Alethic (Conceptual) Modalities

Necessary truth, Contingent truth, Logical entailment, Logical equivalence, Logical incompatibility

Table of Contents

Modalities

Modalities are the concepts of possibility, impossibility, contingency, necessity, and more.
- merriam-webster.com/dictionary/modality
  - the classification of logical propositions according to their asserting or denying the possibility, impossibility, contingency, or necessity of their content

Different Kinds:
- Alethic (conceptual, logical, metaphysical) modality:
  - A person can’t be taller than themself means that it’s conceptually impossible that a person be taller than themself, i.e. the idea is contradictory.
- Physical (nomic, nomological) modality:
  - Nothing can travel faster than light means that it’s physically impossible for anything to travel faster than light, i.e. it’s contrary to the laws of nature.
- Epistemic modality:
  - It’s not possible McDougall is in Alaska, based on the evidence, means that it is certain McDougall is not in Alaska, based on the evidence, e.g. because he’s here with me in Texas.
  - View Epistemic Status

The concern of this page is with the alethic modalities: necessary truth, contingent truth, contingent falsehood, necessary falsehood, logical entailment, logical equivalence, logical incompatibility, and logical contradictories.

Uses of Alethic Modalities in Philosophy

Principles
- Philosophic principles are typically set forth as necessarily truths, for example the Principle of Alternative Possibilities:
  - A person is morally responsible for something they’ve done only if they could have avoided doing it.
- The principle is set forth as a necessary truth, thus subject to refutation by a logically possible counterexample.
Conceptual Analysis
- Conceptual analysis is the analysis of a concept into its conceptual components, for example, the analysis of knowledge as justified true belief:
  - A person knows that P if and only if
    - the person believes that P
    - it’s true that P
    - the person is rationally justified in believing that P.
  - The analysis is put forth as a necessary truth, thus subject to a logically possible counterexample, either
    - a logically possible scenario where a person knows that P but not all three conditions of the analysis are met
      - or
    - a logically possible scenario where a person does not know that P yet all three conditions are met.
Family Connections
- Light can be thrown on a family of words and phrases by setting forth their logical interconnections, for example:
  - “it is certain that P” logically entails “it is beyond a reasonable doubt that P,” but not the reverse;
  - “it is beyond a reasonable doubt that P” logically entails “there is a preponderance of evidence that P,” but not the reverse.

The Alethic Modalities

Necessary and Contingent Truths

Gottfried Wilhelm Leibniz introduced calculus into mathematics, kinetic energy into physics, and the distinction between necessary and contingent truths into philosophy, what he called truths of reason and truths of fact.
- From the Monadology (1714) Part II 33
  - “There are two kinds of truths, those of reasoning and those of fact. Truths of reasoning are necessary and their opposite is impossible: truths of fact are contingent and their opposite is possible. When a truth is necessary, its reason can be found by analysis, resolving it into more simple ideas and truths, until we come to those which are primary.”
- Truths of reason, Leibniz said, are grounded on the principle of “contradiction, in virtue of which we judge false that which involves a contradiction, and true that which is opposed or contradictory to the false.”
Later, in An Enquiry Concerning Human Understanding (1748), David Hume made the same distinction, calling the concepts relations of ideas and matters of fact.
- Relations of Ideas
  - Relations of ideas are propositions that are “either intuitively or demonstratively certain. That the square of the hypotenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would forever retain their certainty and evidence. “ (EHU 20)
- Matters of Fact
  - “Matters of fact …are not ascertained in the same manner [as relations of ideas]; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. That the sun will not rise tomorrow is no less intelligible a proposition, and implies no more contradiction than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply a contradiction, and could never be distinctly conceived by the mind.” (EHU 21)
A proposition is necessarily true if its negation implies a contradiction or, following Hume, that it’s true solely by virtue of the relations among the ideas of which it’s composed.
Necessary truths include propositions such as:
- Everything is what it is and not something else.
- All squares are rectangles.
- If a person is taller than a second and the second taller than a third, the first is taller than the third.
- 3 x 5 = 30 / 2
- The sum of two odd integers is (always) even.
- The square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides.
A proposition is contingently true if it’s true and not necessarily true.
Examples:
- The sun will rise tomorrow;
- Some molecules have three atoms;
- Harper Lee wrote To Kill a Mockingbird;
- The total energy of a closed physical system remains constant.

