In the labyrinth of human thought, few concepts are as captivating and confounding as logical entailments and the idea of truth. Let’s attempts to demystify these intellectual enigmas, armed with the torch of philosophy and the compass of science.

## Understanding Logical Entailments:

At its core, logical entailment refers to the relationship between propositions, where one proposition logically follows from another. Picture it like a domino effect of ideas: if proposition A is true, then proposition B must also be true because of the logical connections between them.

### Entailment Symbols

- The entailment from A to B is represented by the symbol ⊢, therefore A⊢B means A entails B. This is a derivable entailment rather than an necessary one.
- If the entailment is necessarily true, this is represented by the symbol ⊨. Therefore A⊨B means A necesarily entails B.
**→**or –> are symbols of a conditional nature. A–>B means If A then B.

## Necessary Truths:

Think of necessary truths as the bedrock of reality, the unshakable foundations upon which the edifice of knowledge stands. These truths are true in all possible worlds, like 2 + 2 = 4 or “All bachelors are unmarried.” They’re not just true in this world; they’re true in any conceivable world.

Some contrarians might suggest a possible world the word bachelor means married, but that is changing what bachelor refers to, this is known as changing the referent.

Consider if there was a culture where their word for dog was cat. That doesn’t make dogs cats, it’s just two different labels for the same referent. It’s important to remember that labels are not necessary truths, but possible truths.

## Possible Truths:

On the other hand, possible truths are a bit more flexible. They’re statements that could be true given certain conditions or circumstances. For example, “It’s possible for it to rain tomorrow” or “It’s possible for humans to colonize Mars.” These truths depend on the specifics of our world and the conditions within it.

The reason lables are possible truths is because it is possible that there is a culture that uses the word dog to refer to a cat, but it is not something true right now. If labels were necessary truths, the same labels would be used universally. Whilst this might be easier for communicaiton, is not how language works.

A possible truth can also refer to something truth in “some possible world”. For something to be a possible truth in this scenario it would mean that it would be something, given a set of circumstances, would be possible. That means that it would have to be logically possible, in otherwords, without contradiction.

## Contingent Truths:

A contingent truth, on the other hand, is a statement that is true in the actual world but could have been false under different circumstances. It is not necessarily true in all possible worlds. For example, the statement “It is raining in Bournemouth today” could be a contingent truth; it’s true today, but it might not be true tomorrow or in another possible world where the weather is different.

### To further clarify:

- A
**necessary truth**is a statement that must be true in all possible worlds. It could not have been otherwise under any circumstances. For instance, mathematical truths such as 2+2=4 are necessary truths because they hold in all possible worlds. - A
**contingent truth**is true in the actual world but did not have to be true; it is true as things happen to be. It is true in some possible worlds but not in others. An example could be “The Eiffel Tower is in Paris”; while it’s true in our world, one can imagine a possible world where the Eiffel Tower was built in a different city. - A
**possible truth**may not be true in the actual world, but is possible in some world.

The exploration of these concepts helps us understand the nature of truth and the potential variability of truth across different scenarios or worlds. Philosophers like Leibniz and Quine have contributed significantly to the discussion of necessary and contingent truths, each offering different perspectives on the subject.

## Navigating the Terrain:

Now, here comes the fun part: exploring how necessary truths and possible truths dance together in the grand ballroom of logic. Picture this: if something is necessarily true, it’s automatically a possible truth because, well, it’s true in all possible scenarios. However, not all possible truths are necessarily true. For instance, it’s possible for it to rain tomorrow, but that doesn’t mean it’s necessarily going to happen.

The same can be said for contingent truths. They are true in this world, so they are possible. They may be true in another world world as well, but they don’t have to be. The point of the contingent truth is that it is only possible if certain other things are true.

## The Role of Science and Philosophy:

Science and philosophy are like two peas in a pod when it comes to unraveling the mysteries of logical entailments. Science helps us navigate the empirical realm, testing hypotheses and theories against the observable world. Philosophy, on the other hand, delves into the realm of abstract thought and logic, pondering the nature of existence and truth itself.

In fact, the question ‘What is Truth?‘ can pose some very different answers and there are multiple theories of truth with various answers.

When we lock down the concept of truth, science is one of, if not the, best tool to find said truth (at least where it pertains to matters of an empirical nature).

