# 6.5 First-order axioms for waterworld

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The domain axioms of WaterWorld in first-order logic.

We summarize the details of how we choose to model WaterWorld boards in first-order logic: exactly what relations we make up, and the formaldomain axioms which capture the game's rules.

This will follow almost exactly the same pattern as our WaterWorld model in propositional logic . However, we will take advantage of the additional flexibility providedby first-order logic.

Rather than modeling only the default 64 WaterWorld board;, we will be able to model any board representable by our relations.This will allow boards of any size and configuration, with one major constrainteach location can have at most three neighboring pirates.

## Domain and relations

Our domain is simply the set of all board locations. This set can be arbitrarily largeeven infinite!

The board configuration is given by the binaryneighborrelation $\mathrm{nhbr}$ .

The next relations correspond directly to the propositions in the propositional logic model.

• Whether or not a location contains a pirate: $\mathrm{safe}$ . This is a unary relation.
We choose not to include a redundant relation $\mathrm{unsafe}$ .
• Unary relations indicating the number of neighboring pirates: $\mathrm{has0}$ , $\mathrm{has1}$ , $\mathrm{has2}$ , and $\mathrm{has3}$ .
Thus, we have our restriction to three unsafe neighbors. This will also be reflected in our domain axioms below. See also this problem for a discussion of how to avoid this restriction.

In addition, to have encode the domain axioms for an arbitrary domain, we also need an equality relation over our domain of locations.As is traditional, we will use infix notation for this relation, for example, $x=y$ . Furthermore, we will allow ourselves to write $x\neq y$ as shorthand for $\neg (x=y)$ . Thus, we do not need a distinct inequality relation.

Note that these relations describe the state of the underlying boardthe modeland not our particular view of it. Our particular view will be reflected in which formulaswe'll accept as premises. So we'll accept $\mathrm{has2}(A)$ as a premise only when $A$ has been exposed and shows a 2.

## The domain axioms

Many of our axioms correspond directly, albeit much more succinctly, with those of the propositional model. In addition, we have axioms that specify that our neighbor and equalityrelations are self-consistent.

Axioms asserting that the neighbor relation is anti-reflexive and symmetric:

• $\forall x\colon \neg \mathrm{nhbr}(x, x)$
• $\forall x\colon \forall y\colon \mathrm{nhbr}(x, y)\implies \mathrm{nhbr}(y, x)$

Axioms asserting that=truly is an equality relation, i.e. , it is reflexive, symmetric, and transitive.

• $\forall x\colon x=x$
• $\forall x\colon \forall y\colon (x=y)\implies (y=x)$
• $\forall x\colon \forall y\colon \forall z\colon ((x=y)\land (y=z))\implies (x=z)$

Axioms asserting that the neighbor counts are correct. Each of these is of the formif location $x$ has $n$ neighboring pirates, then there are $n$ distinct unsafe neighbors of $x$ , and any other distinct neighbor $x$ is safe.We use the equality relation to specify the distinctness of each neighbor.

• $\forall x\colon \mathrm{has0}(x)\implies \forall y\colon \mathrm{nhbr}(x, y)\implies \mathrm{safe}(y)$
• $\forall x\colon \mathrm{has1}(x)\implies \exists a\colon \mathrm{nhbr}(x, a)\land \neg \mathrm{safe}(a)\land \forall y\colon (\mathrm{nhbr}(x, y)\land (a\neq y))\implies \mathrm{safe}(y)$
• $\forall x\colon \mathrm{has2}(x)\implies \exists a\colon \exists b\colon \mathrm{nhbr}(x, a)\land \mathrm{nhbr}(x, b)\land (a\neq b)\land \neg \mathrm{safe}(a)\land \neg \mathrm{safe}(b)\land \forall y\colon (\mathrm{nhbr}(x, y)\land (a\neq y)\land (b\neq y))\implies \mathrm{safe}(y)$
• $\forall x\colon \mathrm{has3}(x)\implies \exists a\colon \exists b\colon \exists c\colon \mathrm{nhbr}(x, a)\land \mathrm{nhbr}(x, b)\land \mathrm{nhbr}(x, c)\land (a\neq b)\land (a\neq c)\land (b\neq c)\land \neg \mathrm{safe}(a)\land \neg \mathrm{safe}(b)\land \neg \mathrm{safe}(c)\land \forall y\colon (\mathrm{nhbr}(x, y)\land (a\neq y)\land (b\neq y)\land (c\neq y))\implies \mathrm{safe}(y)$