Logical Possibility and Impossibility

A proposition is logically impossible if it’s necessarily false; and logically possible otherwise,
Examples of logical impossibilities:
- Some people are taller than themselves
- 2 + 2 = 5
- An angle can be trisected with a straightedge and a compass.
  - mathworld.wolfram.com/GeometricProblemsofAntiquity.html

Logical Entailment

A set S of propositions logically entails a proposition Q if it’s logically impossible that the propositions of S are all true and Q false.
- Thus, the proposition that only natural-born citizens are eligible to be president but McDermot is not a natural-born citizen logically entails that McDermot is not eligible to be president.
Deductive validity is logical entailment:
- The argument P₁, P₂, …, P_n therefore Q is valid just in case P₁, P₂, …, P_n logically entail Q.
Entailment is useful in determining relative logical strength, for example:
- It is certain that Q logically entails it is beyond a reasonable doubt that Q, but not vice versa.
- It is beyond a reasonable doubt that Q logically entails it is very likely that Q, but not vice versa.
- Thus, in order of strength:
  - It is certain that Q
  - it is beyond a reasonable doubt that Q
  - it is very likely that Q

Logical Equivalence and Contradictories

Propositions P and Q are logically equivalent just in case each entails the other, which is to say that P and Q necessarily have the same truth-value, either both true or both false.
For example,
- Not all swans are white is logically equivalent to some swans are not white.
- It is probable that P is logically equivalent to it’s likely that P
- Stealing is morally wrong is logically equivalent to it’s morally obligatory not to steal.
The opposite of equivalence is contradictory pairs of propositions.
- Two propositions are contradictories if each logically entails the negation of the other, which is to say that they necessarily have opposite truth-values.
  - For example, all swans are white and some swans are not white

Compatibility, Incompatibility, Contraries, and Subcontraries

Two propositions are logically incompatible (or contraries) just in case it’s logically impossible that both are true; and logically compatible otherwise.
- Thus, McCormick is tall is logically incompatible with McCormick is short and compatible with McCormick is under seven feet tall.
Two propositions are subcontraries if it’s logically impossible that both are false.
- E.g. no tachyons have mass and tachyons exist.
The traditional problem of free will and determinism presupposes that free will and determinism are logically incompatible. One proposed solution has been to claim that there is no problem by arguing that free will and determinism are compatible.

Truth Table Definitions of Relational Modalities

You can read off the table that:
- Equivalence is mutual entailment
  - Superimposing the Impossible rows of “P entails Q” onto the rows of “Q entails P” yields the rows of “P and Q are equivalent.”
- Two propositions are contradictories if and only if they are both contraries and subcontraries.
- Equivalence and contradictories are opposites.
  - The rows of “P and Q are equivalent” and “P and Q are contradictories” are opposite (one blank, the other Impossible).

Possible Worlds

Possible worlds afford a way of reasoning about modalities.
- A possible world, intuitively, is a complete and logically possible way everything could have been.
- Specifically, a possible world is a set of propositions such that
  - the set is complete in the sense that, for any proposition, either it or its negation belongs to the set;
  - it’s logically possible that all propositions of the set are true.
- A proposition is true in a possible world if it belongs to that world’s set of propositions.
- The actual world is the possible world whose propositions are all true.
Thus:
- A proposition is necessarily true if and only if it’s true in all possible worlds.
- A proposition is necessarily false if and only if it’s false in all possible worlds.
- A proposition is contingently true if and only if it’s true in the actual world but false in some possible world.
- Proposition P logically entails Q if and only if Q is true in every possible world in which P is true
- Propositions P and Q are logically equivalent if and only if P and Q are true in exactly the same possible worlds

Modalities Defined in Terms of Necessity

Primitive Concepts

It is necessarily true that P
- Alternatively:
  - it is logically necessary that P
  - it is metaphysically necessary that P
  - it is conceptually necessary that P
- In terms of possible words, P is true in all possible worlds
- For example, the sum of two odd integers is (always) even.
The tilde ~ means it is false that.
The ampersand & means and.

