5

Truth-functionally valid: there is no truth value assignment for which the premises (of an argument) are true yet the conclusion is false.

I am told that there are arguments which are logically valid but not truth-functionally valid. I'm having trouble imagining such an argument. Does such an argument even exist?

Edit: To clarify, I am wondering if that argument exists in the context of sentential logic.

9
  • 3
    Is it possible they're just talking about something being "truth-functionally valid" in the context of sentential logic, which is more limited than first-order logic? (this was one of the first results when I googled "truth functionally valid", the phrase is used on p. 3 of the pdf) Or are you certain they talking about "truth functionally valid" vs. "logically valid" within the same system of logic?
    – Hypnosifl
    Dec 9 '19 at 23:11
  • I believe they were talking about truth-functional validity in the context of Sentential Logic. Sorry, should have clarified that originally.
    – iyankv
    Dec 9 '19 at 23:12
  • 3
    But in that case, are you sure they were also talking about logical validity within the context of sentential logic (which is apparently another name for propositional logic)? Sentential/propositional logic lacks the "For all x" and "there exists an x" symbols and their associated rules that are found in first-order logic, so there can be deductions using those symbols which would be logically valid in first-order logic, but couldn't be expressed as truth-functionally valid deductions in sentential logic.
    – Hypnosifl
    Dec 9 '19 at 23:16
  • 2
    A proposition in first-order logic with quantifiers would't even obey the syntax of sentential logic, so it wouldn't have a truth value in that system--are you asking if there could be some way to "translate" a proposition in first-order logic involving quantifiers into one or more propositions in sentential logic without them? If so I can't think of any way that could be done, even in special cases...
    – Hypnosifl
    Dec 9 '19 at 23:29
  • 4
    Given the definitions of valid and "truth functionally valid" in the first comment, which correspond to syntactically vs semantically valid in the usual terminology, a syntactically valid but semantically invalid argument would mean that the corresponding proof system is unsound (and hence uninteresting). The converse, on the other hand, can and does happen in interesting systems, sentential or otherwise, some semantically valid inferences are unprovable, Goedel sentences, for example.
    – Conifold
    Dec 10 '19 at 0:29
1

If you stick to ordinary classical logic, then it is not possible, given the way you have defined truth-functional validity. An argument in the propositional (or sentential) fragment of classical logic is valid just in case every valuation under which the premises are all true also gives the value true to the conclusion. So there is no room for a sentential argument to be classically valid, but not truth-functionally so in that sense.

If you move away from classical logic, it mostly doesn't help. Non-classical logics such as intuitionistic logic and the relevance logics impose additional conditions on what qualifies as a valid argument. As such, their propositional fragments prove a proper subset of the theorems of classical propositional logic. You might be able to use the logic of paradox (LP) to get a valid argument with true premises and a conclusion that is both true and false. But LP is rather weird and relatively few people other than Graham Priest use it.

Another possibility is that you could have a logic that is not truth-theoretic at all. That is to say, it does not concern itself with formulas that can be interpreted as being 'true' or 'false' but with some other property or modality. For example, one could have a logic of obligation that does not commit one to supposing that statements of obligation are capable of being true or false.

I suspect you may have misconstrued what you were told. As Hypnosifl says in the comments, an argument in first-order logic with quantifiers may be valid but in a way that cannot be expressed using sentential logic. For example,

(∀x)Φx  therefore  ¬(∃x)¬Φx 

is valid in first-order logic, but expressed propositionally it just has the form P therefore Q, which of course is not valid. We might say of such an argument that it is not truth-functionally valid in the sense that you cannot prove it using truth tables. It does not mean there are arguments with true premises and a false conclusion.

