## Invalid arguments with true premises and true conclusion

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I'm am study some classical logic and I am having trouble with argument validity. I learned to think of validity as determined by the definition "If the premises are true, then the conclusion must be true". Keeping the truth table of conditionals in the back of my mind has served fairly well in my exercises of evaluating arguments. However, I came across this example on the internet, "All cats are mammals, All tigers are mammals, Therefore all tigers are cats". The site says that this argument has true premises and a true conclusion yet it is invalid. Thinking of the conditional definition would't this be a true implying a true, which means the argument is valid? I appreciate any clarification.

Question was closed 2014-10-18T11:08:42.200

Is invalid not a synonym for false? – Neil Meyer – 2014-10-17T10:03:46.493

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Your question is basically the same as this one: What is the logical form of the definition of validity? . And my answer is a less formal version of what Hunan is telling you.

an argument is valid if having its premises be true necessarily leads to a true conclusion.

The necessarily / must element in the definition makes it so that we are not looking at whether the claims are in fact true but rather whether the forms of the claims are such that their truth implies the truth of the conclusion. Thus, we need to check to see if there is any truth value for the variable involved whether or not it is possible that the premises end up being true and the conclusion being false.

To do so involves several steps and there are multiple methods.

"All cats are mammals, All tigers are mammals, Therefore all tigers are cats".

This gives us three statements and three variables. To make it first order logic, we need understand "all" to mean if it is an A, then it is a B:

(1) C -> M
(2) T -> M
Therefore (3) T -> C


As you rightly point out, all 3 claims turn out to be true (assuming by "cats" we mean something other than felinis familiaris).

Why then is the argument invalid? The key is that the validity looks at if it is possible for the argument to have true premises and a false conclusion.

# Test Method #1

We can test this with several methods of varying difficulty to grasp. One of the easier ones to understand for many people is the exhaustive truth table. Here, we are going to create rows where we assign whether variables are true or false and look to see if the claims are true or false. If we end up with a situation where the premises are true and the conclusion is false, then the argument is invalid.

In our case, we have three variables. Per the law of the excluded middle, each variable can be true or false. Thus, either it is true that it is a cat or it is false that it is a cat. [etc] This will give us 8 rows as follows:

C T M
-----
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F


This represents every way that these variables could be related -- regardless of how they are related in this world. We then look at whether each claim is true. A conditional is true when the antecedent (left part) is true and the consequent (right part) is true. OR when the left part is false, it is true regardless of the right part. Or to put it another way, it is only false when the antecedent is true and the consequent is false.

C T M   C -> M  T->M    T->C
-----
T T T   T       T       T
T T F   F       F       T
T F T   T       T       T
T F F   F       T       T
F T T   T       T       F
F T F   T       F       F
F F T   T       T       T
F F F   T       T       T


If you look the fifth line has true premises and a false conclusion. Thus it is possible for the argument to have had true premises and a false conclusion. It turns out that in our world the premises and conclusion are true, but the logic behind the premises does not compel the conclusion we are drawing. So the argument is invalid.

# Test Method 2

There's a faster way called the short circuit method where you accomplish the same thing as the above method but cheat. Instead of making every row, we just set the conclusion to false and figure out how we can make the premises true if that's the case. If we can make all of the premises true, we've proven it is invalid.o

So we begin like this:

C T M   C -> M  T->M    T->C
-----
F


We then ask what it takes for T -> C to be false. The answer is that T must be true and C must be false. (due to the way conditionals work).

C T M   C -> M  T->M    T->C
-----
F T                     F


If C is false, then C -> M is true regardless of the value of M.

C T M   C -> M  T->M    T->C
-----
F T     T               F


The question then is if we can make T -> M true with these values for C and T already set. The answer is that we can -- if we set M to true.

C T M   C -> M  T->M    T->C
-----
F T T   T       T       F


Thus, we've shown invalidity -- because we can have true premises and a false conclusion. Note again, this does not mean that we do.

# Test Method 3

Finally, we can show the same thing using rules of inference (which I will leave out here but is probably the most common method in philosophy and math).

