Modal logic, necessity, and science fiction

A necessary truth is one that is true in all possible universes. We can capture the concept of necessary truth with the 4 rules of S4 modal logic (where □ is read “necessarily”):

  • if P is any tautology, then  □ P
  • if  □ P  and  □ (PQ)  then  □ Q
  • if  □ P  then  □ □ P
  • if  □ P  then  P

For those who prefer words rather than symbols:

  • if P is any tautology, then P is necessarily true
  • if P and (P implies Q) are both necessarily true, then Q is necessarily true
  • if P is necessarily true, then it is necessarily true that P is necessarily true
  • if P is necessarily true, then P is true (in our universe, among others)

The first rule implies that the truths of mathematics and logic (□ 2 + 2 = 4, etc.) are necessary truths (they must obviously be so, since one cannot consistently imagine an alternate universe where they are false). The second rule implies that the necessary truths include all logical consequences of necessary truths. The last two rules imply that  □ P  is equivalent to  □ □ P,  □ □ □ P,  etc. In other words, there is only one level of “necessary” that needs to be considered.

As it stands, these rules only allow us to infer the truths of mathematics and logic (such as  □ 2 + 2 = 4). One must add other necessary axioms to get more necessary truths than that. A Christian or Muslim might, for example, add “Necessarily, God exists,” and spend time exploring the logical consequences of that.

Countless things that are true in our universe are not necessarily true, such as “Water freezes at 0°C” or “Trees are green” or “Bill Clinton was President of the United States in the year 2000.”

For historical truths like the latter, it’s obvious that they are contingent on events, rather than being necessary. There is a substantial body of “alternate history” fiction which explores alternatives for such contingent truths, such as these four novels (pictured above):

  • Fatherland (Robert Harris, 1992): a detective story set in a universe where Hitler won the war; it is the week leading up to his 75th birthday (3.99 on Goodreads)
  • The Peshawar Lancers (S.M. Stirling, 2002): European civilisation is destroyed by the impact of comet fragments in 1878; a new Kiplingesque Anglo-Indian steampunk civilisation arises (3.86 on Goodreads)
  • SS-GB (Len Deighton, 1978): Hitler defeats Britain in 1941; British police face moral dilemmas cooperating with the SS (3.74 on Goodreads)
  • Romanitas (Sophia McDougall, 2005): the Roman Empire is alive and well in present-day London; slaves are still crucified (3.24 on Goodreads; first of a trilogy)

Three plant pigments: green beech, brown kelp, and red gracilaria algae (cropped from photographs by Simon Burchell, Stef Maruch, and Eric Moody)

The truths of biology are just as contingent as the truths of history. Trees are (mostly) green, but even on our own planet, brown and red are viable alternative colours for plants. From an evolutionary perspective, Stephen Jay Gould expresses the contingency this way:

any replay of the tape [of life] would lead evolution down a pathway radically different from the road actually taken.” (Stephen Jay Gould, Wonderful Life: The Burgess Shale and the Nature of History, 1989)

(some of his colleagues would take issue with the word “radically,” but still accept the word “different”). From a Christian point of view, the contingency of biology follows from the doctrine of the “Free Creation” of God, independently of any beliefs about evolution. To quote Protestant theologian Louis Berkhof:

God determines voluntarily what and whom He will create, and the times, places, and circumstances, of their lives.” (Louis Berkhof, Systematic Theology, Part I, VII, D.1.c)

The Catholic Church shares the same view, as none other than Thomas Aquinas makes clear (using the terminology of necessary truth):

It seems that whatever God wills He wills necessarily. For everything eternal is necessary. But whatever God wills, He wills from eternity, for otherwise His will would be mutable. Therefore whatever He wills, He wills necessarily. … On the contrary, The Apostle says (Ephesians 1:11): ‘Who works all things according to the counsel of His will.’ Now, what we work according to the counsel of the will, we do not will necessarily. Therefore God does not will necessarily whatever He wills.” (Summa Theologiae, Part I, 19.3)

Having taken this line, one might ask why mathematical truths are necessary rather than contingent. The astronomer Johannes Kepler resolves this problem this by telling us that they are not created:

Geometry existed before the Creation, is co-eternal with the mind of God.” (Johannes Kepler, Harmonices Mundi)

In fiction, alternative biologies are normally explored in the context of some other planet, because alternate earths are pretty much logically equivalent to other planets. Here are four examples of fictional biology:

  • Out of the Silent Planet (C.S. Lewis, 1938): written from a Christian perspective, this novel has three intelligent humanoid alien species living on the planet Mars (3.92 on Goodreads; see also my book review)
  • The Mote in God’s Eye (Larry Niven and Jerry Pournelle, 1974): this novel is one of the best alien-contact novels ever written (4.07 on Goodreads)
  • the xenomorph from the film Aliens (1986)
  • the Klingon character Worf from the TV series Star Trek: The Next Generation (1987–1994)

The truths of physics are contingent as well; our universe could have been set up to run on different rules. Science fiction authors often tweak the laws of physics slightly in order to make the plot work (most frequently, to allow interstellar travel). Fantasy authors invent alternate universes which differ from ours far more dramatically:

  • Dune (Frank Herbert, 1965): faster-than-light travel is a feature of the plot; it follows that interstellar navigation requires looking into the future (4.25 on Goodreads; see also my book review)
  • Great North Road (Peter F. Hamilton, 2012): “Stargate” style portals are a key feature of this novel (4.07 on Goodreads)
  • The Many-Coloured Land (Julian May, 1981): a science fiction incorporating psychic powers (4.07 on Goodreads; first of a series)
  • Magician (Raymond E. Feist, 1982): a classic fantasy novel which explores some of the internal logic of magic along the way (4.31 on Goodreads; first of a series)

Because mathematical truths are necessary truths, they are potentially common ground with intelligent aliens. This is one theme in the book (later film) Contact:

‘No, look at it this way,’ she said smiling. ‘This is a beacon. It’s an announcement signal. It’s designed to attract our attention. We get strange patterns of pulses from quasars and pulsars and radio galaxies and God-knows-what. But prime numbers are very specific, very artificial. No even number is prime, for example. It’s hard to imagine some radiating plasma or exploding galaxy sending out a regular set of mathematical signals like this. The prime numbers are to attract our attention.’” (Carl Sagan, Contact, 1985; 4.14 on Goodreads)

Of course, Carl Sagan or his editor should have realised that 2 is prime. Even intelligent beings can make mistakes.


In this post series: logic of necessary truth, logic of belief, logic of knowledge, logic of obligation


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3 thoughts on “Modal logic, necessity, and science fiction

  1. Pingback: Modal logic, belief, and the “flat earth” | Scientific Gems

  2. Pingback: Modal logic, knowledge, and an old joke | Scientific Gems

  3. Pingback: Modal logic, ethics, and obligation | Scientific Gems

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