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  • Hmm, I'm curious where you came across non-classical logic. It's not a classical topic in the usual university curriculum. Oct 2 at 0:45
  • I suppose it depends where and what you study. In my experience, mathematicians stick pretty rigidly to classical logic, with maybe a slight nod towards intuitionistic logic when doing constructive mathematics. Philosophers are often more open to using non-classical logics. I can remember learning about intuitionistic, relevance and even dialethic logics as an undergraduate.
    – Bumble
    Oct 2 at 0:55
  • I'd heard of quantum logic just after finishing high school in our local library. I didn't think much of it, because it didn't appear to have many useful results to speak of. Compared to the long tradition of physical thinking, the results quantum logic had seemed fairly basic and it seemed merely a very speculative attempt at an alternative logic and to be honest, didn't seem much like logic itself: that is, considered as reasoning about reasoning. In fact, it seemed pretty pathetic. Two mathematicians were responsible for that, von Neumann and Birkhoff ... Oct 2 at 1:02
  • ... moreover, the theory of types, also developed by two mathematicians - Whitehead and Russell - after leading an underground movement under the radar of the rigidly educated classical logicians has resurfaced as a revolutionary way at looking at mathematics and also physics as in the theory of toposes. For example, cohomology can be understood in that framework. Here, it's higher types that are important and the logical language internal to the topos is higher intuitionistic logic - this was originally developed by a mathematician - Brouwer. Oct 2 at 2:57
  • So far from mathematicians being rigid it seems they were leading the revolutionary charge against a creaky old system that hadn't seen any new thought for some time. Oct 2 at 2:58
0

The Tao opens up with the sentence:

The way that can be walked is not the way.

This is true if one understands the Tao. But it is not deductively, that is, logically true as the sentence rescinds the deductive meaning given it. A way, normatively thought, is somethimg that can be walked. But then the sentence turns around and says this is not the way. Logically speaking, it reduces to a contradiction or paradox. Butv it simply says, all the ways that can be thought about the Tao does not exhaust the Tao. In Aristotle, a similar definition was given of the Infinite but not in such paradoxical language as Aristotle is not given to such poetic language.

In a papal encyclical about Catholicism relations to other major religions, this is called the via negativa towards the divine sphere. In other words: the god that can be defined is not god.

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  • "The way that can be walked is not the way." This is merely slipshod language. This can be expressed properly, logically, without equivocation: The way that can be walked is not The Way. Oct 2 at 10:23
  • @speakpigeon: This is completely wrong. The language is not slipshod or misleading. And you are being misleafing because by say 'The Way' you are suggesting that The Way can be defined. This is the crux of the matter in the metaphysics of the Tao because this concepts exceeds all deginitions made for it, it is always larger. You're thinking in terms of rigid classical thinking. Oct 2 at 23:21
  • I am not interested in what you think about the Tao since you have essentially just admitted that you don't know what it is. I was merely redacting your statement to make it logical. My redacted statement "The way that can be walked is not The Way" may be true or false because it is not illogical, while your statement "The way that can be walked is not the way" is neither true nor false because it is illogical (since it is either self-contradictory or committing the fallacy of equivocation). Rigid classical thinking? Well, merely logical thinking, which is the point. Oct 3 at 10:44
  • For your information, here is a good dictionary definition of the Tao: 1. In Taoism, the basic, eternal principle of the universe that transcends reality and is the source of being, non-being, and change. 2. In Confucianism, the right manner of human activity and virtuous conduct seen as stemming from universal criteria and ideals governing right, wrong, and other categories of existence. Oct 3 at 10:46
  • @SpeakPigeon: Yeah, and because it transcends reality, it transcends being defined, and all definitions, hence the way that can be defined (walked) is not the way. Oct 3 at 10:50
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By " logically valid " I mean here " deductively valid". This is a broad sense of logical validity which requires only one thing : namely, that it is logically impossible for the conclusion to be false in case the premise(s) is/are true.

A classification of logically valid arguments

(1) formally valid ( valid in virtue of its form alone, the meaning of the non-

logical symbols/expressions playing no role in the validity).

 (1) A/ truth functionally valid

 (2) B/ non truth functionally valid 

(2) valid, but not formally ( valid in virtue of the meaning of non logical words).

Examples

Of 1 A)

If I think, therefore I am.

I think.

Therefore I am.

Of 1 B)

All men are mortal

Socrates is a man

Therefore Socrates is mortal.

Of (2)

Peter is a pianist.

Therefore, Peter is a musician.

The form of this reasoning is

(1) P (a) (2) Therefore , M(a)

in the context of predicate logic.

The form is

(1) P (2) Therefore M

in the context of sentential logic.

Consequently, it is easy to see that the reasoning is not valid in virtue of its form.

It is logically valid however since it is logically impossible fo the conclusion to be false if the premise is true.

-2

There exist arguments such that following two statements are true:

  • the argument is valid (not "seems" valid, but definitely is 100% valid)
  • the argument appears to be invalid if you use formal logic and truth-tables to represent the argument.

Specially, it is a mistake to...