## Examples in Action:

An example should help solidify our understanding. Imagine we have the statement: “All humans are mortal.” This statement is a necessary truth because it holds true in all possible worlds. Now, consider the statement: “Socrates is mortal.” This statement follows logically from the previous one because Socrates is a human (a necessary truth), therefore, he must be mortal.

This particular deductive argument can be written in the form of a syllogism:

P1. All humans are mortal.

P2. Socrates is human.

C. Socrates is mortal.

### Breaking Down the Syllogism

This argument is logically valid because the conclusion is a necessary entailment of the the premises being true. This argument is sound because the premises are true, which we know from the science.

Being human entails necessarily entails mortality: Human ⊨ Mortal. (an immortal human would be changing the referent of what a human is).

If someone is Human then someone Mortal: Human -> Mortal.

#### Symbolic Logic

We can use symboloic logic to simplify the argument. The symbols, when you don’t know them, can be quite overwhelming. Plain language is actually my preferred method of communicating these concepts, however, as we having a disussion about logical entailments, it would be prudent to understand how this argument could be contructed an a more methmatical form.

##### Standard Use of Logic Symbols

**∀** = For All**x** = Variable. A variable can be populated with, or represents, any thing or individual.**()** = a Grouping or Set. To be specific, the brackets denote the container for a set, a member of a set, and/or describe qualities within that set.**s** = a specific Someone or Subject.

–> = if … then …

Logic symbols are not limited to this, but they are the ones we are using below.

##### Our Additions to the Symbols

When creating an argument, it is also prudent to define any custom terms used for the argument.

**H** = human **M** = mortal**s** = Socrates. We have filled our somone with a specific individual.

We then can populate our set: (*H*(*x*)–>*M*(*x*))

##### Application of Symbolic Logic

We know that H ⊨ M, so all things that are human are necessarily mortal.

We could also state this as: ∀*x*(*H*(*x*)→*M*(*x*)).

For all [∀] things [*x*] in the set of humans [*H*(*x*)] are mortal [*M*(*x*)].

Therefore if someone [*s*] is human [*H*] or *H*(*s*) then somone is mortal [*M*] or all together:*H*(*s*)→*M*(*s*).

To replace the syllogysm completely with symboloic logic:

- ∀
*x*(*H*(*x*)→*M*(*x*)) - 𝐻(𝑠)
- Therefore, 𝑀(𝑠).

In fact, we can go one step further and write it like this:

- The premises {∀𝑥(𝐻(𝑥)→𝑀(𝑥)),𝐻(𝑠)} entail the conclusion 𝑀(𝑠):

{∀𝑥(𝐻(𝑥)→𝑀(𝑥)),𝐻(𝑠)}⊢𝑀(𝑠) - If the premises are true, the conclusion necessarily follows:

{∀𝑥(𝐻(𝑥)→𝑀(𝑥)),𝐻(𝑠)}⊨𝑀(𝑠)

#### Necessarily True Entailments

These truths are true in all possible worlds, like 2 + 2 = 4 or “All bachelors are unmarried.” They’re not just true in this world; they’re true in any conceivable world.

## Entailments of knowledge and belief

If we understand knowledge in the propositional sense, rather than know-how or know-of sense, we know this knowledge is a subset of belief.

This means if we have knowledge-that proposition being true, we necessarily believe it is true. If we believe it is true, we don’t believe it is false. If we don’t believe it is false we don’t have knowledge it is false.

Now, we might have knowledge-of a proposition being true, that’s knowledge-that a proposition is considered true but not knowledge-that the proposition is true. We might think the proposition is possibly true. We may even believe it is true. But we don’t have knowledge-that it is true.

Therefore we can have knowledge-of something being (considered) true without the necessary entailment of believing that it is true.

We can equally believe something is true without the necessary entailment of knowledge-that it is true. Propositional knowledge is a subset of belief, not the other way round.

If we accept K = knowledge, B = belief, p = proposition, ⊨ = entails ¬ = not

And Kp is knowledge-that something is true, Bp is believing something is true, ¬Kp is not having knowledge something is true, K¬p is having knowledge something is false, and so on then:

Kp ⊨ Bp ⊨ ¬B¬p ⊨ ¬K¬p

K¬p ⊨ B¬p ⊨ ¬Bp ⊨ ¬Kp

Bp ⊨ ¬B¬p ⊨ ¬K¬p

B¬p ⊨ ¬Bp ⊨ ¬Kp

¬Bp ⊨ ¬Kp

## Can you work out what those statements mean? If you’d like to check your work, or just want the answer, click the down arrow to expand the selection.

