Invalid argument with true premisses and true conclusion

Joshua asked

Can an argument have all true premises and a true conclusion, yet not be deductively valid?

Answer by Craig Skinner

Yes it can.

Just state some unconnected true statements:

Example 1

P1 Grass is green
P2 Paris is the capital of France
C A poodle is a dog

Ps and C all true, but argument not deductively valid.

If you object that this doesn’t count as an argument because there is no connexion between the Ps or between the Ps and the C, try

Example 2

P1 All atoms are tiny
P2 The smallest particle of helium gas is tiny
C The smallest particle of helium gas is an atom

All true, but not deductively valid. To see this, substitute ‘oxygen’ for ‘helium’ (the smallest part of oxygen gas is a molecule not an atom, so C false) or ‘pollen’ for ‘helium gas’ (the smallest particle of pollen is a grain, C false)

I find a Venn diagram useful here.
Draw a circle (T) to represent tiny things. Draw a smaller circle inside it (P) to represent smallest particles. You have illustrated that some, but not all, tiny things are particles. Now put another, yet smaller, circle (A) inside P to represent atoms, illustrating that some smallest particles are atoms. You can instantly see that all atoms are smallest particles, but that a smallest particle need not be an atom: and that all smallest particles are tiny, but not all tiny things are smallest particles.

As well as arguments that have true premises/conclusion but are invalid, we can have arguments that are valid and have a true conclusion but are unsound because a premise is untrue.

Example:

P1 Craig is a Scot
P2 All Scots are drunks
C Craig is a drunk

Here, P1 can be true, C follows from the Ps (validity), C can be true, but the argument is unsound because P2 is false. So, although the C is true we can’t rely on the argument to establish it. It is an unsound argument.

So, distinguish between validity and soundness.

Validity means that the C follows from the Ps. If the Ps are true, the C is necessarily true. Validity is a purely formal matter. Whether or not the Ps are really true doesn’t come into deciding validity.

Soundness combines validity and truth: the argument is valid, the premises are true, the conclusion must be true. Sound arguments are what we want in philosophy and in life.

So, useful to distinguish truth, validity and soundness when you evaluate logical (deductive) arguments.

Finally, not all arguments are deductive. Some are inductive, some abductive. There is much to be said about these, but here I’ll only say that induction and abduction lack the logically watertight character of deduction.

Answer by Shaun Williamson

The answer to your question is yes. Consider this argument.

Premise 1 The earth orbits around the sun
Premise 2 The moon orbits around the earth.
Conclusion Some Birds can fly.

All these things can be accidentally true but that doesn’t make the argument
logically valid.

The definition of a LOGICALLY (or deductively) VALID argument is that it is any argument where IF the premises are true then it is IMPOSSIBLE for the conclusion to be false.

In our world, in the argument given above, the premises and the conclusion are all true but we can easily imagine a world where premise 1 and premise 2 are true but the conclusion is false.

Consider this logically valid argument.

Premise 1 If Frank is a rabbit then Frank wears white gloves.
Premise 2 Frank is a rabbit.
Conclusion Frank wears white gloves.

This argument is logically valid because we cannot imagine a world where
premise 1 and premise 2 are TRUE but the conclusion ‘Frank wears white gloves’ is false.

3 thoughts on “Invalid argument with true premisses and true conclusion

  1. “To see this, substitute ‘oxygen’ for ‘hydrogen’ (the smallest part of oxygen gas is a molecule not an atom, so C false) or ‘pollen’ for ‘hydrogen gas’ (the smallest particle of pollen is a grain, C false)”

    Just doesn’t work. If you’re going to assign “particle” to “atom” in one case, you can’t deny the connection elsewhere just because it fits your argument. Atoms definitely fit the definition of “particle,” so yes, the smallest particle of oxygen *gas* (another mistake, since you started with an element and then changed to a gas) is an atom. The smallest particle of pollen is also an atom. (Actually, we should probably say “quark,” but whatever.)

  2. Grass is green.
    Paris is the capital of France.
    Poodles are dogs.
    Therefore, Apples are fruit.

    If “validity” is a property of an only of deductive arguments; and validity is defined as a deductive argument that CANNOT have true premises and a false conclusion, then how is the above “argument” an example of a valid DEDUCTIVE argument if there is no claim of inference between premises to entail the conclusion?
    In determining whether someone is giving an argument or just stringing a series of independent statements together is not solely up to the presenter; there must at least be some ambiguity as to whether something COULD be an argument or, say, an explanation. If a series of statements (propositions) CANNOT be rationally understood as specifically a DEDUCTIVE argument–as there is no effort to create any claim of inference–then there is no ambiguity, and thus, a series of independent statements cannot claim to be a deductive argument, and thus, cannot be used as an example of deductive validity.

  3. I find that the responses “Yes, it can” quite ambiguous: does that mean, “Yes they CAN be deductively valid,” or “Yes, then cannot be deductively valid?” Then Skinner states that it CANNOT BE deductively valid, but states an observation that “you object that this doesn’t count as an argument because there is no connexion between the Ps or between the Ps and the C. There is no objection because he has already stated that it cannot be deductively valid because there is no claim of inference. (?!) He seems to imply that the example IS deductively valid an anticipates an objection when he has already stated that it cannot be deductively valid.
    As for Williamson’s response, the initial ambiguity of his “yes” is clarified, but his reasoning is incorrect: ‘In our world, in the argument given above, the premises and the conclusion are all true but we can easily imagine a world where premise 1 and premise 2 are true but the conclusion is false.” That is not why the series of statements do not create a deductively valid argument, but rather that there is no claim of inference being made.

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