In addition, we want the implications to go the opposite way. Otherwise, each of $\mathrm{has0}$ , $\mathrm{has1}$ , $\mathrm{has2}$ , and $\mathrm{has3}$ could always be false, while still satisfying the above!For brevity, we elide the details in the following list:

• $\forall x\colon \forall y\colon \mathrm{nhbr}(x, y)\implies \mathrm{safe}(y)\implies \mathrm{has0}(x)$
• $\forall x\colon \text{}\implies \mathrm{has1}(x)$
• $\forall x\colon \text{}\implies \mathrm{has2}(x)$
• $\forall x\colon \text{}\implies \mathrm{has3}(x)$

Axioms asserting that the neighbor counts are consistent. While redundant, including axioms like the following can be convenient.

• $\forall x\colon \mathrm{has0}(x)\implies \neg (\mathrm{has1}(x)\lor \mathrm{has2}(x)\lor \mathrm{has3}(x))$
• $\forall x\colon \mathrm{has1}(x)\implies \neg (\mathrm{has0}(x)\lor \mathrm{has2}(x)\lor \mathrm{has3}(x))$
• $\forall x\colon \mathrm{has2}(x)\implies \neg (\mathrm{has0}(x)\lor \mathrm{has1}(x)\lor \mathrm{has3}(x))$
• $\forall x\colon \mathrm{has3}(x)\implies \neg (\mathrm{has0}(x)\lor \mathrm{has1}(x)\lor \mathrm{has2}(x))$

Note that this set of axioms is not quite complete, as explored in an exercise .

#### Questions & Answers

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Robin Reply
Psychology Doctorate. The main difference between PsyD and PhD Psychology is that PsyD mainly focuses on clinical sessions and PhD focuses on research.
Shrajan
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Bridget
Hi, I have a question. What factors influence social facilitation and social inhibition in groups?
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Lametra
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Lametra
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Lametra
Do you know of any studies done to show social facilitation and social inhibition
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social facilitation is often influenced by the person's perception of the situation and their appraisal of the task. for example, a track athlete who views their competition as a challenge rather than a threat, will tend to run the fastest they have ever ran under a large crowd.
fady
Another condition is if the person has mastered the ability to perform that task. A track athlete who trains everyday, will perform above average under a large crowd, however, someone who does not run, will tend to do worse than their actual ability under a large crowd
fady
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fady
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Philip
“ Every child is left to evaluate his experiences for himself, and to take care of his own personal development outside the classroom. There is no tradition for the acquisition of a true knowledge of the human psyche. The science of human nature thus finds itself today in the position that chemis
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Junalyn Reply
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war
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Astaroxche
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Ana
297 according to the DSM 5
fady
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I'm guessing it's in the tens of thousands, maybe hundreds if you include counter interactions and stuff like it. although I'm sure it's almost impossible to know for sure, unless you're very rich and connected to the right people. but as I said: guessing.
Beenie
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Collins
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shine
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Balaji
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lavanya
Both men and women are capable of expressing emotions. But women has the highest percentage of doing that. Because our society is conditioned by nature in such a way that men are expected to suppress their emotions and motivate them through their acts or thoughts, which has it's side effects of....
Santos
denial of any emotion which they feel that useless at that point. Women in the other hand were encouraged/not controlled to suppress their emotions and let them out what they feel about it. I feel that's the wonderful superpower of the women.
Santos
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Mahmoud
Men's emotion comes mostly with memories triggered by senses. That means, they thoughts have the power to decide whether to let go of emotions or not.
Santos
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Mark
because women tend to be more agreeable than men.
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The question is "Why women Viewed are as far more emotional than men ?" it's not a question whether women are more emotional than men. This is more an issue about the point of view from the observer, his/her assumption what emotional behavior is or what emotional behavior is.
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Mehmet
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Parmizan
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Any stream in UG can go for M.Sc (Psychology)
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chinedu Reply
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Robin Reply
Sigmund
Dentriodite
sigmund
Margie
Sigismund Schlomo Freud
Mohammad
Sigmund Freud
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sigmund
Ankita
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Uzma
sigmund Freud
Harpreet
Lev Vygotsky was the founder of socio-cultural theory
Jo
So psychology was based off of the Greek gods I that right or no?
Tanya Reply
psychology is the study. anything -ology is the study of a certain field. Psyche is a mortal woman who becomes divine in Greek mythology. The etymology of the word "psyche" in Greek means "spirit" or "soul"
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Source:  OpenStax, Intro to logic. OpenStax CNX. Jan 29, 2008 Download for free at http://cnx.org/content/col10154/1.20
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