# Can you make a valid inference invalid by adding extra premises?

From An introduction to formal logic by Peter Smith there is a question:

Can you make a valid inference invalid by adding extra premises?

I think it's true. If I add premises that contradict other premises that will make the argument invalid.

Reference

Smith, P. (2003). An introduction to formal logic. Cambridge University Press.

• Think about the definition of validity: if the premises are true, then the conclusion is true as well. To show that an argument is invalid, you need to find an interpretation that makes the premises true but the conclusion false. If you start with a valid argument, can you add further premises that make all of the premises true but the conclusion false? – possibleWorld Jun 3 '17 at 13:02
• In classical logic no, but you can make a sound argument unsound by adding false premises. The difference between soundness and validity is usually ignored colloquially, but logical validity is neutral on the truth of the premises, it only cares whether inference would preserve that truth. Contradictory premises, however, would make any argument from them unsound. Moreover, in classical logic due to the law of explosion anything can be validly inferred from contradictory premises. – Conifold Jun 4 '17 at 22:03
• Your intuition is captured by non-monotonic logics:"The term “non-monotonic logic” (in short, NML) covers a family of formal frameworks devised to capture and represent defeasible inference, i.e., that kind of inference in which reasoners draw conclusions tentatively, reserving the right to retract them in the light of further information". In the light of new premises some of the original premises may be defeased (annuled), i.e. become untrustworthy, thus invalidating the original argument. – Conifold Jun 4 '17 at 22:07
• In a nutshell, the def of validity of an argument is: every "model" of the premises (i.e. every interpretation satisfying them) is also a model of the conclusion. If we "add" a new premise, we restrict the number of models (because the new premise will "cut off" all the previous models that do not satisfy it): thus, it cannot invalidate a valid argument. – Mauro ALLEGRANZA Jun 5 '17 at 9:28

No.

In propositional logic, an argument is valid IFF (1) it is inconsistent to assert all the premises and the negation of the conclusion (semantic validity), or (2) the rules of inference allow you to derive the conclusion from the premises (syntactic validity).

Let's go with definition (2) first. Suppose you have a valid argument P, Q |- R. That means you can derive R from P and Q. Adding extra premises, S, T, cannot prevent you from deriving R from P and Q.

Now let's go with definition (1). Suppose you have a valid argument, P, Q |= R. That means that {P, Q, ~R} is inconsistent. Adding extra premises cannot make that set consistent.

No, you can't.

If an argument is valid, then (by definition) it's impossible for its premises to be true and its conclusion false at the same time. Adding an extra premise cannot change that. By adding premises you can only go from invalid to valid, and not vice versa.

Another way to think about it: suppose you have an invalid argument. That means there's a possibility that the premises are true and the conclusion false. Can you eliminate that possibility (and thus make the argument valid) by removing a premise? No, you can't.

• "its premises to be true" - but two contradictive premises can't be both true. So, you made an error somewhere. – rus9384 Jul 28 '18 at 19:25
• @rus9384 What's your point? If the premises cannot be all true at the same time then the argument cannot be invalid. So that doesn't change the answer. – Eliran Aug 1 '18 at 11:42
• In fact, you are confusing the words. Inference can be valid whatever its premises are. But not the argument. Argument can't be valid when premises are wrong. – rus9384 Aug 1 '18 at 12:51
• @rus9384 Now I remember why I stopped coming here. Look up the definition of validity in a logic textbook. – Eliran Aug 2 '18 at 13:50

As a preliminary consider the definitions of validity and jointly logically consistent provided by the authors of forall x: Calgary Remix.

Here is their definition of "valid" (page 8):

An argument is valid if and only if it is impossible for all of the premises to be true and the conclusion false.

Here is their definition of "jointly logically consistent" (page 72):

A1,A2,...,An are jointly logically consistent iff there is some valuation which makes them all true.

Derivatively, sentences are jointly logically inconsistent if there is no valuation that makes them all true.

Consider the question: Can you make a valid inference invalid by adding extra premises?

Arguments are valid or invalid. Sets of premises can be jointly logically consistent or jointly logically inconsistent. Adding premises to a set of premises used in a valid argument may make the larger set of premises jointly logically inconsistent but that does affect the validity of arguments using a smaller set of premises.

On page 265 they have a chart answering questions about validity of an argument and consistency of premises. In summary, an argument is valid if we can give a proof using the premises and reaching the conclusion. It is invalid if we can find an interpretation where all of the premises are individually true but the conclusion is false.

Consider this an example. Let "P" and "Q" be premises. Try to derive "P ∧ Q". Here is the proof: The two premises are assumed in lines 1 and 2. By using the conjunction introduction rule (∧I), I reach the goal in line 3.

Suppose I add ¬P to the premises to see if this expanded set of premises affects the validity of this argument. Since I am not using ¬P in the proof the proof checker shows that it does not. Although the particular argument remains valid, something changed. I can easily find an interpretation for the first argument using "P" and "Q" as premises, but I will not be able to find any for the second argument using "P", "Q", and "¬P".

How would I be able to tell if the premises are jointly consistent or not? That is also answered on page 265. To show the premises are jointly consistent I need to provide an interpretation where all of the premises are true. Note, it says nothing about the conclusion of an argument. To show the premises are jointly inconsistent I need to prove that these premises lead to a contradiction. In other words, to show that the premises, "P", "Q", and "R", are jointly inconsistent, I would need to prove this argument: "P, Q, ¬P ∴ ⊥"

Here is that proof using contradiction introduction (⊥I): In summary Peter Smith's question attempts to help the reader distinguish between the validity of an argument and the joint consistency of a set of premises.

References

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

Only if you add premises that make the system inconsistent, because then the whole logical system foes out the window.

E.g. the premises 1. If A, then B 2. A

Put together, you can infer

B

1. If A, then (not B)

Then your system is inconsistent and you can prove anything you want.

• Just a small point, but it's not the logical system that's inconsistent, but merely the premises of the argument. – possibleWorld Jun 3 '17 at 12:59
• Sorry, by system I meant the system of inferences created by the premises, not the formal system – Franz Jun 3 '17 at 13:35

Can you make a valid inference invalid by adding extra premises?

YES, always it will become invalid. If you add a dimension (extra premises) then the inference (conclusion, theory) has to change. If you analyze a subject based on two dimensions and draw your inference but then somebody shows that it is a three dimensional case then the original inference will become wrong. Dimensions are nothing but additional features.

As an example, Kepler said – satellites go around earth in an elliptical path and then produced a position formula for the satellite at any given time. His premises were - static, two dimensional, and on a paper. But in reality the earth is moving in 3D space around the sun, the sun is also moving in the same 3D space around the galaxy, the galaxy itself is moving in the universe. Thus when you add these dimensions, the Kepler’s law becomes invalid. Path of the satellite never meets and therefore will never be elliptical. The complexity of the path is beyond our comprehension.

However, the answer to the problem will always be NO, if the conclusion is based on the observation of nature, just like Galileo did. Kepler was wrong because he did not observe the nature. All inferences are independent of premises when based on observation of nature.

For more on the laws of nature and on universal truths take a look at the free book at the blog site https://theoryofsouls.wordpress.com/