built or claim ownership no matter how a wall gets built? Mar 11 '17 at 3:45
• @Mr.Kennedy In the book you mentioned, it says ' all A is A ' is a premiseless syllogism. It also states that a premiseless syllogism is valid iff it's impossible for its conclusion to be false. First of all I don't understand how it can be a syllogism since it not only it doesn't have two premises but none premises. Secondly, considering what it says is true, can you give an example when a premiseless syllogism is false. Also before that it gives an example of a single premise syllogism Mar 11 '17 at 15:33
• @DanielWells also, if you've found the question or any answers useful, please do vote them up! Mar 11 '17 at 20:42
• @katipra Found the passage you refer to in Gensler. See my edits to the answer. If you find my answer or others useful, please do vote them up & don't forget to accept the answer you find most useful (the check mark). Mar 11 '17 at 22:52