Definitions

Nec(…) ≣ it is necessarily true that …
It is logically possible that P ≣ ~Nec(~P)
- That is, it is not necessarily true that P is false
- In terms of possible words, P is true in some possible world
- For example, some things travel faster than light.
Pos(…) ≣ it is logically possible that …
It is necessarily false that P ≣ ~Pos(P)
- Alternatively, it is logically impossible that P
- In terms of possible words, P is false in all possible worlds
- For example, some people are taller than themselves
It is contingently true that P ≣ P & Pos(~P)
- In terms of possible words, P is true in the actual world but false in some possible world.
- For example, Harper Lee wrote To Kill a Mockingbird
It is contingently false that P ≣ ~P & Pos(P)
- In terms of possible words, P is false in the actual world but true in some possible world.
- For example, the sun will not rise tomorrow
It is contingent that P ≣ Pos(P) & Pos(~P)
- Alternatively, P is either contingently true or contingently false.
  - That is, P & Pos(~P) v ~P & Pos(P), where v means or.
- In terms of possible words, P is true in some possible world and false in some other possible world.
- For example, the amount of energy in the universe remains the same.
P logically entails Q ≣ ~Pos(P&~Q)
- That is, it’s logically impossible that P is true and Q false.
- Alternatively:
  - Q is a necessary consequence of P
  - Q is a logical consequence of P
- In terms of possible words, Q is true in every possible world in which P is true
- For example, only natural-born citizens are eligible to be president but McDougall is not a natural-born citizen logically entails McDougall is not eligible to be president.
P and Q are logically equivalent ≣ ~Pos(P&~Q) & ~Pos(Q&~P)
- Alternatively, P and Q necessarily have the same truth-value, either both true or both false
- Alternatively, P and Q logically entail each other
- In terms of possible words, P and Q are true in the very same possible worlds
- For example, Not all swans are white is logically equivalent to some swans are not white.
P and Q are contradictories ≣ ~Pos(P&Q) & ~Pos(~P&~Q)
- Alternatively, P and Q necessarily have different truth values, one true and the other false
- Alternatively, each entails that the other is false.
- In terms of possible words, there is no possible world in which P and Q have the same truth value
- For example, all swans are white and some swans are not white are contradictories.
P and Q are logically incompatible ≣ ~Pos(P&Q)
- Alternatively, P and Q are contraries
- In terms of possible words, there is no possible world in which P and Q are both true
- For example, McCormick is over 6 feet tall is incompatible with McCormick is under 5 feet tall.
P and Q are subcontraries ≣ ~Pos(~P&~Q)
- That is, it’s logically impossible that both P and Q are false
- In terms of possible words, there is no possible world where P and Q are both false
- For example, McCormick is under 6 feet tall and McCormick is over 5 feet tall are subcontraries.

Necessary Truth and Knowability a Priori
AND
Contingent Truth and Knowability Only A Posteriori

Knowability a Priori and Knowability Only A Posteriori

A proposition is knowable a priori if it is knowable independent of (“prior to”) experience, i.e. if it can be known without being based on empirical evidence (which includes the evidence of perception, testimony, memory, and introspection).
The truth of such propositions is either directly evident or can be derived from from directly-evident truths using deductive logic.
- That all squares are rectangles is directly evident and is thus known a priori.
- That the sum of two odd integers is (always) even can be derived from directly evident truths using deductive logic and is thus known a priori.
  - View Quick Proof
A proposition is knowable only a posteriori if it is knowable only from (“after”) experience, i.e. if it can be known based only on empirical evidence (which includes the evidence of perception, testimony, memory, and introspection).. For example,
- I have a headache
- I saw a Shelby Cobra yesterday
- Black swans exists
- Nothing can travel faster than light.
Knowability a priori and knowability only a posteriori are conceptually distinct from necessary truth and contingent truth. The first pair are epistemic concepts, relating to knowledge. The second pair are metaphysical, concerning propositions.
Since Hume, many philosophers assumed that the pairs of concepts coincide, that is:
- A proposition is knowable a priori if and only if it is necessarily true.
- A proposition is knowable only a posteriori if and only if it is contingently true.
It turns out they were wrong, that is:
- Some necessary truths are knowable only a posteriori.
- Some contingent truths are knowable a priori.

