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.

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.

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 – 2019-12-09T23:11:21.833I believe they were talking about truth-functional validity in the context of Sentential Logic. Sorry, should have clarified that originally. – iyankv – 2019-12-09T23:12:49.967

2

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 – 2019-12-09T23:16:52.497I assumed they were also talking about logical validity within the context of sentential logic since I was told this in a discussion involving only sentential logic. Out of curiosity, if I provided an argument involving quantifiers, could I prove that said argument is not truth-functionally valid in the context of sentential logic? – iyankv – 2019-12-09T23:21:28.570

1A 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 – 2019-12-09T23:29:43.563

I figured such an argument wouldn't obey the syntax of sentential logic. In any case, my original question still stands. Does there exist an argument (in sentential logic) that is logically valid but not truth functionally valid? – iyankv – 2019-12-09T23:32:46.617

3

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 – 2019-12-10T00:29:55.993In the context of sentential logic, see Modal Logic.

– Mauro ALLEGRANZA – 2019-12-10T08:32:34.267@Conifold You should put that as an answer! – Bertrand Wittgenstein's Ghost – 2021-01-29T04:29:19.490