Conceptual Modalities

The conceptual modalities include necessary truth, contingent truth, logical entailment, logical equivalence, and logical incompatibility

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
  • Three Kinds:
    • Physical or nomological modality:
      • Nothing can travel faster than light means that it’s physically impossible for anything to travel faster than light, i.e. contrary to the laws of nature.
    • Conceptual, logical or 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. that the idea is internally inconsistent.
    • Epistemic modality:
      • It’s not possible McDougall won the election means that it is certain McDougall did not win the election, based on the evidence.
      • View Epistemic Probability

My concern is with the conceptual modalities: necessary truth, contingent truth, contingent falsehood, necessary falsehood, logical entailment, logical equivalence, logical incompatibility, and logical contradictories.

Uses of Conceptual 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 an action only if they could have avoided doing it.
    • This version of PAP, as it’s referred to, is subject to counter-examples.  But there are more sophisticated versions.
  • Conceptual Analysis
    • Conceptual Analysis is the analysis of a concept into its conceptual components. Perhaps the most famous example is the analysis of knowledge.
  • Family Connections

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.”
  • 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)
  • 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
  • Contingent truth can be defined in terms of necessary truth:
    • It is contingently true that P ≣ P is true and it’s not necessarily true that P
  • 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:

Logical Entailment

  • A proposition P logically entails a proposition Q if it’s logically impossible that P be 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, …, Pn therefore Q is valid just in case P1, P2, …, Pn 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.
  • Logical equivalence is a necessary condition for successful conceptual analysis.  Plato, for example, set forward the analysis of knowledge as justified, true belief.  The analysis is correct only if:
    • A person S knows that P is logically equivalent to S believes P, P is true, and S is justified in believing P.
  • Sadly, in the 1960s, counterexamples were developed showing that the right side of the equivalence does not logically entail the left side.  Thus, knowledge is not simply justified, true belief. 
  • The opposite of equivalence is contradictory pairs.
    • Two propositions are contradictories if they necessarily have opposite truth-values.
      • E.g. All swans are white and some swans are not white are contradictories

Compatibility, Incompatibility, Contraries, and Subcontraries

  • Two propositions are logically incompatible (or contraries) just in case it’s logically impossible that they both be 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 they both be 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
    • Two propositions are contradictories if and only if they are both contraries and subcontraries.
    • Equivalence and contradictories are opposites.

Possible Worlds

  • Possible worlds afford a way of reasoning about modalities.
    • A possible world is a complete and logically possible way the universe could have been (or is).  One of these worlds is the actual world.
  • Thus we can say for example that:
    • 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 where 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 Concept
  • It is necessarily true that P
    • AKA: it is logically necessary that P
    • AKA: it is metaphysically necessary that P
    • AKA: it is conceptually necessary that P
    • Eg, everything is what it is and not something else
    • PW: P is true in all possible worlds
Definitions
  • nec(P)  ≣ it is necessarily true that P 
  • It is logically possible that P ≣ ~nec~(P)
    • Eg Some things travel faster than light.
    • PW: P is true in some possible world
  • pos(P) ≣ It is logically possible that P
  • It is necessarily false that P ≣ ~pos(P)
    • AKA, it is logically impossible that P
    • Eg Some people are taller than themselves
    • PW: P is false in all possible worlds
  • It is contingently true that P ≣ P & pos~(P)
    • Eg Harper Lee wrote To Kill a Mockingbird
    • PW: P is true in the actual world but false in some possible world.
  • It is contingently false that P ≣ ~P & pos(P)
    • Eg The sun will not rise tomorrow
    • PW: P is false in the actual world but true in some possible world.
  • It is contingent that P ≣ pos(P) & pos~(P)
    • Ie P is either contingently true or contingently false.
    • Eg The amount of energy in the universe remains the same.
    • PW: P is true in some possible world and false in some other possible world.
  • P logically entails Q ≣ ~pos(P&~Q)
    • AKA Q is a necessary consequence of P
    • AKA Q is a logical consequence of P
    • Eg 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.
    • PW: Q is true in every possible world where P is true
  • P and Q are logically equivalent ≣ ~pos(P&~Q) & ~pos(Q&~P)
    • AKA P and Q are necessarily equivalent 
    • Ie P and Q necessarily have the same truth-value, either both true or both false
    • Ie P and Q logically entail each other
    • Eg Not all swans are white is logically equivalent to some swans are not white.
    • PW: P and Q are true in the very same possible worlds
  • P and Q are contradictories ≣ ~pos(P&Q) & ~pos(~P&~Q)
    • Ie P and Q necessarily have different truth values, one true and the other false
    • Ie Each entails the other is false.
    • Eg All swans are white and some swans are not white are contradictories.
    • PW: There is no possible world where P and Q have the same truth value
  • P and Q are logically incompatible ≣ ~pos(P&Q)
    • AKA:  P and Q are contraries
    • Eg McCormick is tall is incompatible with McCormick is short
    • PW: There is no possible world where P and Q are both true
  • P and Q are subcontraries ≣ ~pos(~P&~Q)
    • Eg The propositions that no tachyons have mass and tachyons exist are subcontraries.
    • PW: There is no possible world where P and Q are both false

Necessary Truth and Knowability a Priori

  • A proposition is knowable a priori if it is knowable independent of (“prior to”) experience
  • Such propositions are either directly evident (all squares are rectangles) or derivable from self-evident truths (the sum of two odds integers is even).
  • A proposition is knowable only a posteriori if it is knowable only from (“after”) experience.  For example,
    • I have a headache
    • Black swans exists
    • Nothing can travel faster than light.
  • Knowable a priori and knowable only a posteriori are conceptually distinct from necessarily true and contingently true.  The first pair are epistemic concepts, relating to knowledge.  The second pair are metaphysical, concerning propositions.
  • It was long thought that the pairs of concepts coincide:
    • 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.
  • But there is good reason to believe that some necessary truths are knowable only a posteriori. It has to do with names and naming.
  • 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 presented a persuasive 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. So “Phosphorus” and “Hesperus” rigidly designate the same object
    4. Therefore they designate the same object in all possible worlds where they exist.
    5. Hence, it’s necessarily true that Hesperus = Phosphorus.
  • Argument #1 why Kripke’s theory is better:
    • Suppose Venus changes orbit so it no longer appears near the sun in the morning. According to the CT theory, Hesperus has ceased to exist. On the NT theory Hesperus has simply changed orbit.
  • Argument #2 why Kripke’s theory is better:
    • If Hesperus = Phosphorus is contingently true, there’s a possible world where 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.
  • Kripke’s argument that Hesperus = Phosphorus is necessarily true can be adapted to establish that the following are also necessarily true (assuming the terms involved are rigid designators):
    • Water is H2O
    • Heat is mean molecular kinetic energy
    • Gold has atomic number 79.
  • It also seems that some contingent truths are knowable a priori.
    • An example by Gareth Evans’s (1982) in plato.stanford.edu/entries/modality-varieties/:
      • Suppose I stipulate that ‘Julius’ is a rigid designator for whoever invented zip codes, if such a person exists. So Julius is a guy who happened to have invented zip codes Thus, Julius = the inventor of zip codes is contingently true. Yet I know Julius = the inventor of zip codes a priori since it’s true by definition.