Kripke and Rigid Designators

Necessary Truths Knowable Only a Posteriori

Some necessary truths are knowable only a posteriori. The reason has to do with names and how they refer to the objects they name.
The ancient Greeks named the heavenly body that appears near the sun in the morning “Phosphorus.” And they named the heavenly body that appears near the sun in the evening “Hesperus.”
It was discovered later that both bodies were the planet Venus. It was thus learned that
- Hesperus = Phosphorus
The identity of Hesperus and Phosphorus is therefore known a posteriori.
The proposition Hesperus = Phosphorus would seem contingently true. Here’s the argument:
1. Names are defined in terms of definite descriptions. Thus,
  - “Phosphorus” means the heavenly body that appears near the sun in the morning.
  - “Hesperus” means the heavenly body that appears near the sun in the evening.
2. Therefore Hesperus = Phosphorus means:
  - The heavenly body that appears near the sun in the morning = the heavenly body that appears near the sun in the evening.
3. That proposition is contingently true.
4. Therefore Hesperus = Phosphorus is contingently true.
But in Naming and Necessity (1980), Saul Kripke presents a compelling argument that Hesperus = Phosphorus is necessarily true:
1. Names are rigid designators
  - A rigid designator designates the same object in all possible worlds in which that object exists and never designates anything else. (stanford.edu/entries/rigid-designators/)
  - So names attach necessarily to their objects, sticking to them through possible worlds.
2. Therefore,
  - “Phosphorus” rigidly designates Phosphorus.
  - “Hesperus” rigidly designates Hesperus.
3. Since Hesperus = Phosphorus, “Phosphorus” and “Hesperus” rigidly designate the same object
4. Therefore “Phosphorus” and “Hesperus” designate the same object in all possible worlds where they exist.
5. Hence, it’s necessarily true that Hesperus = Phosphorus.
Argument #1 supporting Kripke’s view:
- Suppose Venus changes orbit so it no longer appears near the sun in the morning. According to the definite description theory of names, Hesperus has ceased to exist. Per Kripke’s rigid designator theory, Hesperus has simply changed orbit.
Argument #2 supporting Kripke’s view:
- If Hesperus = Phosphorus is contingently true, there’s a possible world in which Hesperus is not Phosphorus. Perhaps in that world Hesperus = Mercury. Then Hesperus could have been Mercury. But a thing could not have been a different thing. (I could not have been my brother.). “Hesperus” could have named a different object. But Hesperus could not have been a different thing.
Rigid designators include not only names but also nouns that refer to natural kinds such as water, heat, and gold. Thus the following, like “Hesperus = Phosphorus,” are necessarily true but knowable only a posteriori:
- Water is H₂O
- Heat is mean molecular kinetic energy
- Gold has atomic number 79.

Contingent Truths Knowable a Priori

It also seems that some contingent truths are knowable a priori.
Here’s an updated version of one of Kripke’s examples:
- A meter is defined as the length of the path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second. “One meter” thus rigidly designates that length. Consider the statement:
  - The length of the path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second is one meter.
- We know a priori that the statement is true, based on the definition of “meter.” But the statement is only contingently true, since it’s false in a possible world where the speed of light is not 299,792,458 meters per second. For example, in a possible world where the speed of light is half of its actual speed (i.e. 149,896,229 m/s), the length of the path traveled by light in a vacuum during 1/299,792,458 of a second is half a meter.

Addendum

Proof that the sum of two odd integers is even

An integer is even if it’s two times some positive integer.
An integer is odd if it’s two times some positive integer + 1.
Let x and y be two odd integers.
Then, x = 2a + 1, for some integer a.
And y = 2b + 1, for some integer b.
So x + y = 2a + 1 + 2b + 1.
Thus, x + y = 2a + 2b + 2.
And so x + y = 2(a + b + 1).
Thus x + y = two times some integer (a + b + 1) and is therefore even.