**Kp ⊨ Bp ⊨ ¬B¬p ⊨ ¬K¬p**- 𝐾𝑝: Knowledge that 𝑝 is true.
- 𝐵𝑝: Belief that 𝑝 is true.
- ¬𝐵¬𝑝: Not believing that 𝑝 is false.
- ¬𝐾¬𝑝: Not knowing that 𝑝 is false.
**Translation:**- If you know that 𝑝 is true, then you believe that 𝑝 is true.
- If you believe that 𝑝 is true, then you do not believe that 𝑝 is false.
- If you do not believe that 𝑝 is false, then you do not know that 𝑝 is false.

**K¬p ⊨ B¬p ⊨ ¬Bp ⊨ ¬Kp**- 𝐾¬𝑝: Knowledge that 𝑝 is false.
- 𝐵¬𝑝: Belief that 𝑝 is false.
- ¬𝐵𝑝: Not believing that 𝑝 is true.
- ¬𝐾𝑝: Not knowing that 𝑝 is true.
**Translation**:- If you know that 𝑝 is false, then you believe that 𝑝 is false.
- If you believe that 𝑝 is false, then you do not believe that 𝑝 is true.
- If you do not believe that 𝑝 is true, then you do not know that 𝑝 is true.

**Bp ⊨ ¬B¬p ⊨ ¬K¬p**- 𝐵𝑝: Belief that 𝑝 is true.
- ¬𝐵¬𝑝: Not believing that 𝑝 is false.
- ¬𝐾¬𝑝: Not knowing that 𝑝 is false.
**Translation:**- If you believe that 𝑝 is true, then you do not believe that 𝑝 is false.
- If you do not believe that 𝑝 is false, then you do not know that 𝑝 is false.

**B¬p ⊨ ¬Bp ⊨ ¬Kp**- 𝐵¬𝑝: Belief that 𝑝 is false.
- ¬𝐵𝑝: Not believing that 𝑝 is true.
- ¬𝐾𝑝: Not knowing that 𝑝 is true.
**Translation:**- If you believe that 𝑝 is false, then you do not believe that 𝑝 is true.
- If you do not believe that 𝑝 is true, then you do not know that 𝑝 is true.

**¬Bp ⊨ ¬Kp**- ¬𝐵𝑝: Not believing that 𝑝 is true.
- ¬𝐾𝑝: Not knowing that 𝑝 is true.
**Translation:**- If you do not believe that 𝑝 is true, then you do not know that 𝑝 is true.

**In summary:**- Knowing something is true entails believing it, not believing it is false, and not knowing it is false.
- Knowing something is false entails believing it is false, not believing it is true, and not knowing it is true.
- Believing something is true entails not believing it is false and not knowing it is false.
- Believing something is false entails not believing it is true and not knowing it is true.
- Not believing something is true entails not knowing it is true.

## Conclusion:

Logical entailments, necessary truths, and possible truths are like the three musketeers of reason, each playing a crucial role in our quest for understanding. By grasping these concepts, we unlock the doors to deeper insights into the fabric of reality itself. So, dear reader, let your curiosity be your guide as you navigate the labyrinth of logical reasoning, for therein lies the beauty of intellectual exploration.

## References and Further Reading

- Varieties of Modality (Stanford Encyclopedia of Philosophy)
- Contingency (philosophy) – Wikipedia
- Necessary/contingent truths – Oxford Reference
- logic – What is the difference between ⊢ and ⊨? – Mathematics Stack Exchange
- List of logic symbols – Wikipedia
- Beliefs, Language, and Logic
- Impossibly Possible – Logical and Metaphysical Possibility Explained
- Validity (logic) – Wikipedia
- Validity | Reasoning, Argument, Evidence | Britannica
- Validity (statistics) – Wikipedia
- Definition and Examples of Valid Arguments (thoughtco.com)
- Logical consequence – Wikipedia
- Logical Entailment (stanford.edu)
- Entailment (linguistics) – Wikipedia

I’m Joe. I write under the name Davidian, not only because it is a Machine Head song I enjoy but because it was a game character I used to role-play that was always looking to better himself.

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