  • directly translate English "not" into logical "not"
  • directly translate English "and" into logical "and"
  • directly translate English "or" into logical "or"

The thing is, logical operators are a model of English.

A scientist might make a model of a baseball flying through the air such that:

  • the baseball is represented a mathematically perfect sphere. For example, the threads which baseball pitchers use to grab the ball don't actually stick up above the ball surface, etc..)
  • There is no air resistance in the scientist's model.

baseball picture

The problem is that models are flawed; they imperfect paintings of reality.

Formal logic is an imperfect model of English. As such, a translation from English to formal logic can make a valid argument appear invalid.

Consider the formal fallacy of "Affirming a Disjunct"

The following is an an outline of affirming a disjunctive:

(P or Q) is true (line 1)
P is true (line 2)
Q is false (from lines 1 and 2)

affirming a disjunctive is an invalid argument.
The following is an example of "Affirming a Disjunct"

1... (You are currently on planet mars) or (You are currently on planet earth)
2... (You are currently on planet earth)
3... NOT (You are currently on planet mars) (from lines 1 & 2)

Note that concluding NOT (You are currently on planet mars) is logically fallacious.
Affirming a disjunctive is supposed to be incorrect reasoning; poor quality thinking; major no-no

1... (You could have had sausage) or (You could have had bacon)
2... (You had bacon)
3... NOT (You had sausage) (from lines 1 & 2)

The example involving sausage seems valid, even though, from a formal standpoint, it is invalid. How do we explain this?

Well... one explanation is that:

  • English "or" is "exclusive"
  • logical "or" is "inclusive."

If P and Q are both true, then

  • (P) logical or (Q) is true.
  • (P) English or (Q) is false.

If English or was inclusive, then the phrase “and/or” would not exist.
Newspapers and legal documents abound that use the phrase "and/or" to indicate inclusive-or

Below is an example of English “and/or” at work:

Violation of medical privacy is cause for disciplinary action, criminal prosecution, and/or personal liability for a civil suit.

The previous English sentence means that at least one of the following may be the result of violation of medical privacy:

  • disciplinary action
  • criminal prosecution
  • personal liability for a civil suit

Affirming a Disjunct is a fallacious argument-form because logical-OR is inclusive.

1. ..... (P and/or Q) is true ..... axiom
2. ..... P is true ..... axiom
3. ..... Therefore, Q is false ......from lines 1 and 2

Affirming a Disjunct is fallacious because it is possible that both P and Q might be true. One being true does not discount the other.

However, in English, the word “or” often (not always, but often enough) means that exactly one of the options must be true and all other options must be false.

The following two arguments follow the “affirming a disjunctive” pattern. For some people, the following arguments are supposed to be examples of bad thinking. However, they are perfectly valid arguments as long as “or” means “exactly one of the following things is true.”

(Joey will take ceramics as an elective) or (Joey will take pole-vaulting as an elective)
Joey will take ceramics as an elective
Therefore, Joey will not take pole-vaulting as an elective

I am sure you're getting tired of examples by now, but let's have one last example:

(Sarah’s Car is painted Blue) or (Sarah’s Car is painted Red)
Sarah’s Car is painted Blue
Therefore, it is not the case that (Sarah’s car is painted Red)

Logical “or,” means “at least one option is true.”
The Logical-disjunctive operator (logical-or) is often a very bad model of the English word “or.”

Guess what? It turns out that the "exclusive or" operator from formal logic is ALSO a bad model of English "or"!

A xor B xor C xor D xor E xor F is an example of inputting 6 statements into exclusive or.
The result will be true if and only if an odd number of statements are true.
That is exactly one of the following must be the case:

  • exactly 1 statement is true ("exactly" means at least one and no more than 1)
  • exactly 3 statements are true
  • exactly 5 statements are true.

2 or 4 inputs true makes the output false.

When someone writes a long list delimited by the word “or” how often do they mean “an odd number of these statements are true?” The answer is: basically never.

Consider the following example:

> All sandwiches are served with sweet potato fries, quinoa salad, or a Cesar salad.

>  You can have Pepsi, diet Pepsi, Dr. pepper, sprite, root-beer, or coke.

Do you think that a person describing the restaurant menu means, “you can have a mix of 1, 3, or 5 of the soda flavors, but absolutely not a mix of 2 flavors, 4 flavors, or 6 flavors?”

