# How should one symbolize “and then” in logic?

Graham Priest observed that not all uses of "and" in English commute: (page 15)

...according to the truth table for &, 'a and b' always has the same value as 'b and a', namely, they are both true if a and b are both true, and false otherwise. But consider the sentences:

1. John hit his head and fell down.
2. John fell down and hit his head.

The first says that John hit his head and then fell down. The second says that John fell down and then hit his head. Clearly, the first could be true whilst the second is false, and vice versa. Thus, it is not just the truth values of the conjuncts that are important, but which conjunct caused which.

I am not sure how one would symbolize such sentences where "and" means "and then" nor which logic to use to best represent them, hence the question: How should one symbolize "and then" in logic?

Priest, G. (2017). Logic: a very short introduction (Vol. 29). Oxford University Press.

• It is well-known that the conditional symbolizing the truth-functional "if..., then..." does not convey a "causal" relationship. You have to read Priest's book : maybe he will explain so-called "non-classical" implications. – Mauro ALLEGRANZA Sep 27 '19 at 13:19
• @FrankHubeny Clearly, in classical mathematical propositional logic, you can't, which shows that it isn't a good model at all of human deductive logic since we all understand these simple sentences and the causal link they imply. – Speakpigeon Sep 27 '19 at 15:28
• In classical logic, those commute. Time is not a factor in traditional mathematical deductive systems unless it is injected as part of the subject matter, rather than the model. Classical logic is one of those. If you want time in your logic, you need to inject time into the content by recording the implications made about time as facts. The most basic approach is to type propositions as events, with a time. So A = John hit his head at time t_A, B = John fell down at time t_B. A & B & t_A < t_B or vice versus. It does not change the logic at all, only the information collected. – user9166 Sep 27 '19 at 17:33
• You'll need a second order predicate After(a,b), where a and b are statements (a="John hit his head", b="John fell down"). Then 1. is a&b&After(a,b) and 2. is a&b&After(b,a). Alternatively, you can have time stamped statements, and use < on the stamps to express succession. – Conifold Sep 27 '19 at 23:35

I don't think you can do what you want to do within classical propositional or quantificational logic. At least not elegantly, there's some solutions you can do, such as letting a sentence like "John fell down after hitting his head" be an atomic formula, represented by a variable like "P". It seems that no one uses propositional variables to represent complex sentences, particularly those using connectives like and/or. But I don't see why it should be a problem if you're just using propositional logic, since that system can't "see inside" the variables like "P" and "Q", and you define these as true or false according to your own needs.

Temporal logic might be what you're looking for. I don't have a detailed knowledge about the area, but I can tell you it's one of the standard "adaptions" of alethic modal logic to various philosophically interesting concepts. Alethic modal logic has the operators of "possibly" and "necessarily". A good example of this is epistemic logic, where "necessarily" is interpreted as "knows".

I don't have much else to say on the matter, so here's some links:

https://en.wikipedia.org/wiki/Temporal_logic

https://plato.stanford.edu/entries/logic-temporal/

• Thanks! I will look into temporal logic. – Frank Hubeny Sep 27 '19 at 14:18
• A good example of this is epistemic logic, where "possibly" is interpreted as "believes" and "necessarily" is interpreted as "knows"". That's not quite accurate. Elementary epistemic logic usually does not have a notion of belief. Doxastic epistemic logics that extend the language with a belief operator usually define this belief operator in other ways, not as simple analogue to a possibility operator. – transitionsynthesis Sep 27 '19 at 18:13
• @transitionsynthesis Thanks for that. I really don't know much about these kinds of sstems, so I just wanted to present a bit of context before being a sign post to relevant sources of information. I was aware of epistemic logic only due to the "KK principle" debate (does knowing require knowing that you know), where axiom S4 of "necessarily P then Necessarily necessarily P" is used to model the principle. I assumed the possibility rules would be adapted to belief in the same way as necessity rules are adapted to knowledge – Daniel Prendergast Sep 27 '19 at 18:26
• @DanielPrendergast Yes, I wasn't trying to be pedantic, I thought it was a good answer on the whole. – transitionsynthesis Sep 27 '19 at 19:54
• In propositional logic the atomic sentences may be represented abstractly by symbols like p and q, but they are usually understood to stand for statements that assign properties to particular entities (like 'Socrates is mortal'), as on p. 1-2 of this pdf about a logic teaching program called 'Tarski's world' where an example of an atomic sentence is Cube(a), meaning the object labeled "a" has the property of being a cube. So there's no reason you can't have temporal properties referred to in atomic sentences of propositional logic. – Hypnosifl Sep 28 '19 at 15:45

Sequences of events need a separate variable, or some other formal system like Minkowski spacetime if you want to see tense built into the system. In normal propositional logic, tense is generally ignored, and not built into the logic operators, but one could assume the chronologies are unstated propositions and produce an additional proposition.

P1. John fell down. P2. John hit his head. P3. John fell before hitting his head. P4. John hit his head before falling.

Now examine the relationships and arguments according. P1 is independent of P2 logically, but P3 and P4 are logical contradictions. Thus:

P1, P2, P3 -><- P4

Another way to do it would be to use multivariable predicate logic.

HitHead(John,time_1) = T, FellDown(John,time_2) = T, OccursBefore(time_1,time_2) = T

Of course, there are fancier logics that use modal operators and temporal logic that make things interesting.

Time composed of hours, minutes, and seconds is on the whole in everyday life generally treated as a linear variable in a modular arithmetic, so axioms regarding that arithmetic could simply be wed to other forms of logic!

• Thanks! I will at these links. – Frank Hubeny Sep 27 '19 at 14:43