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]]>One thing has to be that it is necessary that I was born of my parents. For nobody else, this is necessarily true, except my siblings.

The essence of humans is the laws of nature. The only true objects in the universe that we know of are particles. What’s below the sub-atomic realm, we don’t know. We can’t know at this stage.

I certainly reject any notion of identity beyond the biological categories. But even species is kind of arbitrary. The most common definition of *species* is “able to beget fertile offspring”, however, this is not true for all species. It’s more of a general rule, in that sense.

The only truths about knowledge that we know of are the truths of mathematics and logic, which are stipulated to be true once proven to be true. The logical law of identity A=A is the only type of identity, therefore, that I accept to be true. But again this is circular because that hinges on what we believe constitutes something being an entity possessing an identity. Again, only the most basic objects in the universe (physical particles) are objects.

Tables, chairs, bottles, concepts, etc. etc. are convenient ways of thinking about the world, and are also useful for pragmatists, but ultimately these non-physical and non-scientific ontological categories aren’t defensible.

You could point out, however, that even logical and mathematical truths are just a feature of the universe, something that culminates in our understanding of these ideas through the firing of neurons, which are also based in physics, even though mathematics has some very unique properties and are sublimely beautiful in many ways and seemingly mysterious, there is no reason to assume that there is a platonic realm of ideas of concepts where abstract entities and ideas are found. I’m, if you haven’t already guessed, strictly a physicalist.

Yes, this is reductionistic (and typical of non-continental philosophical discourse), but that’s just the way things are from what we understand about science. I don’t agree with constructing our ontology based on constructivist ideas outside of textual analysis and the analysis of language. When we are talking about the physical universe, this is not helpful or useful. When we are talking about why people believe what they do, what ideologies they adhere to, etc. then constructivist ontologies can be very useful in challenging and understanding the motivations behind why people act in the ways that they do.

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]]>A. J. Ayer (1936), *Language, Truth, and Logic*, London: Gollancz, 2^{nd} Edition, 1946.

W. V. Quine (1951), “Two Dogmas of Empiricism”, *Philosophical Review*, 60 (1951): 20–43; reprinted in *From a Logical Point of View*, pp. 20–46.

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]]>If you’ve studied some formal logic, you may have seen how a conclusion can be true when the premise of that argument is false.

How could this be? you might be wondering . . . This article will spell out the reasoning behind that and what’s known as “logical implication”.

Logical implication can take the form of:

Sometimes different symbols are used to represent logical implication:

are all logically equivalent.

*p*is generally thought of also as the**antecedent**or premise of the argument.*q*is generally thought of also as the**consequent**or conclusion of the argument.

The statement “p implies q” means that if p is true, then q must also be true. The statement “p implies q” is also written “if p then q” or sometimes “q if p.” Statement p is called the

premiseof the implication and q is called theconclusion.

(Source: http://www.math.niu.edu)

When thinking about logical implication, it is useful to be first reacquainted with **necessity **and** sufficiency**.

To say that p is a *necessary condition* for q is to say that **it is impossible to have q without p**.

(NB: it is not necessarily the case that if you have q, then you must also have p—this is known as “affirming the consequent“)

Said another way: It is true if you do not have p you are guaranteed not to have q otherwise this logical implication does not hold true. (i.e. when *not-p* implies *q *this is false)

Examples:

1. “To get a high distinction on the exam, you must study hard”.

(NB: You could study hard but not also get a high distinction on the exam.)

2. “To become the president of the United States, you must be born in the United States”.

(NB: You could be born in the United States but not become president.)

3. “A prime number must be odd if it is a whole number greater than 2”.

(NB: A whole number could be greater than two and also be odd but not necessarily be a prime number.)

To say that p is a *sufficient condition* for q is to say that **the presence of p is enough for q to be present but it is not necessary**.

Said another way: It is possible to have p when q is also present, but the presence of p does not guarantee the presence of q.

Examples:

1. It is a sufficient that you can graduate by attending your graduation ceremony once you have completed the requirements for your degree.

(NB: You do not have to attend your graduation ceremony to have your degree conferred.)

2. It is a sufficient condition that if the president of the United States wins the Nobel Peace Prize he or she will be remembered by many people.

(NB: The president of the United States can still be remembered by many people even if he or she does not win the Nobel Peace Prize.)

3. It is a sufficient condition that every even integer greater than 2 can be expressed as the sum of two prime numbers (Goldbach’s conjecture).

(NB: Every even integer greater than 2 can also be expressed as the sum of non-primes, i.e. two even numbers added together will always equal an even number.)

For one statement to be a **necessary and sufficient condition of another then one statement must be true when the other is also true**.

The truth of one statement guarantees the truth of the another statement.

It is true that:

are all logically equivalent and p is true **if and only if** q is true.

Examples:

1. “If you get a high distinction on the exam, you will score 85% or above, **and** will have performed well on the exam”.

(NB: There is no way that you can score 85% or above, not get a high distinction on the test, and have performed well.)

2. “Barack Obama was the president of the United States, **and** he was inaugurated as president”.

(NB: There is no way that Barack Obama was not the president of the United States and inaugurated as president)

3. “Prime numbers are greater than 2, only divisible by themselves and 1, **and** there are infinite prime numbers”.

(NB: There is no way that prime numbers are greater than 2, are not divisible by themselves and 1, and there aren’t infinitely many prime numbers)

The truth table for logical implication is as follows:

It is also the case that:

To illustrate with an example:

There is never an instance where somebody is the president of the United States but was also not born in the United States. This is the ONLY time that this proposition is false (the second row).

Therefore, the statement “If somebody is president of the United States then they will be born in the United States” is only violated or false when somebody becomes the president when they weren’t born in the United States.

Below are some objections to the use of logical implication, if you are still not satisfied with what has been covered above:

Further reading: Indicative Conditionals on the *SEP*.

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