Venn diagrams can also be a useful test method--the premises imply that circles for both "cats" and "tigers" must lie within the larger circle "mammals", but don't tell you whether "cats" lies within "tigers", or "tigers" lies within "cats", or if the two circles partly overlap but also have non-overlapping sections, or if the two circles don't overlap at all. – Hypnosifl – 2019-08-09T16:06:48.197

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Consider the following, classical, definition of validity:

An argument from premises P, P′, ... to conclusion C is valid iff (P ∧ P′ ∧ ...) → C is a logical truth.

Let's apply that definition to your example argument with cats and tigers to see what happens. Let C, T, M be the predicates 'is a cat', 'is a tiger', and 'is a mammal', respectively. In first-order logic we can express the cat-tiger argument as:

1. x Cx → Mx.
2. x Tx → Mx.
3. x Tx → Cx.

The argument from (1) and (2) to (3) is valid iff [(1) ∧ (2)] → (3) is a logical truth. But that's not the case, because there is at least one possible world (which is, in fact, the actual world) where (1) and (2) are satisfied but (3) is not. Had the relation been an identity (=) instead of a conditional (→), we'd have a valid argument because identity is symmetrical. The conditional is transitive, but not symmetrical.

"The argument from (1) and (2) to (3) is valid iff [(1) ∧ (2)] → (3) is a logical truth" is false, material implication does not reflect inferential validity. For example, if (1) ("sky is blue") and (2) ("grass is green") are true statements unrelated to (3) ("water flows"), which is also true, the implication is true, but the inference is invalid. – Conifold – 2014-10-20T20:07:09.760

Notice that it says "logical truth" not simply "truth". "If (sky is blue and grass is green) then water flows" has true interpretations, but is not a logical truth (i.e. there is at least one truth-assignment to "sky is blue", "grass is green" and "water flows" that makes that conditional false, namely the one that assigns true to the first two and false to "water flows".) – Hunan Rostomyan – 2014-10-20T23:32:51.133

These sentences are meant as logical constants with truth values fixed to T, they can not be reassigned because there is a single interpretation. So by your definition with "logical truth" (which seems to mean "tautology") the inference is valid. And there is a bigger problem since the only valid inferences it allows are the purely analytic ones. So Euclidean proofs in geometry are all invalid under it whenever geometric axioms are used. – Conifold – 2014-10-21T17:40:45.757

@Conifold Which sentences are logical constants? – Hunan Rostomyan – 2014-10-21T17:55:11.167

The ones about grass, sky and water. – Conifold – 2014-10-21T18:54:34.220

@Conifold Those sentences are denoted by whatever primitive propositional symbols the underlying language has, say by 'p', 'q', and 'r'. These are not logical constants and their interpretation is not fixed. In the world we live in, those symbols are indeed assigned the truth-value T, but that doesn't mean that they're true in all possible worlds. It slightly puzzles me why you would think that those arbitrary propositional letters have a standard interpretation and one that assigns true to all three. – Hunan Rostomyan – 2014-10-21T19:01:55.767

They are logical constants if that is how one wishes to introduce them, or are we only supposed to have theories where all symbols have multiple interpretations? Or not have inferences in the rest? – Conifold – 2014-10-21T22:29:26.497

@Conifold I did not say that all symbols have multiple interpretations; I said that the propositional letters (p, q, r,...) do. Among the symbols that do have fixed interpretations are the logical symbols, such as '~', 'v', '&', and so on. Sometimes among the logical symbols we also find the epsilon symbol, which can stand for set membership, type, and so on. What makes these symbols special is that we have fixed, sometimes even standard, (set of) interpretations for them. I say "set of", because there are alternative semantics of negation, implication, membership, and so on. – Hunan Rostomyan – 2014-10-21T23:24:43.193

I can have a formal theory with non-logical constants entering predicates, and with axioms that declare the resulting sentences true (arithmetic is one). That makes them sentences with fixed truth values for which your definition of inference doesn't work because it only allows logical tautologies as derivation rules. – Conifold – 2014-10-22T00:05:09.050

@Conifold In arithmetic, 'N' is a logical constant. Sentences like "2 belongs to N" are logically true in the sense that once the usual logical symbols ('and', 'or', 'is in', etc.) are given their standard interpretation and N is defined according to Peano's Axioms, "2 belongs to N" becomes true. In other words, all interpretations that satisfy the Peano Axioms make "2 belongs to N" true. The same cannot be said of "snow is white", "sky is blue", or "water flows". – Hunan Rostomyan – 2014-10-22T00:16:46.790