Neither logical OR not logical XOR correctly model English OR

Notably:

  • English OR is usually exclusive (so, do not model it using inclusive logical "OR")
  • English or doesn't care about even or odd inputs. Logical XOR has to do with odd and even.
  • English or is not binary: it can accept 2, 3, 4, ... 281 inputs.

80% of the time English or means "exactly one of the following is true." The other 20% of the time, English or means "at least one of the following is true." There is no "an odd number of the statements are true."

The fallacy of Affirming a Disjunct is frustrating because if someone is accused of committing the fallacy, they almost always have not committed the fallacy.

Instead, English “OR” has been incorrectly modeled logical “OR.”

Logical operators are horrible horrible models of natural language constructs.
The “or” in logic is not the same “or” as in English (or other languages, such as French, German, Mandarin, etc...).

Therefore, an argument can be valid even if a truth-table interpretation says the argument is invalid.

It is not just "OR." Unary negation (NOT), and all of the others aren't perfect either.

Introductory logic teachers will tell you that the negation of

Bob's car is red"
is
"Bob's car is NOT red."

That is WRONG.

The negation of
"Bob's car is red"
is
"Bob's car does not exist or Bob's car is not red."

Consider:

Felicia's daughter's landlord's law degree's paint color is heavy

The negation is

NOT (Felicia's daughter's landlord's law degree's paint color is heavy)

In English, it is not valid to distribute the "NOT" operator from the outside of a heavily nested statement to the innermost statement.

The negation is not:

Felicia's daughter's landlord's law degree's paint color is NOT heavy.

The correct negation is:

at least one, and at most one, of the following statements is true:

  • Felicia does not exist
  • Felicia exists but Felicia does NOT have a daughter
  • Felicia's daughter exists, but the daughter does NOT have a landlord
  • Felicia's daughter has a landlord, but the landlord doesn't have a law degree
  • Felicia's daughter's landlord's exists and has law degree, but the law degree does NOT have a paint color Felicia's daughter's landlord's law degree has a paint color, but the paint color is NOT "heavy"

Out of all binary logical operators, "If... then" is probably has worst mismatch between logic and English.

Notably, you cannot determine when English "If P then Q" is true from knowing the truth values of statements P and Q alone. Additional context is required.
In logic, "If P then Q" commits the fallacy of "correlation implies causation."

The following are both true:

  • My name is "Sam"
  • Queen Elizabeth of England was born on April 21, 1926

In logic P and Q implies if P then Q.

If (My name is "Sam") then (Queen Elizabeth of England was born on April 21, 1926)

How about Q implies if P then Q?

I am depressed. Therefore, If I brush my teeth, then I will be depressed.

Teeth brushing causes depression? According to formal symbolic logic, correlation implies causation.

None of that makes any sense.... why?

BECAUSE LOGICAL IMPLICATION IS TERRIBLE, HORRIBLE, AWFUL, BAD MODEL OF "If...then" IN ENGLISH

Basically:

  • Logic is dead simple
  • English, French, German, Hangul/Korean, Mandarin, etc... are all horribly complicated languages

The model is usually simpler than reality. There is no air-resistance in some engineer's model of a baseball flying. Hence the errors...

Saul Kripke (a very famous logician) argued that English sentences of the form if...then always contain a necessity operator from modal logic. Some day, humans might develop good logical models of English, but the current models have some problems.

2
  • Great answer. Language is use.
    – CriglCragl
    Jul 19 '20 at 21:26
  • "the argument is valid (not "seems" valid, but definitely is 100% valid) the argument appears to be invalid if you use formal logic and truth-tables to represent the argument." What is "validity" anyways? Jan 29 at 4:27
-3

I believe it is possible for arguments to be logically valid but not truth-functionally valid, in that they appeal to logic but are false in reality.

For example, consider how a herd of bison will run toward anything that scares them. Logically, if something is scared it either runs away or stays very still, right? And this is true for nearly all living things (except plants, of course). But through the eye witness experiences of numerous folks, it has been proven that bison contradict this logic and run toward the things that scare them.

Another fact about bison: Who would guess a 1600 to 2000 lb. animal, that is 10 to 12 feet long, can run up to 40 mph. and jump 6 feet of the ground? This fact does not appeal to logic either, although it has been proven to be reality.

I have little doubt that similar scenarios, of which the facts contradict logic, exist in many areas of science. Likely many of them have been discovered and many more are, as of yet, unknown to humanity.

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