Actually, N is not a constant of arithmetic and things like "2 belongs to N" are not expressible in arithmetic, only in set theory. But one can certainly have a theory with non-logical constants called "snow", "sky" and "water", with predicates white, blue and flows, and with these very sentences being axioms (or theorems). Along with many other predicates and axioms that incorporate our understanding of snow, sky and water. – Conifold – 2014-10-22T00:25:08.420

@Conifold Although I'm not in complete disagreement about your last comment, and in general do share your liberal attitude about what could count as a logical constant, this discussion isn't very productive. – Hunan Rostomyan – 2014-10-22T00:38:54.900

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Having you chosen an example from syllogism we can go back to the source : Aristotle's Logic, in order to point to the fallacy in your argument.

You have forgotten the "formal" aspect (see : formal logic) :

a valid argument must be so "by virtue of form alone".

In Aristotle's logic :

A deduction is speech (logos) in which, certain things having been supposed, something different from those supposed results of necessity because of their being so [emphasis added]. (Prior Analytics I.2, 24b18-20)

The core of this definition is the notion of “resulting of necessity” . This corresponds to a modern notion of logical consequence: X results of necessity from Y and Z if it would be impossible for X* to be false when **Y and Z are true. We could therefore take this to be a general definition of “valid argument”.

Aristotle proves invalidity by constructing counterexamples. This is very much in the spirit of modern logical theory: all that it takes to show that a certain form is invalid is a single instance of that form with true premises and a false conclusion. However, Aristotle states his results not by saying that certain premise-conclusion combinations are invalid but by saying that certain premise pairs do not “syllogize”: that is, that, given the pair in question, examples can be constructed in which premises of that form are true and a conclusion of any of the four possible forms is false.

"All cats are mammals"

"All tigers are mammals"

therefore : "All tigers are cats"

the premises "do not syllogize”; in order to show this, consider the following couple of premises:

"All cats are mammals"

"All whales are mammals"

therefore : "All whales are cats"

what has gone wrong ? Exactly the "form" of the argument; having found a counter-example (i.e. a couple of true premises of the requested "form" with a false conclusion of the requested "form"), we have to conclude that the argument with that "form" is invalid, i.e., with the invalidity of the following argument :

"All P are Q"

"All R are Q"

therefore "All R are P".

In modern logic, the "formal" aspects are treated mathematically :

(i) using a formalized language able to express the "form" in a perspicuous way; "All P are Q" is expressed as :

∀x(P(x) → Q(x)).

(ii) with the concept of interpretation of a formula

(iii) with a rigorous definition of the relation of logical consequence :

"All R are P" is a logical consequence of "All P are Q" and "All R are Q" if and only if, for all interpretation, it is not the case that "All P are Q" and "All R are Q" are true and "All R are P" is false.

The previous counter-example shows that this is not the case; thus, the purported conclusion : "All R are P" is not a logical consequence of the premises (they "do not syllogize").

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Right, propositional validity and rhetorical sense are different things.

In propositional logic, any falsehood implies anything you wish. Whereas we know from experience that you can only get from certain lies to certain other lies without doing something obviously problematic.

Likewise, in propositional logic anything true implies any established truth, whereas we know that specific truths have to be reached by defensible paths.

The interpretation of "A implies B" as "B or not A" is an idealization of real rhetorical logic into a form that assumes all true deductions have already been established before the question is asked. It is tense-free and ignores the ordering of premises or logical steps.

It is a safe way to learn to argue and to vet arguments because it is conservative in this fashion. Anything provable from a rhetorical point of view is provable in the idealized system, and vice versus, but some of its actual acceptable arguments will come across as unconvincing to humans.

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Maybe I'm just crazy, but I don't think he's asking about temporal logic (http://plato.stanford.edu/entries/logic-temporal/).

– virmaior – 2014-10-17T00:40:50.730

Sorry for the outdated terminology. I have fixed it. – None – 2014-10-17T13:09:08.553